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Properties of Division of Integers are here. Check all the rules, methods, and explanations of integer division properties. Follow simple tricks and tips to remember all rules of division of integer properties. With the Dividing Integers Properties, you can easily simplify and answer the series of operations. Refer to various properties of the division of integers and apply them while solving the problems. Scroll to the below sections to know more details regarding the division of integer properties.

Properties of Division of Integers – Importance

Knowing the properties of the division of integers is important because most of the candidates often get confused between the properties of multiplication and division. Both the operations have different properties and are mandatory to follow all the rules. The properties help to solve many problems in an easy manner. To recall, integer numbers are positive or negative numbers, including zero. All the properties for multiplication, subtraction, division, and addition are applicable to integers. Integers are denoted by the letter “Z”.

Rules for Division of Integers

Rule 1: The quotient of 2 positive(+) integers is positive(+).

Rule 2: The quotient of a positive(+) integer and a negative(-) integer is negative(-).

Rule 3: The quotient of 2 negative(-) integers is positive(+).

In brief, if the signs of the two integers are the same, then the result will be positive. If the signs of the two integers are different, then the result will be negative.

Integers Division Properties

The division is the inverse operation of multiplication.

Let us take the example of whole numbers,

24/4 which means dividing 24 by 4 is nothing but finding an integer when multiplied with 4 that gives us 24, such integer is 6.

So, 24/6=4 and 24/4=6

Hence, for each whole-number multiplication statement, there are two division statements.

1. Division of negative integer by positive integer

Whenever a negative number is divided by a positive number, the result will always be negative.

Steps:

1. Divide the given number as the whole number first.
2. Then add, minus symbol before the quotient. Thus we get the final result as a negative integer.

Example:

(-10)/(2) = (-5)

(-32)/(8) = (4)

2. Division of positive integer by negative integer

Whenever a positive integer is divided by a negative integer, the result will always be negative.

Steps:

1. Divide the given number as the whole number first.
2. Then add, minus symbol before the quotient. Thus we get the final result as a negative integer.

Example:

81/(–9) = –9

60/(–10) = –6

3. Divide a negative integer by a negative integer

Whenever a negative integer is divided by a negative integer, then the result will always be positive.

Steps:

1. Divide the given number as the whole number first.
2. Then add, plus symbol before the quotient. Thus we get the final result as a positive integer.

Example: 

(–15)/(– 3)=5

(–21)/(– 7)=3

4. Closure Under Division Property

Generally, the closure property is, if there are 2 integers, then the addition or subtraction of those integer results in an integer. But integers division does not follow closure property.

Example:

Let us consider the pair of integers.

(-12)/(-6) = 2 (Result is an integer)

(-5)/(-10) = -1/2 (Result is not an integer)

From the above examples, we conclude that integers are not closed under division.

5. Commutative Property of Division

The commutative property states that swapping or changing the order of integers does not affect the final result. Integers division does not follow commutative property also.

Let us consider the pairs of integers.

(– 14)/(– 7)=2

(– 7)/(– 14)=1/2

(– 14)/(– 7)≠(– 7)/(– 14)

From the above examples, we conclude that integers are not commutative for integers.

6. Division of Integer by Zero

Any integer divided by zero gives no result or meaningless result.

Example:

5 ÷ 0 = not defined

When zero is divided by an integer other than zero it results in zero.

Example:

0 ÷ 6 = 0

7. Division of Integer by 1

When an integer is divided by 1, it gives the result as 1.

Example:

(– 7)/1=(– 7)

The above example shows that when a negative integer is divided by 1, it gives the same negative integer.

Dividing Integers Properties Examples

Question 1:

Verify that a/(b+c)≠(a/b)+(a/c) when a = 8, b = – 2, c = 4.

Solution:

L.H.S=a/(b + c)

= 8/(-2+4)

= 8/2=4

R.H.S=(a/b)+(a/c)

= [8/(-2)]+(8/4)

= (-4)+2

= -2

Therefore, L.H.S≠R.H.S

Hence the above equation is verified.

Question 2:

(– 80)/(4) is not the same as 80/(–4). True/False

Solution:

(– 80)/(4)=-20

80/(–4)=-20

As (– 80)/(4) = 80/(–4), therefore the above statement is false.

Question 3:

Evaluate [(– 8)+4)]/[(–5)+1]

Solution: [(– 8)+4)]/[(–5)+1]

= (-4)/(-4)

= 1

Question 4:

Find the quotient of

(-15625)/(-125)

Solution:

From the question, (-15625)/(-125)

= (-15625/-125)

= (15625/125)

= 125

Question 5:

Find the value of [32+2*17+(-6)]/15

Solution:

[32+2*17+(-6)]/15

= [32+34+(-6)]/15

= (66 – 6) ÷ 15

= 60/15

= 4

How to Divide Whole Numbers?

1. After dividing the first digit of the dividend by divisor, if the divisor is a larger number than the first digit of the dividend, then divide the first 2 digits of the dividend by divisor and so on.
2. Always, write the quotient above the dividend.
3. Multiply the quotient value by the divisor and write the product value under the dividend.
4. Subtract the product value from the dividend and bring down the next digit of that dividend.
5. Repeat solving from Step 1 until there are no digits left in the dividend.
6. Finally, verify the solution by multiplying the quotient times the divisor.

Hope you can now get the complete information on the Properties of the Division of Integers. Get the latest updates on all types of mathematical concepts like Integers, Time and Work, Pipes & Cisterns, Ratios and Proportions, Variations, etc. Follow all the articles to get complete clarity on the Integers topic. Stay tuned to our site to get the complete data or information regarding mathematical concepts.

Parallelogram consists of a flat shape that has two opposite & parallel sides with equal length. The parallelogram has four sides and also it is called a quadrilateral. The pair of parallel sides are always equal in length of a parallelogram. Furthermore, the interior opposite angles also equal in measurement. While adding the adjacent angles of a parallelogram, you will get 180 degrees.

The area of the parallelogram always depends on its base and height. Also, the perimeter of the parallelogram depends on the length of its four sides.

List of Parallelogram Concepts

Have a glance at the list of Parallelogram Concepts available below and use them for your reference. All you need to do is simply tap on the quick links and avail the underlying concept within. Practice as much as you can and solve all the problems easily.

• Properties of a Rectangle Rhombus and Square
• Problems on Parallelogram
• Practice Test on Parallelogram

Parallelogram Definition

A parallelogram called a quadrilateral that has two pairs of parallel sides. The interior angles of the parallelogram on the same side of the transversal are supplementary. The sum of all the interior angles becomes 360 degrees in a parallelogram.

