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Lines and Angles - Angle Sum Property of a Triangle

Grade 9CBSE

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

🔑Concepts

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Angle Sum Property: The sum of the interior angles of a triangle is always 180∘180^{\circ}. If you have a triangle △ABC\triangle ABC, then ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}. Visually, if you were to cut off the three corners of any triangle and place them together at a single point, they would form a straight line.

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Exterior Angle Theorem: If a side of a triangle is produced (extended), then the exterior angle so formed is equal to the sum of the two interior opposite angles. For example, if side BCBC of △ABC\triangle ABC is extended to point DD, the exterior angle ∠ACD\angle ACD is equal to ∠A+∠B\angle A + \angle B.

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Linear Pair with Exterior Angle: An exterior angle of a triangle and its adjacent interior angle form a linear pair. This means their sum is always 180∘180^{\circ}. Visually, these two angles sit side-by-side on the straight line formed by the extended side of the triangle.

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Right-Angled Triangle Property: In a right-angled triangle, one angle is 90∘90^{\circ}. Consequently, the sum of the other two acute angles must be 90∘90^{\circ} (they are complementary). In a visual representation, the side opposite the 90∘90^{\circ} angle is the longest side, known as the hypotenuse.

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Equilateral Triangle Angles: In an equilateral triangle, all three sides are equal in length, which forces all three interior angles to be equal. Since the total sum must be 180∘180^{\circ}, each angle in an equilateral triangle is exactly 60∘60^{\circ}.

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Isosceles Triangle Property: In an isosceles triangle, the angles opposite to the equal sides are equal. If two sides of a triangle are equal, the base angles (the angles adjacent to the third side) will have the same measure. Visually, this creates a symmetric shape.

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Triangle Inequality and Angles: The largest angle in a triangle is always opposite the longest side, and the smallest angle is always opposite the shortest side. This provides a visual way to check if the calculated angle measures are consistent with the triangle's geometry.

📐Formulae

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Exterior Angle=Sum of Interior Opposite Angles\text{Exterior Angle} = \text{Sum of Interior Opposite Angles}

∠Ext+∠Intadj=180∘\angle \text{Ext} + \angle \text{Int}_{\text{adj}} = 180^{\circ}

In △ABC, if side BC is extended to D, then ∠ACD=∠CAB+∠ABC\text{In } \triangle ABC, \text{ if side } BC \text{ is extended to } D, \text{ then } \angle ACD = \angle CAB + \angle ABC

Each angle of an equilateral triangle=180∘3=60∘\text{Each angle of an equilateral triangle} = \frac{180^{\circ}}{3} = 60^{\circ}

💡Examples

Problem 1:

The angles of a triangle are in the ratio 2:3:42:3:4. Find the measure of each angle of the triangle.

Solution:

Step 1: Let the angles of the triangle be 2x2x, 3x3x, and 4x4x. Step 2: According to the Angle Sum Property, the sum of these angles must be 180∘180^{\circ}. So, 2x+3x+4x=180∘2x + 3x + 4x = 180^{\circ}. Step 3: Combine the terms: 9x=180∘9x = 180^{\circ}. Step 4: Solve for xx: x=180∘9=20∘x = \frac{180^{\circ}}{9} = 20^{\circ}. Step 5: Calculate each angle: Angle 1: 2x=2×20∘=40∘2x = 2 \times 20^{\circ} = 40^{\circ} Angle 2: 3x=3×20∘=60∘3x = 3 \times 20^{\circ} = 60^{\circ} Angle 3: 4x=4×20∘=80∘4x = 4 \times 20^{\circ} = 80^{\circ}. Verification: 40∘+60∘+80∘=180∘40^{\circ} + 60^{\circ} + 80^{\circ} = 180^{\circ}.

Explanation:

This problem uses the Angle Sum Property. By representing the ratios as algebraic terms, we can set up a linear equation that sums to 180∘180^{\circ} to find the unknown multiplier.

Problem 2:

In △PQR\triangle PQR, the side QRQR is produced to SS. If the exterior angle ∠PRS=105∘\angle PRS = 105^{\circ} and ∠Q=45∘\angle Q = 45^{\circ}, find the measure of ∠P\angle P.

Solution:

Step 1: Identify the given values: Exterior angle ∠PRS=105∘\angle PRS = 105^{\circ} and one interior opposite angle ∠Q=45∘\angle Q = 45^{\circ}. Step 2: Apply the Exterior Angle Theorem: Exterior Angle=Sum of Interior Opposite Angles\text{Exterior Angle} = \text{Sum of Interior Opposite Angles}. Step 3: Write the equation: ∠PRS=∠P+∠Q\angle PRS = \angle P + \angle Q. Step 4: Substitute the values: 105∘=∠P+45∘105^{\circ} = \angle P + 45^{\circ}. Step 5: Solve for ∠P\angle P: ∠P=105∘−45∘=60∘\angle P = 105^{\circ} - 45^{\circ} = 60^{\circ}.

Explanation:

The Exterior Angle Theorem is the most efficient way to solve this. It relates the outside angle directly to the two non-adjacent inside angles, bypassing the need to find the adjacent interior angle ∠PRQ\angle PRQ first.