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Lines and Angles - Pairs of Angles

Grade 9CBSE

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

🔑Concepts

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Adjacent Angles: Two angles are adjacent if they share a common vertex and a common arm, and their non-common arms lie on opposite sides of the common arm. Visually, they appear side-by-side like two rooms sharing a single wall.

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Linear Pair of Angles: If the non-common arms of two adjacent angles form a straight line, they are called a linear pair. The sum of the measures of angles in a linear pair is always 180∘180^{\circ}. Visually, a ray stands on a straight line, dividing the straight angle into two parts.

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Vertically Opposite Angles: When two lines intersect at a point, the angles formed opposite to each other are called vertically opposite angles. These angles are always equal. 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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Complementary Angles: A pair of angles is called complementary if the sum of their measures is equal to 90∘90^{\circ}. Visually, these two angles fit together to form a right angle or an 'L' shape.

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Supplementary Angles: A pair of angles is called supplementary if the sum of their measures is equal to 180∘180^{\circ}. While they do not have to be adjacent, visually, when placed together, they form a straight line.

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Linear Pair Axiom: This axiom states that if a ray stands on a line, then the sum of the two adjacent angles so formed is 180∘180^{\circ}. Conversely, if the sum of two adjacent angles is 180∘180^{\circ}, then the non-common arms of the angles form a straight line.

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Angles at a Point: The sum of all the angles formed at a point (around a vertex) is 360∘360^{\circ}. Visually, these angles together complete a full circle around the central vertex point.

📐Formulae

∠A+∠B=90∘\angle A + \angle B = 90^{\circ} (For Complementary Angles)

∠A+∠B=180∘\angle A + \angle B = 180^{\circ} (For Supplementary Angles or Linear Pairs)

If lines ABAB and CDCD intersect at OO, then ∠AOC=∠BOD\angle AOC = \angle BOD and ∠AOD=∠BOC\angle AOD = \angle BOC

∑Angles on a straight line=180∘\sum \text{Angles on a straight line} = 180^{\circ}

∑Angles around a point=360∘\sum \text{Angles around a point} = 360^{\circ}

💡Examples

Problem 1:

Two angles form a linear pair. If one angle is twice the size of the other, find the measure of both angles.

Solution:

Let the smaller angle be xx. Since they form a linear pair, the other angle is 2x2x. According to the Linear Pair Axiom: x+2x=180∘x + 2x = 180^{\circ} 3x=180∘3x = 180^{\circ} x=180∘3=60∘x = \frac{180^{\circ}}{3} = 60^{\circ} The angles are 60∘60^{\circ} and 2×60∘=120∘2 \times 60^{\circ} = 120^{\circ}.

Explanation:

We use the property that the sum of angles in a linear pair is 180∘180^{\circ} to set up an algebraic equation.

Problem 2:

In a figure, lines XYXY and MNMN intersect at OO. If ∠POY=90∘\angle POY = 90^{\circ} and a:b=2:3a:b = 2:3, find cc where aa and bb are adjacent angles on the line XYXY and cc is vertically opposite to the angle formed by XYXY and MNMN.

Solution:

Since XYXY is a straight line, ∠POX+∠POY=180∘\angle POX + \angle POY = 180^{\circ}. Given ∠POY=90∘\angle POY = 90^{\circ}, then ∠POX=90∘\angle POX = 90^{\circ}. Thus, a+b=90∘a + b = 90^{\circ}. Given a:b=2:3a:b = 2:3, let a=2xa = 2x and b=3xb = 3x. 2x+3x=90∘  ⟹  5x=90∘  ⟹  x=18∘2x + 3x = 90^{\circ} \implies 5x = 90^{\circ} \implies x = 18^{\circ} So, b=3×18∘=54∘b = 3 \times 18^{\circ} = 54^{\circ}. Since MNMN is a straight line, b+c=180∘b + c = 180^{\circ} (Linear pair). $$54^{\circ} + c = 180^{\circ} \implies c = 180^{\circ} - 54^{\circ} = 126^{\circ}$.

Explanation:

This problem uses the Linear Pair Axiom twice and the concept of ratios to find the unknown angle measures step-by-step.