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Lines and Angles - Related Angles (Complementary, Supplementary, Adjacent, Linear Pair, Vertically Opposite)

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Complementary Angles: Two angles are said to be complementary if the sum of their measures is 90∘90^\circ. Each angle is called the complement of the other.

Complementary angles adding up to 90 degrees
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Supplementary Angles: Two angles are supplementary if the sum of their measures is 180∘180^\circ. They form a straight angle when placed adjacently.

Supplementary angles on a straight line
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Adjacent Angles: Two angles are adjacent if they have a common vertex, a common arm, and their non-common arms are on opposite sides of the common arm.

Adjacent angles with a common vertex and arm
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Linear Pair: A pair of adjacent angles is called a linear pair if their non-common arms are opposite rays. The sum of angles in a linear pair is always 180∘180^\circ.

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Vertically Opposite Angles: When two lines intersect, the angles formed opposite to each other at the vertex are called vertically opposite angles. These angles are always equal.

Vertically opposite angles marked 1, 2, 3, 4

📐Formulae

Sum of Complementary Angles=90∘\text{Sum of Complementary Angles} = 90^\circ

Sum of Supplementary Angles=180∘\text{Sum of Supplementary Angles} = 180^\circ

If ∠A and ∠B form a Linear Pair, then ∠A+∠B=180∘\text{If } \angle A \text{ and } \angle B \text{ form a Linear Pair, then } \angle A + \angle B = 180^\circ

If two lines intersect, ∠1=∠3 and ∠2=∠4 (Vertically Opposite Angles)\text{If two lines intersect, } \angle 1 = \angle 3 \text{ and } \angle 2 = \angle 4 \text{ (Vertically Opposite Angles)}

Complement of angle x=(90−x)∘\text{Complement of angle } x = (90 - x)^\circ

Supplement of angle x=(180−x)∘\text{Supplement of angle } x = (180 - x)^\circ

💡Examples

Problem 1:

Find the measure of an angle which is 24∘24^\circ more than its complement.

Solution:

Let the angle be xx. Its complement will be (90−x)(90 - x). According to the problem: x=(90−x)+24x = (90 - x) + 24 x+x=90+24x + x = 90 + 24 2x=1142x = 114 x=1142=57∘x = \frac{114}{2} = 57^\circ The angle is 57∘57^\circ.

Explanation:

We use the definition of complementary angles (x+y=90∘x + y = 90^\circ) to set up a linear equation based on the given condition that one angle is 2424 units larger than the other.

Problem 2:

Two lines ABAB and CDCD intersect at point OO. If ∠AOC=50∘\angle AOC = 50^\circ, find the measures of ∠AOD\angle AOD and ∠BOD\angle BOD.

Solution:

  1. Since ABAB is a straight line, ∠AOC\angle AOC and ∠BOC\angle BOC form a linear pair. However, it's easier to use ∠AOC\angle AOC and ∠AOD\angle AOD: ∠AOC+∠AOD=180∘ (Linear Pair)\angle AOC + \angle AOD = 180^\circ \text{ (Linear Pair)} 50∘+∠AOD=180∘50^\circ + \angle AOD = 180^\circ ∠AOD=180∘−50∘=130∘\angle AOD = 180^\circ - 50^\circ = 130^\circ 2. ∠BOD\angle BOD and ∠AOC\angle AOC are vertically opposite angles: ∠BOD=∠AOC=50∘\angle BOD = \angle AOC = 50^\circ

Explanation:

We apply the Linear Pair Postulate to find the adjacent supplementary angle and the Vertically Opposite Angles property to find the angle across the vertex.

Problem 3:

In the following figure, ∠POR\angle POR and ∠QOR\angle QOR form a linear pair. If a−b=80∘a - b = 80^\circ, find the values of aa and bb.

Linear pair diagram with angles a and b

Solution:

Given: a−b=80∘—(i)\text{Given: } a - b = 80^\circ \quad \text{---(i)} Since ∠POR and ∠QOR form a linear pair, a+b=180∘—(ii)\text{Since } \angle POR \text{ and } \angle QOR \text{ form a linear pair, } a + b = 180^\circ \quad \text{---(ii)} Adding (i) and (ii):\text{Adding (i) and (ii):} (a−b)+(a+b)=80∘+180∘(a - b) + (a + b) = 80^\circ + 180^\circ 2a=260∘2a = 260^\circ a=130∘a = 130^\circ Substituting a=130∘ in (ii):\text{Substituting } a = 130^\circ \text{ in (ii):} 130∘+b=180∘130^\circ + b = 180^\circ b=180∘−130∘=50∘b = 180^\circ - 130^\circ = 50^\circ Thus, a=130∘ and b=50∘.\text{Thus, } a = 130^\circ \text{ and } b = 50^\circ.

Explanation:

We use the property that angles in a linear pair sum to 180∘180^\circ to create a system of two equations with two variables.

Problem 4:

Find the values of xx, yy, and zz in the given figure where two lines intersect.

Intersecting lines showing vertically opposite angles

Solution:

From the figure, x and 45∘ are vertically opposite angles.\text{From the figure, } x \text{ and } 45^\circ \text{ are vertically opposite angles.} So, x=45∘\text{So, } x = 45^\circ Also, x and y form a linear pair:\text{Also, } x \text{ and } y \text{ form a linear pair:} x+y=180∘x + y = 180^\circ 45∘+y=180∘45^\circ + y = 180^\circ y=180∘−45∘=135∘y = 180^\circ - 45^\circ = 135^\circ Finally, y and z are vertically opposite angles:\text{Finally, } y \text{ and } z \text{ are vertically opposite angles:} z=y=135∘z = y = 135^\circ Thus, x=45∘,y=135∘,z=135∘.\text{Thus, } x = 45^\circ, y = 135^\circ, z = 135^\circ.

Explanation:

Identify vertically opposite angles to find xx. Use the linear pair property (angles on a straight line) to find yy, and then use vertically opposite angles again for zz.