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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 exactly 90∘90^\circ. Visually, if you place two complementary angles adjacent to each other, their non-common arms form a right angle resembling the letter 'L'.

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Supplementary Angles: Two angles are called supplementary if the sum of their measures is 180∘180^\circ. When placed side-by-side, these angles form a straight line. For example, a 60∘60^\circ angle and a 120∘120^\circ angle are supplementary.

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Adjacent Angles: These are a pair of angles that have a common vertex and a common arm but no common interior points. Visually, they sit next to each other like two rooms sharing a single wall.

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Linear Pair: A linear pair is a pair of adjacent angles whose non-common arms are opposite rays, meaning they form a straight line. The sum of angles in a linear pair is always 180∘180^\circ. Visually, it looks like a straight line with a single ray protruding from a point on that line.

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Vertically Opposite Angles: When two lines intersect at a point, the angles formed opposite to each other at the vertex are called vertically opposite angles. These angles are always equal in measure. Visually, they form an 'X' shape where the top angle equals the bottom angle, and the left angle equals the right angle.

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Identification of Pairs: To identify these angles, look for specific geometric signatures: an 'L' shape for complementary (90∘90^\circ), a straight line for supplementary or linear pairs (180∘180^\circ), and an 'X' shape for vertically opposite angles (equality).

📐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.