A rectangle and square also consist of similar properties of a parallelogram. If in case, the sides of the parallelogram become equal, then it treats as a rhombus. Or else, if a parallelogram has one parallel side and the other two sides are non-parallel, then it treats as a trapezium.

From the above figure, ABCD is a parallelogram, where CD || AB and BC || AD. Also, CD = AB and BC = AD.
And, ∠A = ∠C & ∠B = ∠D
Also, ∠A & ∠D are supplementary angles. Because ∠A & ∠D are interior angles present on the same side of the transversal. Similarly, ∠B & ∠C are supplementary angles.

Therefore, ∠B + ∠C = 180, ∠A + ∠D = 180.

Shape of Parallelogram

A parallelogram shape is a two-dimensional shape. It consists of four sides and two pairs of parallel sides. All the parallel sides of the Parallelogram are equal in length. Also, the interior angles of the parallelogram should always equal.

Angles of Parallelogram

The Parallelogram has four angles. Its opposite interior angles always equal and the angles on the same side of the transversal are always supplementary with each other. When you add the same side of the transversal angles, you can get 180 degrees. Furthermore, the sum of the interior angles is 360 degrees.

Properties of Parallelogram

Check the below properties of a parallelogram and solve the related problems easily by applying the same. They are as follows

• The opposite angles of a parallelogram are congruent.
• Also, the opposite sides are parallel and congruent
• The consecutive angles are supplementary
• Furthermore, The two diagonals bisect each other
• If one angle of a parallelogram is a right angle, then all other angles are right angles.
• Parallelogram law: The sum of squares of all the sides of a parallelogram is always equal to the sum of squares of its diagonals.
• Each diagonal bisects the parallelogram into two congruent triangles.

Area of Parallelogram

The area of the Parallelogram totally depends on the Base and height of the Parallelogram.
The area of the Parallelogram can be calculated as Area = Base × Height

Perimeter of Parallelogram

The perimeter of a parallelogram is calculated as the total distance of the boundaries of the parallelogram. By knowing the length and breadth of the parallelogram, we can get the perimeter of a parallelogram.

Perimeter = 2 (a+b) units where a and b are the length of the sides of the parallelogram.

Types of Parallelogram

Depends on the angles and sides of the Parallelogram, mainly four types of Parallelogram are considered.
Let ABCD is a Parallelogram,
1. If AB = BC = CD = DA are equal, then it is called a rhombus. The properties of the rhombus and parallelogram are equal.
2. Rectangle, Square are also types of the parallelogram.

Parallelogram Theorems

Theorem: Prove that in a parallelogram, the opposite sides are equal; the opposite angles are equal; diagonals bisect each other.

Proof:
Let ABCD be a parallelogram. Draw its diagonal AC.
In ∆ ABC and ∆ ACD, ∠1 = ∠4 (alternate angles)
∠3 = ∠2 (alternate angles)
and AC = CA (common)

Therefore, ∆ ABC ≅ ∆ ACD (by ASA congruence)
⇒ AB = CD, BC = AD and ∠B = ∠D.
Similarly, by drawing the diagonal BD, we can prove that
∆ ABD ≅ ∆ BCD
Therefore, ∠A = ∠C
Thus, AB = CD, BC = AD, ∠A = ∠C and ∠B = ∠D.
This proves (i) and (ii)
In order to prove (iii) consider parallelogram ABCD and draw its diagonals AC and BD, intersecting each other at O.
In ∆ OAB and ∆ OCD, we have
AB = CD [Opposite sides of a parallelogram]
∠AOB = ∠ COD [Vertically opposite angles]
∠OAB = ∠OCD [Alternate angles]
Therefore, ∆ OAB ≅ ∆ OCD [By ASA property]
⇒ OA = OC and OB = OD.
This shows that the diagonals of a parallelogram bisect each other.

Therefore, in a parallelogram, the opposite sides are equal; diagonals bisect each other; the opposite angles are equal.

A Transversal Line is a line that crosses at least two lines (Parallel Lines) and intersects with them. Find Problems on Transversal Lines with Solutions for a better understanding of the concept. Check concept and examples of Lines and Angles on our website for free.

From the above figure, A and B are parallel lines and line C is intersecting both lines at points D and E. The line C is called as Transversal Line.

From the above figure, The line M intersecting Lines N and O at a point c. But the Lines N and O are not parallel lines. Therefore, Line M is not a Transversal Line.

Angles made by the Transversal with Two Lines

From the above figure, Line C intersecting Line A and B (parallel lines). The angles formed around the Transversal line are called Transversal angles. From the given figure, ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, ∠8 are the angles with special names.

Interior Angles

The angles that present in the area enclosed between two parallel lines and that are intersected by a transversal are also called interior angles. From figure, ∠3, ∠4, ∠5, ∠6 are interior angles.

Exterior Angles

The angles that present outside two parallel lines and that are intersected by a transversal are called Exterior angles. From the figure, ∠1, ∠2, ∠7, ∠8 are Exterior angles.

Corresponding Pair of Angles

The corresponding pair of angles present on the same side of the transversal. In the Pair of corresponding angles, if one is an interior angle, the other will be an exterior angle. Also, they do not form a linear pair. From the figure, (∠2, ∠6); (∠1, ∠5); (∠3, ∠7); (∠4, ∠8) are the Corresponding pair of angles.

Pair of Alternate Angles

Pair of alternate angles present on the opposite sided of the transversal. Both Pair of Alternate Angles either interior angles or both are exterior angles. They will not form a linear pair. From the figure, (∠4, ∠6) and (∠3, ∠5) are interior alternate angles. Also, (∠1, ∠7) and (∠2, ∠8) are exterior alternate angles.

Pair of Co-interior or Conjoined or Allied Angles

Pair of co-interior or Conjoined or Allied Angles are pairs of interior angles that present on the same side on the transversal. From the figure, (∠3, ∠6); (∠4, ∠5) are co-interior angles.

Two Parallel Lines are cut by the Transversal

When two parallel lines A and B are intersected by a Transversal line C, then

• the pairs of alternate angles are equal ∠4 = ∠6, ∠ 1 = ∠7, ∠3 =∠5, ∠2 = ∠8.
• the pairs of corresponding angles are equal ∠2 = ∠6, ∠1 = ∠5, ∠3 = ∠7, ∠4 = ∠8.
• Interior angles on the same side of transversal are supplementary ∠4 + ∠ 5 = 180°, ∠6 = 180°.

Converse

When two parallel lines are intersected by a Transversal line

• pairs of alternate angles are equal
• pairs of corresponding angles are equal
• interior angles on the same side of the transversal are supplementary.

