Edu Simulation

📐 Exterior Angles of a Triangle

Click below to start exploring examples or test yourself

Example 1

Find the value of x from the following figure:

A triangle has two interior angles of 50° and 65°. An exterior angle x is formed at the third vertex. Find x.

Solution

Step 1 — Apply the Exterior Angle Theorem An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

x = 50° + 65° [ ∵ Exterior angle = sum of two opposite interior angles ]

Step 2 — Evaluate x = 50° + 65°
x = 115°

✅ Answer ∴ x = 115°

Example 2

Find the value of x from the following figure:

In triangle PQR, the exterior angle at P (extended to S) is (8x + 25)°. The two non-adjacent interior angles are ∠PQR = (2x + 10)° and ∠PRQ = (5x + 20)°. Find x and ∠SPR.

Solution

Step 1 — Read Off the Angles from the FigureFrom the figure:
∠SPR = (8x + 25)°
∠PQR = (2x + 10)°
∠PRQ = (5x + 20)°

Step 2 — Apply the Exterior Angle Theorem∠PQR + ∠PRQ = ∠SPR

(2x + 10) + (5x + 20) = (8x + 25)

Step 3 — Simplify the Left Side7x + 30 = 8x + 25

Step 4 — Solve for x30 − 25 = 8x − 7x
x = 5

Step 5 — Find ∠SPR∠SPR = 8(5) + 25 = 40 + 25
∠SPR = 65°

✅ Answerx = 5 and ∠SPR = 65°

Example 3

Find all interior angles of the triangle:

The exterior angle of a triangle is 125° and the two opposite interior angles are in the ratio 3 : 2. Find all interior angles of the triangle.

Solution

Step 1 — Identify Given InformationExterior angle = 125°
Two non-adjacent interior angles are in the ratio 3 : 2

Step 2 — Assume the AnglesLet the two non-adjacent interior angles be 3x and 2x.

Step 3 — Apply the Exterior Angle TheoremExterior angle = Sum of two opposite interior angles

3x + 2x = 125°

Step 4 — Solve for x5x = 125°
x = 25°

Step 5 — Find the Two Opposite Interior AnglesFirst angle = 3x = 3 × 25° = 75°
Second angle = 2x = 2 × 25° = 50°

Step 6 — Find the Third Interior AngleThird angle = 180° − Exterior angle
= 180° − 125° = 55°

✅ AnswerThe three interior angles are: 75°, 50°, 55°

Example 4

Find the values of x and y from the following figure:

In the given triangle, one angle is 80°, another angle is 50°, and the remaining two angles are marked as y and x. Find the values of x and y.

Solution

Step 1 — Identify Vertically Opposite AnglesFrom the figure, angle y is vertically opposite to the 80° angle.
Vertically opposite angles are equal.
∴ y = 80°

Step 2 — Apply Triangle Angle Sum PropertyIn a triangle, the sum of all three interior angles is 180°.
The triangle has angles: y, 50°, and x.
So, y + 50° + x = 180°

Step 3 — Substitute the Value of yWe found y = 80°.
80° + 50° + x = 180°

Step 4 — Simplify and Solve for x130° + x = 180°
x = 180° − 130°
x = 50°

✅ Answerx = 50° and y = 80°

Example 5

Find the values of x and y from the following figure:

In the given triangle, two interior angles are 50° and 60°. The angle adjacent to the exterior angle is marked as y, and the exterior angle itself is marked as x. Determine x and y.

Solution

Step 1 — Identify Given AnglesFrom the diagram, the triangle has interior angles: 50°, 60°, and y.

Step 2 — Apply Triangle Angle Sum PropertySum of all interior angles of a triangle = 180°
Therefore: 50° + 60° + y = 180°

Step 3 — Solve for y110° + y = 180°
y = 180° − 110°
y = 70°

Step 4 — Use Linear Pair to Find xAt the vertex where the exterior angle x is drawn, the interior angle is y.
A linear pair sums to 180°: x + y = 180°

Step 5 — Substitute y and Solve for xx + 70° = 180°
x = 180° − 70°
x = 110°

✅ Answerx = 110° and y = 70°

Example 6

Find the values of x and y from the following figure:

In the given triangle, one interior angle is 50° and the exterior angle is 120°. The angle adjacent to the exterior angle is marked as y, and the remaining interior angle is marked as x. Determine x and y.

Solution

Step 1 — Identify Given AnglesInterior angles: x (apex), 50° (right base), y (left base).
Exterior angle at left base = 120°.

Step 2 — Apply Linear Pair Propertyy + 120° = 180°

Step 3 — Solve for yy = 180° − 120°
y = 60°

Step 4 — Apply Triangle Angle Sum Propertyx + 50° + y = 180°
x + 50° + 60° = 180°

Step 5 — Solve for xx + 110° = 180°
x = 70°

✅ Answerx = 70° and y = 60°

Example 7

Find the value of x from the following figure:

In triangle PQR, the exterior angle at P (extended to S) is (8x - 11)°. The two non-adjacent interior angles are ∠PQR = (2x + 6)° and ∠PRQ = 5x. Find the value of x and the measure of the exterior angle.

Solution

Step 1 — Identify Given Angles from the Figure∠SPR = (8x - 11)°
∠PQR = (2x + 6)°
∠PRQ = 5x

Step 2 — Apply the Exterior Angle Theorem∠PQR + ∠PRQ = ∠SPR
(2x + 6) + 5x = (8x - 11)

Step 3 — Simplify the Equation7x + 6 = 8x - 11

Step 4 — Solve for x6 + 11 = 8x - 7x
x = 17

Step 5 — Find the Exterior Angle∠SPR = 8(17) - 11 = 136 - 11 = 125°

✅ Answerx = 17 and ∠SPR = 125°

Example 8

Find the values of x, y, and z from the following figure:

In the given geometric configuration, angles are marked as x°, y°, z°, 120°, and 130°. Use the exterior angle theorem and linear pair properties to determine x, y, and z.

Solution

Step 1 — Identify Linear Pairs At point C: y + 120° = 180° → y = 60°

At point B: z + 130° = 180° → z = 50°

Step 2 — Use Triangle Angle Sum x + y + z = 180°
x + 60° + 50° = 180°
x = 70°

✅ Final Answers x = 70°, y = 60°, z = 50°

🎉

Excellent Work!

You solved all steps correctly. Keep it up!