Parallel Lines are two lines in a plane, they do not intersect and parallel to each other. The distance between the parallel lines is always the same. The symbol to denote parallel lines is ||. Learn and practice all Lines and Angles concepts and problems on our website.

From the above figure, Line A and Line B are parallel lines. We can write it as A || B and we can read it as A is parallel to B.

Properties Of Angles Associated with Parallel Lines

From the above figure, A and B are two parallel lines and C is passing through two parallel lines by intersecting them at a point. C is a transversal line. If a line intersects two or more lines at distinct points then it is known as a transversal line. When two lines meet at a point in a plane, they are known as intersecting lines.

• The pair of corresponding angles is equal (∠4 = ∠8); (∠3 = ∠7); (∠2 = ∠6); (∠1 = ∠5).
• The pair of exterior alternate angles is equal (∠2 = ∠8); (∠1 = ∠7).
• The pair of interior alternate angles is equal (∠3 = ∠5); (∠4 = ∠6).
• Interior angles present on the same side of the transversal are supplementary, i.e., ∠4 + ∠5 = 180° and ∠3 + ∠6 = 180°.

Example:

Look at the above figure and check out the parallel lines, transversal line, and etc.,
From the above figure, AB and CD are parallel lines and MN is a transversal line that cuts the parallel lines AB and CD.
(i) Interior and exterior alternate angles are equal.
i.e. ∠4 = ∠5 and ∠3 = ∠6 [Interior alternate angles]
∠2 = ∠7 and ∠1 = ∠8 [Exterior alternate angles]
(ii) Corresponding angles are equal.
i.e. ∠2 = ∠6; ∠1 = ∠5; ∠3 = ∠7 and ∠4 = ∠8
(iii) Co-interior or allied angles are supplementary.
i.e. ∠4 + ∠6 = 180° and ∠3 + ∠5 = 180°

Conditions of Parallelism

Two straight lines are cut by a transversal, and if

• the two pairs of corresponding angles are equal and also the lines are parallel to each other.
• if the pair of interior angles on the same side of the transversal are supplementary, then the two straight lines are parallel.
• the pair of alternate angles are equal, then the straight lines are parallel to each other.

Parallel Rays

Parallel Rays are two rays that do not intersect each other and have the same distance every time. A ray consists of one endpoint. If AB and CD are two rays and parallel rays, then it is represented as ray AB ∥ ray CD.

Parallel Segments

If two line segments are parallel then those two line segments are called Parallel Segments. The line segment is a line that consists of two endpoints. If AB and CD are two line segments and parallel, then it is represented as line segment AB ∥ line segment CD.

Learn what are Vertically Opposite Angles and how to calculate them in this article. Vertically Opposite Angles are the angles formed when two lines intersect each other. The vertically opposite angles are always equal to each other. The adjacent angles of vertically opposite angles are always supplementary angles and they form 180º when we add them. Find Lines and Angles concepts and problems on our website.

If one angle is 35º, then the other angle is 180º – 35º.

From the above figure, the line segment PQ and RS meet at the point O and these two lines are intersecting lines.

In a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. From the figure, ∠SOQ and ∠POR form a pair of vertically opposite angles.

∠SOQ = ∠POR
∠SOP = ∠ROQ

Note: A vertical angle and its adjacent angle is supplementary to each other. It means they add up to 180 degrees.

Vertical Angles Theorem and Proof

Theorem: In a pair of intersecting lines the vertically opposite angles are equal.

Proof: Consider two lines PQ and RS which intersect each other at O. The two pairs of vertical angles are:
i) ∠SOQ = ∠POR
ii) ∠SOP = ∠ROQ

It can be seen that ray OS stands on the line PQ and according to Linear Pair Axiom, if a ray stands on a line, then the adjacent angles form a linear pair of angles.
Therefore, ∠SOQ + ∠SOP = 180° —(1) (Linear pair of angles)
Similarly, Ray OP stands on the line SR.
Therefore, ∠SOP + ∠POR = 180° —(2) (Linear pair of angles)
From (1) and (2),
∠SOQ + ∠SOP = ∠SOP + ∠POR
⇒ ∠SOQ = ∠POR —(3)
Also, OQ stands on the line SR.
Therefore, ∠SOQ + ∠QOR = 180° —(4) (Linear pair of angles)
From (1) and (4),
∠SOQ + ∠SOP = ∠SOQ + ∠QOR
⇒ ∠SOP = ∠QOR —(5)
Thus, the pair of opposite angles are equal.

Hence, proved.

Vertically Opposite Angles Examples

1. In the given figure, find the measure of unknown angles.

Solution:
From the given figure
(i) ∠C = 60° vertically opposite angles
(ii) ∠B = 90° vertically opposite angles
(iii) ∠B + ∠A + 60° = 180° (straight angle)
90° + ∠ A + 60° = 180°
150° + ∠ A = 180°
Therefore, ∠A = 180° – 150° = 30°
(iv) ∠A = ∠D vertically opposite angles

Therefore, ∠D = 30°

2. In the given figure, lines AB, CD, EF intersects at O. If M : N : P = 1 : 2 : 3, then find the values of M, N, P.

Solution:
We know that the sum of all the angles at a point is 360°.
∠AOC = ∠DOB = M° (Pair of vertically opposite angles are equal.)
∠FOB = ∠AOE = N° (Pair of vertically opposite angles are equal.)
∠EOD = ∠COF = P° (Pair of vertically opposite angles are equal.)
Therefore, ∠AOE + ∠AOC + ∠COF + ∠FOB + ∠BOD + ∠DOE = 360°
N + M + P + N + M + P = 360°
⟹ 2M + 2N + 2P = 360°
⟹ 2(M + N + P) = 360°
⟹ M + N + P = 3̶6̶0̶°/2̶
⟹ M + N + P = 180° ——— (i)
Let the common ratio be a.
Therefore, M = a, N = 2a, P = 3a
Therefore, from the equation (i) we get;
a + 2a + 3a = 180°
⟹ 6a = 180°
⟹ a = 1̶8̶0̶°/6̶
⟹ a = 30°
Therefore, M = a, means M = 30°
N = 2a, means N = 2 × 30 = 60°
P = 3a, means P = 3 × 30 = 90°

Therefore, the measures of the angles are 30°, 60°, 90°.

Do you want to know What is linear pair of angles? Linear Pair of Angles are similar to adjacent angles but only the difference is adding two angles we get 180º in linear angles. Linear Pair of Angles will also have a common vertex, common arm like adjacent angles. Their interiors will also do not overlap. Remember that all adjacent angles may not form a linear pair. Learn all the Lines and Angles concepts in one place on our website.

Properties of Linear Pair of Angles

Check the below Linear Pair of Angles properties listed below.

• They have a common vertex.
• They have a common arm.
• Also, they do not overlap.
• The sum of the angles is 180°.

From the above figure, AO and OC are two opposite rays and ∠AOB and ∠BOC are the adjacent angles. Therefore, ∠AOB and ∠BOC form a linear pair.

Explanation for Linear Pair of Angles

The angle between the two straight lines is 180° and they form a straight angle. The line segment is the portion of a line that consists of two endpoints. Also, a line with one endpoint is known as a ray. If you consider a line mn and the middle point is o. A ray OP dividing the two lines MO and ON. The complete angle MOP and angle PON becomes 180º to form a linear pair of angles.

∠MOP + ∠PON = ∠MON = 180°

We have also given Lines and Angles concepts and problems for free of cost on our website.

Axioms

Axiom 1: If a ray stands on a line then the adjacent angles form a linear pair of angles.

From the above figure, there are different line segments available that are passing through point O. From the figure, we can say that Angle BOD and Angle BOC are Linear Pair of Angles.

Linear Pair of Angles Examples

In the given figure, ∠AOC and ∠ BOC form a linear pair if x – y = 50°, find the value of x and y.

Solution:
Given x – y = 50° ………… (i)
We know that, x + y = 180° ………… (ii)
Adding (i) and (ii)
x – y + x + y = 180° + 50°
2x = 230°
x = 240°/2
Therefore, x = 115°

Since, x – y = 50°
or, 115° – y = 50°
or, 115° – 115° – y = 50° – 115°
or, -y = -65°
Therefore, y = 65°

The final answer is x = 115° and y = 65°.

Adjacent Angles are the angles that have a common vertex and common arm. The other arms of the two adjacent angles present on the opposite sides of the common arm. The adjacent angles can be a supplementary angle or complementary angle when they share the common vertex and side. Find other concepts of Lines and Angles along with the adjacent angles on our website with a clear explanation.

From the above figure, YO is the common arm and Y is the common vertex. The above figure shows a pair of adjacent angles. The Ray XY and ray YZ are present on the opposite sides of the common arm YO. Therefore, ∠XYO and ∠OYZ are adjacent angles.

∠XYZ and ∠XYO are not adjacent angles, because their other arms YZ and YO are not on the opposite sides of the common arm XY.

Adjacent Angle Example

If you take a wall clock, you can see a minute hand, second hand, and also an hour hand. The minute hand and second hand of the clock form one angle represented as ∠XYO and the hour hand form another angle with the second hand represented as∠OYZ. Both the angles i.e.∠XYO and ∠OYZ are present on the opposite sides of the common hand. Therefore, these two angles are adjacent angles.

Properties of Adjacent Angles

Check out the Properties of Adjacent Angles mentioned below as a part of your preparation. They are as such

• Adjacent Angles share the common vertex
• Angles do not overlap
• They share the common arm
• There is no common interior-point in Adjacent Angles.
• It may be complementary or supplementary angles.
• There should be a non-common arm on both sides of the common arm.

Adjacent Angles Solved Problems

1. Are the following angles adjacent? Give reasons.

Solution:

(a) From figure a, ∠1, and ∠2 don’t have a common arm. Therefore, ∠1 and ∠2 are not adjacent angles.
(b) Adjacent angles will not overlap each other. From figure b, ∠1 and ∠2 interiors are overlapped. Therefore, ∠1 and ∠2 are not adjacent angles.
(c) Adjacent angles must have a common vertex. from figure c, ∠1, and ∠2 don’t have a common vertex. Therefore, ∠1 and ∠2 are not adjacent angles.
(d) From figure (d), ∠1 and ∠2 are adjacent angles. Because they have a common vertex, a common arm, and their interiors do not overlap.

Complementary and Supplementary Angles are the angles formed by adding two angles. If the sum of the two angles is 90º, they are called Complementary angles. Or else, if the sum of the two angles is 180º, they are called Supplementary angles. Find different problems on Supplementary and Complementary Angles in this article. Also, we have given all the concepts available in Lines and Angles on our website.

Complementary Angles

The two angles are said to be complementary angles if their sum is one right angle i.e. 90°. Each angle is called the complement of the other. If x° is one angle then (90 – x)° is the other angle in Complementary Angles.

For example, if 70° is one angle in Complementary Angles, then the other angle will be 20°. 70° and 20° complement each other.

Example 1: To find the complement of 3y + 52°, subtract the given angle from 90 degrees.
90º – (3y + 52º) = 90º – 3y – 52º = -3y + 38º

The complement of 3y + 52° is 38º – 3y.

Facts of complementary angles

• Two complementary angles are acute but vice versa is not possible
• Two right angles cannot complement each other
• Also, two obtuse angles cannot complement each other

Supplementary Angles

The two angles are said to be Supplementary angles if their sum is 180°. Two angles are added and formed a straight line in Supplementary Angles. If x is one angle then (180 – x)° is the other angle in Supplementary Angles.

For example, if 130° is one angle in Supplementary Angles, then the other angle will be 50°. By adding 130 and 50 degrees, it forms a straight line. In Supplementary Angles, both angles are said to be a supplement to each other.

Solved Examples on Supplementary and Complementary Angles

1. Find the complement of 50 degrees?

Solution:
The given angle is 50 degrees, then, the Complement is 40 degrees.
We know that Sum of Complementary angles = 90 degrees.
By adding 40 degrees to 50 degrees, the total angle becomes 90 degrees.
So, 50° + 40° = 90°

The final answer is 90°

2. Find the Supplement of the angle 1/4 of 200°.

Solution:
Given that the angle 1/4 of 200°
Convert 1/4 of 200°
That is, 1/4 x 200° = 50°
Supplement of 50° = 180° – 50° = 130°
By adding 130 degrees to 50 degrees, the total angle becomes 180 degrees.

Therefore, Supplement of the angle 1/4 of 200° is 130°

3. The measures of two angles are (x + 15)° and (3x + 25)°. Find the value of x if angles are supplementary angles.

Solution:
We know that, Sum of Supplementary angles = 180 degrees.
So, (x + 15)° + (3x + 25)° = 180°
Add the x terms and numbers.
4x + 40° = 180°
Move 40° to the right side and subtract it from 180°.
4x = 140°
Move 4 to the right side and divide it from 140°.
x = 35°

The value of x is 35 degrees.

4. The difference between two complementary angles is 54°. Find both the angles.

Solution:
Given that the difference between two complementary angles is 54°.
Let, the first angle = x degrees, then, Second angle = (90 – x)degrees {as per the definition of complementary angles}
Difference between angles = 54°
Now, (90° – x) – x = 54°
90° – 2x = 54°
– 2x = 54° – 90°
-2x = -36°
x = 36°/2°
x = 18°
Again, Second angle = 90° – 18° = 72°

Therefore, the required angles are 18°, 72°.

5. Find the complement of the angle 4/5 of 90°.

Solution:
Convert 4/5 of 90°
Multiply 4/5 with 90°
4/5 × 90° = 72°
Complement of 72° = 90° – 72° = 18°

Therefore, a complement of the angle 4/5 of 90° = 18°

6. Find the supplement of the angle 2/3 of 90°.

Solution:
Convert 2/3 of 90°
2/3 × 90° = 60°
Supplement of 60° = 180° – 60° = 120°

Therefore, a supplement of the angle 2/3 of 90° = 120°

7. The measure of two complementary angles are (3x – 6)° and (x – 4)°. Find the value of x.

Solution:
According to the problem, (3x – 6)° and (x – 4)°, are complementary angles’ so we get;
(3x – 6)° + (x – 4)° = 90°
or, 3x – 6° + x – 4° = 90°
or, 3x + x – 6° – 4° = 90°
or, 4x – 10° = 90°
Add 10° on both sides.
or, 4x – 10° + 10° = 90° + 10°
or, 4x = 100°
or, x = 100°/4°
or, x = 25°

Therefore, the value of x = 25°.

Supplementary angles are the angles that are added up to 180 degrees. If one angle is 120 degrees then the other angle is 60 degrees in supplementary angles because by adding 120 and 60, we get 180 degrees. By adding two supplementary angles, they form a straight line and a straight angle. One important thing in supplementary angles is the two angles need not be next to each other.

The important geometry concepts Lines and Angles are explained on our website with detailed explanations and solved examples. Learn all the concepts and improve your preparation level easily.

Supplementary Angle Definition

Supplementary Angle form 180º by adding two angles.

From the given figure, Angle AOC is one angle and angle BOC is another angle. By adding the two angles AOC and BOC, we get 180º. ∠AOC and ∠BOC are Supplementary Angles.
∠AOC + ∠BOC = 180°

Properties of Supplementary Angles

Have a look at the important properties of supplementary angles explained in the below modules. They are as such

• The two angles are said to be supplementary angles when they add up to form 180°.
• The two supplementary angles together make a straight line, but the angles need not be together.
• Two acute angles cannot be a supplement by each other.
• “S” of supplementary angles stands for the “Straight” line. This means they form 180°.
• The two right angles are always supplementary.

Adjacent and Non-Adjacent Supplementary Angles

The supplementary angles may be classified as either adjacent or non-adjacent. The adjacent supplementary angles have the common line segment or arm with each other. Also, the non-adjacent supplementary angles do not have the line segment or arm.

Supplementary Angles Theorem

The supplementary angle theorem states that if two angles are supplementary to the same angle, then the two angles are said to be congruent.

Proof:

If ∠a and ∠b are two different angles that are supplementary to a third angle ∠c, such that,
∠a + ∠c = 180 ……. (1)
∠b + ∠c = 180 ……. (2)
Then, from the above two equations, we can say,
∠a = ∠b

Hence proved.

Supplementary Angles Examples

1. Verify if 125°, 55° are a pair of supplementary angles?

Solution:
Add the given two angles and check if the resultant angle is 180° or not.
125° + 55° = 180°

The resultant angle is 180°. Hence, they are a pair of supplementary angles.

2. Find the supplement of the angle (30 + x)°.

Solution:
To find the supplement of (30 + x)°, subtract it from 180°
Supplement of the angle (30 + x)° = 180° – (30 + x)°
= 180° – 30° – x°
= (150 – x) °

3. If angles of measures (y – 2)° and (3y + 7)° are a pair of supplementary angles. Find the measures.

Solution:
Since (y – 2)° and (3y + 7)° represent a pair of supplementary angles, then their sum must be equal to 180°.
Therefore, (y – 2) + (3y + 7) = 180
y – 2 + 3y + 7 = 180
y + 3y – 2 + 7 = 180
4y + 5 = 180
Subtract 5 from both sides
4y + 5 – 5 = 180 – 5
4y = 180 – 5
4y = 175
y = 175/4
y = 43.75°

Therefore, we know the value of y = 43.75°, put the value in place of y
y – 2
= 43.75 – 2
= 41.75°

And again, 3y + 7
= 3 × 43.75° + 7
= 118 + 5
= 138.25°
Therefore, the two supplementary angles are 41.75° and 138.25°.

4. Two supplementary angles are in the ratio 5 : 4. Find the measure of the angles.

Solution:
Let the common ratio be m.
If one angle is 5m, then the other angle is 4m.
Therefore, 5m + 4m = 180
9m = 180
m = 180/9
m = 20
Put the value of m = 20

One angle is 5m
= 5 × 20
= 100°

And the other angle is 4m
= 4 × 20
= 80°

Therefore, the two supplementary angles are 100° and 80°.

5. In the given figure find the measure of the unknown angle.

Solution:
x + 45° + 25° = 180°
The sum of angles at a point on a line on one side of it is 180°
Therefore, x + 70° = 180°
Subtract 70° from both sides
x + 70° – 70° = 180° – 70°
x = 110°

Complementary Angles are the angles with 90°. When we add two angles then the resultant angle should be 90° to call those angles complementary angles. In complementary angles, each angle complements another angle. If an angle is 90 degrees, then it can’t be called a complementary because it doesn’t have any pairs. It consists of only one angle with 90 degrees.

The angles become complementary when the sum of the two angles becomes 90 degrees. Therefore, an angle with 90 degrees cannot be called complementary angles. Find all the Lines and Angles concepts on our website with solved examples.

From the above figure, the ∠AOB and ∠BOC are complementary as ∠AOB + ∠BOC = 35° + 55° = 90°. Therefore, they both are Complementary Angles. Also, angles 35 and angle 55 complements each other.

Example 1: If you take a triangle with one angle of 50 degrees and another angle of 40 degrees, the combined angle will become 90 degrees. The two 40 and 50 angles are called complementary angles.

Example 2: If you take a triangle with one angle of 15 degrees and another angle of 75 degrees, the combined angle will become 90 degrees. The two 15 and 75 angles are called complementary angles.

Example 3: If you take a triangle with one angle of 10 degrees and another angle of 80 degrees, the combined angle will become 90 degrees. The two 10 and 80 angles are called complementary angles.

Observations of Complementary Angles

• If two angles complement each other, then each angle must be an acute angle. But any two acute angles need not be complementary.
  For example, angles of measures 30° and 40° are not a complement to each other.
• Two right angles cannot complement each other.
• Also, two obtuse angles cannot complement each other.

Complementary Angles Examples

1. Find the complement of below angles

(a) 58°

Solution:
To find the complement of 58°, subtract it from 90°
90° – 58° = 32°

Therefore, the complement of 58° is 32°

(b) 37°40′

Solution:
To find the complement of 37°40′, subtract it from 90°
90° – 37°40′
90° = 89°60′
= 89°60′ – 37°40′
= 52°20′
Therefore, the complement of 37°40′ is 52°20′

(c) y + 42°

Solution:
To find the complement of y + 42°, subtract it from 90°
90° – (y + 42°)
= 90° – y – 42°
= 56° – y
Therefore, the complement of y + 42° is 56° – y

2. Find the complement of the angle (20 + x)°

Solution:
To find the complement of (20 + x)°, subtract it from 90°
90° – (20 + x)
= 90° – 20 – x°
= 70° – x° = (70 – x)°
Therefore, the complement of (20 + x)° is (70 – x)°

3. Find the measure of an angle that is 36° less than its complement.

Solution:
Let the unknown angle be y, then measure of its complement = (90° – y)
According to the given question,
(90° – y) – y = 36°
90° – y – y = 36°
90° – 2y = 36°
Subtract 90° from both sides
90° – 90° – 2y = 36° – 90°
-2y = -54°
y = 54/2 = 27°
y = 27°
Therefore, 90 – y (Put the value of y = 27°)
= 90 – 27°
= 63°
Therefore, the pair of complementary angles are 63° and 27°

Have a look at Some Geometric Terms and Results. We have given a useful reference of geometric terms and their definitions along with solved examples. Most of the students feel difficult to understand Some Geometric Terms. So, to help such students, we explained all Geometric Terms in a clear and understandable way. We can get strong and exact results using Certain Geometric Statements. So, make use of the given formulas and get the easy process to solve problems.

We have also given Lines and Angles concepts and problems for free of cost on our website.

Some Geometric Terms and Results

• The sum of all the angles at a point is 360°.

i.e., ∠7 + ∠8 + ∠9 + ∠10 + ∠11 + ∠12 = 360°

• The sum of all the angles about a point on a straight line on one side of if it is 180°.

i.e., ∠5 + ∠6 + ∠7 + ∠8 = 360°

Some Important Geometric Terms

1. Equal Angles

The two angles are considered to be equal if they have the same degree measure.

Given that two angles. The angles are ∠ABC and ∠PQR.
∠ABC = 90°; ∠PQR = 90°.
∠ABC and ∠PQR are equal angles of measure 90°.

2. Bisector of an Angle

A ray that divides the given angle into two equal angles is called an angle bisector.

Given that ∠XYZ along with a ray YO that divides ∠XYZ. In the adjoining figure, the ray YO divides ∠XYZ into two equal angles ∠XYO and ∠OYZ
i.e., ∠XYO = ∠OYZ.

3. Perpendicular Lines

Perpendicular Lines are the lines that intersect each other to form right tangles between them. In the adjoining e, lines AB and CD intersect at 0 such that ∠COB = ∠ COA = ∠AOD = ∠BOD = 90°.
Therefore, we say that AB is perpendicular to CD, i.e., (AB ⊥ CD).

4. Perpendicular Bisector

Perpendicular Bisector is the line that passes through the midpoint of the given line segment and also it is perpendicular to it. Here, the CD is the line segment. AB is the perpendicular bisector as ∠AOB = ∠AOD = 90° and CO = OD.

Related angles are nothing but the pairs of angles and assigned with specific names that we come across. Related angles have some conditions to mention. Learn the detailed concept of Related angles with images and examples in this article. Improve your preparation level by reading the entire concept without missing any subtopic. We have given complete information about Lines and Angles on our website for free of cost.

Different Types of Related Angles

Check different types of Related Angles along with examples to clearly understand the concept. We have explained each of them with definitions, solved examples, etc. Refer to the following modules and get a grip on it.

Complementary Angles

If the sum of the measures of two angles is about 90°, those angles are called complementary angles.

Facts of Complementary Angles

• The two right angles never complement each other.
• Also, the two obtuse angles never complement each other.
• Two complementary angles are always acute but there is no possibility of vice versa.

Example:

Let us take two angles which are complementary angles. If one angle is a, then the other angle is 90° – a.
An angle of 40° and another angle of 50° are complementary angles of each other.

Also, complement of 40° is 90° – 40° = 50°.
And complement of 50° is 90° – 50° = 40°

The ∠XOY is 40° and ∠MON is 50°. By adding two angles ∠XOY and ∠MON, we get 90° as they are the Complementary Angles.

Therefore, the sum of the ∠XOY and ∠MON is 90°
∠XOY + ∠MON = 90°.

Supplementary Angles

Supplementary Angles are the angles when the sum of the measures of two angles is 180°. If the sum of the two angles forms a straight angle, then those angles are called Supplementary Angles. If one angle is a, then the other angle is 180° – a.

Example:

Let us take two angles which are Supplementary angles. If one angle is x, then the other angle is 180° – x.
An angle of 110° and another angle of 70° are supplementary angles of each other. Also, a supplement of 110° is 180° – 110° = 70°.
And the supplement of 60° is 180° – 70° = 110°.

From the above figure, the ∠XOY is 110° and ∠MON is 70°. By adding two angles ∠XOY and ∠MON, we get 180° as they are the Supplementary Angles.

Therefore, the sum of the ∠XOY and ∠MON is 180°
∠XOY + ∠MON = 180°.

Adjacent Angles

The two angles are called to be Adjacent angles when they have a common arm, a common vertex, and also the non-common arms present on the opposite side of the common arm.

From the above figure, ∠ABD and ∠CBD are adjacent angles with the common arm BD. The B is the common vertex and BA, BC is opposite sides of BD.

Linear Pair

When two adjacent angles form a linear pair of angles with the non-common arms are two opposite rays. In other words, the sum of two adjacent angles is 180°.

From the above figure, the ∠XOY and ∠XOZ are two adjacent angles. By adding two angles ∠XOY and ∠XOZ, we get 180°.

Therefore, the sum of the ∠XOY and ∠XOZ is 180°
∠XOY + ∠XOZ = 180°.

Vertically Opposite Angles

The arms of the lines are opposite in direction and both lines are interesting to each other in Vertically opposite angles. The pair of vertically opposite angles are equal.

From the above figure, the ∠MOR and ∠SON and ∠MOS and ∠RON are pairs of vertically opposite angles.

Theorems on Related Angles

1. If a ray stands on a line, then the sum of adjacent angles formed is 180°.

Given: A ray BE standing on (AC) ⃡ such that ∠ABE and ∠CBE are formed.

Construction: Draw BD ⊥ AC.

Proof: Take the angle ABE.
Now ∠ABE = ∠ABD + ∠DBE ……………. (1)
Take the angle CBE.
Also ∠CBE = ∠CBD – ∠DBE ……………. (2)
Now, add equation 1 and equation 2.
Adding (1) and (2),
∠ABE + ∠CBE = ∠ABD + ∠CBD + ∠DBE – ∠DBE
∠DBE – ∠DBE = 0
= ∠ABD + ∠CBD
∠ABD = 90°; ∠CBD = 90°
Substitute ∠ABD and ∠CBD values in ∠ABD + ∠CBD
= 90° + 90°
= 180°

2. The sum of all the angles around a point is equal to 360°.

Given: A point O and rays OA, OB, OC, OD, OE which make angles around O.

Construction: Draw OD opposite to ray OA

Proof: Since, OB stands on DA therefore
∠AOB + ∠BOD = 180°
∠BOD = ∠BOC + ∠COD
Substitute ∠BOD = ∠BOC + ∠COD in ∠AOB + ∠BOD = 180°
∠AOB + (∠BOC + ∠COD) = 180°
∠AOB + ∠BOC + ∠COD = 180° ……………. (i)
Again OE stands on DA, therefore
∠DOE + ∠EOA = 180°
∠EOA = ∠EOF + ∠FOA
Substitute ∠EOA = ∠EOF + ∠FOA in ∠DOE + ∠EOA = 180°
∠DOE + (∠EOF + ∠FOA) = 180°
∠DOE + ∠EOF + ∠FOA = 180° ……………. (ii)
Now, add euqtion (i) and equation (ii)
Adding (i) and (ii),
∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOF + ∠FOA
= 180° + 180°
= 360°

3. If two lines intersect, then vertically opposite angles are equal.

Given: MN and AB intersect at point O.

Proof: OB stands on MN.
Therefore, ∠MOB + ∠BON = 180° ……………. (i)
MO stands on AB
∠MOB + ∠MOA = 180° ……………. (ii)
From (i) and (ii),
∠MOB + ∠BON = ∠MOB + ∠MOA
∠BON + ∠MOA
Similarly, ∠MOB = ∠AON can be proved.

Comparison of Integers is a bit confusing topic and it requires proper understanding to solve correctly. We have provided the necessary information here when comparing or ordering the integer numbers. Know various methods like comparing integers using absolute value and using number line etc. Follow all the tips given and solve the problems related to it easily. Check the below sections to know the detailed information regarding the comparison and ordering of integers.

Comparison of Integers – Introduction

Are you confused while comparing the integers? We are sure you will get clarity on things by the end of this article. Most people feel confused while comparing negative integers, but if you follow an exact procedure, then it is easy for you to solve all the problems. Two Integers are compared mostly using conditional or arithmetic methods. It is also done using the number line method. When the integers are represented on a number line, we observe that, as we are moving towards the right, the value of the number increases. In the same way, when we move towards the left, the value decreases.

The right side consists of whole numbers and the left side consists of a negative number and zero is present in the middle of the number line. Zero is less than all the positive numbers and greater than negative numbers. It is neither a positive nor a negative number. If you have to order the integer numbers from least to highest, then you would order the numbers from the right of the number line to the left.

Symbolic Representation for comparing Integers

Integers can be compared using the symbols ‘<‘, ‘>’ or ‘=’

Procedure for Comparing or Ordering of Integer Numbers

Follow the below-listed steps while comparing or ordering integer numbers. They are along the lines

1. Consider, there are 2 numbers to compare. In that case, one of the numbers is greater than or less than or equal to other numbers.
2. If the count of the first number is higher than the second number, then that first number is greater than the second number. The symbol ‘>’ is used to represent greater than and vice versa. The symbol ‘<‘ is used to represent less than the symbol.
3. For example, consider 15 and 10 as the two integer numbers to be compared. 15 is greater than 10, as 15 has a higher count when compared to the number 10.
4. Using a number line, point all the integer numbers on a number line and thus the comparison can be done easily.

Key Points to Remember while comparing Integers

Do remember the following points while ordering integers or comparing them. These help you to solve your problems easily.

1. As we know, the integer is the set of whole numbers and also their opposites. In other words, Integer numbers are positive or negative whole numbers.
2. Integers do not include fractions and decimals which are known as rational numbers. Integers only include whole numbers.
3. Look at the integers and compare the values and then plot the integers on a number line.
4. Compare the integer values and use the symbols of ‘>’,'<‘,’=’. Drawing the number line will help you to visualize the integers.
5. Then, order the integer values from least to greatest.
6. Arrange in order from ‘Least’ to ‘Greatest’ i.e. left to right. ‘Greatest’ to ‘Least’ is right to left.
7. Consider the absolute value of the number which is the distance from the number line. Absolute values are always positive because it signifies the distance.

Comparing Integers Examples

Example 1:

Josh has $23, his friend Krista has $25 and another friend Mark has $8, his friend Luis has -$5, another friend Tessa has -$10. Josh wants to order and compare the amount of money they have?

Solution:

First, draw the number line and make sure you include all the numbers you have to plot. Plot all the above money values on the number line.

Then, find out the number which is furthest to the left of the number line. -10 is the smallest number which means Tessa is with the least amount of money.

After -10, -5 is the second least number. Therefore, Luis has the second least amount of money. Next comes the Mark with the next least amount of money of $8. Thus the pattern follows with Josh and Krista.

This analysis will give you the numbers that are ordered from least to greatest.

Therefore, the final solution is in the order of -$10,-$5,$8,$23,$25. Thus, Tessa has less amount of money and Krista has the highest amount of money.

Example 2:

Choose the inequality symbol which goes in the blank to make the given statement true: -22 _____ -32

Solution:

First, visualize or draw a number line. Then, plot the points of -22 and -32.

Then, determine where -22 is on the drawn number line when compared to -32. -22 will be to the right of -32.

As it is on the right side, it determines -22 is greater than -32.

The final solution is -22>-32

The inequality symbol which goes in the blank is “>”

Example 3:

Arrange the numbers −4, 2, 8, 9, −11, −5 in order using the number line?

Solution:

First of all, draw a number line and include all the numbers in such a way that the least number -11 is included and the highest number 9 is included.

Once, you draw the number line, start plotting the numbers as per the ascending order. First, start by -11, then -5 followed by -4 and so on.

Continue by reading the points on the plotted number line from left to right which gives you the order of the numbers from least to highest.

The final solution is 11, -5, -4, 2, 8, 9.

Example 4: 

Arrange the numbers 5, −16, 6, -12, 1, 4 in order using the number line?

Solution:

First of all, draw a number line that includes all the numbers given in the above question. The least number must be greater than -16 and the highest number must be greater than 6.

Once, you start drawing the number line, make sure you arrange the numbers in ascending order. The first least number comes as -16, then -12 followed by 1 and so on.

Now, plot the numbers from left to right which you have previously arranged in order. Once you plot the numbers, you will get the list of numbers from least to highest values.

The solution is -16, -12, 1, 4, 5, 6.

Example 5: 

Arrange the numbers −3, −2, −7, −12, −1 in order using the number line?

Solution:

The first step is to draw the number line by including all the numbers that are given in the question. Make sure you draw a number line by including a few extra numbers.

Now, that you are ready with the number line, the next step is to plot the numbers on the graph. Analyze the order of the numbers from least to highest. The least number from the given series is -12. -7 is the second least number, then comes -3, and so on.

Once you are ready with the order of numbers, you can easily plot them on the number line. Plot the numbers from left to right in the order you arranged previously. After plotting the numbers, you will find the final result of numbers from least to highest values.

The final solution is -12,-7,-3,-2,-1.

Here is the detailed information about Comparison of Integers and also how to order the integers. This concept will help you throughout your integers problem-solving methods. I hope, you got clarity on how to order, compare the integers, and also how to include inequality symbols for various integer numbers. For more details regarding other mathematical topics, stay tuned to our site.

The polygon area is the region occupied by the polygon. The basic types of polygons are regular polygon and irregular polygon. Learn about the area of a polygon, polygon definition, central point of a polygon, radius of the inscribed circle, circumscribed circle, and polygon of n sides.

Types of Polygons

Polygon is a closed shape in a two dimensional plane with straight lines. It has an infinite number of sides but all sides are straight lines. The line segments of a polygon are called the sides or edges. The point of intersection of two line segments is called the vertex.

Based on the sides and angles, the polygons are classified into different types. They are regular polygon, irregular polygon, convex polygon, and concave polygon. The regular polygon measures all sides and angles equal. Some of the polygons are triangle, square, pentagon, hexagon, and others.

Central Point of a Polygon:

The inscribed and the circumscribed circles of a polygon have the same center, called the central point of the polygon.

The radius of the Inscribed Circle of a Polygon:

The length of the perpendicular from the central point of a polygon upon any one of its sides is called the radius of the inscribed circle of the polygon. The radius of the inscribed circle of a polygon is denoted by r.

The radius of the Circumscribed Circle of a Polygon:

The line segment joining the central point of a polygon to any vertex is the radius of the circumscribed circle of the polygon. The radius of the circumscribed circle of a polygon is denoted by R.

Properties of Polygon

The polygon’s properties are based on the sides and angles. Refer to the below properties and learn entirely about Polygons. They are as follows

• The sum of all interior angles of an n-sided polygon is (n – 2) x 180°.
• The number of triangles formed by joining the diagonals from one corner of a polygon = n – 2.
• The number of diagonals in a polygon with n sides = n(n – 3)/2.
• The measure of each interior angle of n-sided regular polygon = [(n – 2) × 180°]/n
• The measure of each exterior angle of n sided regular polygon = 360°/n

Area of Regular Polygons Formulas

• If a regular polygon has three sides, then it is called a triangle and its area formula is ½ x base x height
• If the regular polygon has 4 sides, then it called square, and its area formula is side².
• If the regular polygon has 5 sides, then it is called the pentagon and its area formula is /2 x side length × distance from the center of sides to the center of the pentagon.
• If the regular polygon has 6 sides, then it is called the hexagon and its area formula is (3√3)/2 × distance from the center of sides to the center of the hexagon.
• The area of the rectangle formula is length x breadth.
• The area of the rhombus formula is 1/2 x product of diagonals.
• The area of the hexagon is [3(√3)a²/2] square units.
• The area of the octagon is 2a² (1 + √2) square units.
• The area of a polygon is n/2 × a × √(R² – a²/4) square units.

How to Find the Area of N- Sided Polygons?

Follow the below mentioned simple steps and instructions to calculate the area of a polygon having N-Sides easily.

• If the given is a regular polygon, then substitute the values in the formula.
• If the given figure is an irregular polygon or regular polygon having n sides can use the below process.
• Divide the geometric figure into the combination of regular polygons like triangle, square, rectangle, etc.
• Find the area of each shape.
• Add up those areas to get the polygon area in square units.

Area of a Polygon Word Problems

Example 1.

Find the area of a regular pentagon whose perimeter is 40 units and whose apothem is 5 units?

Solution:

Given that,

The perimeter of the regular pentagon = 40 units

Apothem = 5 units

Area of the regular pentagon = ½ (perimeter) (apothem)

= ½ (40) (5)

= 20 x 5 = 100 sq units

Therefore, the area of regular pentagon = 100 sq units.

Example 2.

Find the area of a regular octagon each of whose sides measures 25 cm?

Solution:

Given that,

Side of a regular octagon a = 25 cm

Area of octagon A = 2a² (1 + √2)

Substitute a = 25 in above equation.

A = 2 x 25² (1 + √2)

= 2 x 625 (1 + √2) = 1250 (1 + √2)

= 1250 x 2.414 = 3,017.76 cm²

Therefore, regular octagon area is 3017.76 cm².

Example 3.

The area of a regular pentagon is 215 cm². Find the length of the side of the pentagon?

Solution:

Given that,

Regular pentagon area A = 215 cm²

Pentagon area formula is 1/4 ((√(5 (5 + 2 √5) s²))

Pentagon area = 215 cm²

1/4 ((√(5 (5 + 2 √5) s²)) = 215

(√(5 (5 + 2 √5) s²)) = 215 x 4 = 860

s² = 860/(√(5 (5 + 2 √5))

s² = 860/6.88 = 125

s = √125

s = 11.18

Therefore, side of regular pentagon is 11.18 cm.

Example 4.

Find the area of a regular polygon having 12 sides and each side length is 5 m?

Solution:

Given that,

Number of sides of a polygon n = 12

Side length a= 5 cm

Area of regular polygon A = [n * a² * cot(π/n)]/4

A = [12 x 5² x cot(π/12)]/4

= [12 x 25 x (2 + √3)]/4

= (300 x (2 + √3))/4

= 75 (2 + √3)

= 75 (3.73) = 279.9 cm²

Therefore, the area of a regular polygon is 279.9 cm².