Exterior Angle Property of Triangle

Class 7Class 9Triangles

When one side of a triangle is extended beyond a vertex, the angle formed outside the triangle at that vertex is called an exterior angle. Every triangle has six exterior angles (two at each vertex), but we usually work with one exterior angle at a time.

The exterior angle property connects this exterior angle to the two interior angles that are NOT next to it. These two angles are called remote interior angles (or non-adjacent interior angles).

This property is one of the most useful results in geometry. It helps us find unknown angles in triangles, solve problems involving parallel lines, and prove other important theorems. It follows directly from the angle sum property (the sum of the three interior angles of a triangle is 180°).

In NCERT Class 7 Mathematics, this property appears in the chapter on Triangles. Once you understand this, many angle-finding problems become much simpler.

What is Exterior Angle Property of Triangle?

Definition: An exterior angle of a triangle is the angle formed between one side of the triangle and the extension of an adjacent side.

Key terms:

Exterior angle: The angle formed outside the triangle when one side is extended.
Interior opposite angles (Remote interior angles): The two angles of the triangle that are NOT adjacent to the exterior angle.
Adjacent interior angle: The interior angle that is next to (shares a vertex with) the exterior angle.

The Exterior Angle Property states:

An exterior angle of a triangle is equal to the sum of its two interior opposite angles.

Important:

The exterior angle is always greater than either of the two remote interior angles.
The exterior angle and its adjacent interior angle form a linear pair (they add up to 180°).
At each vertex, the two exterior angles are equal (they are vertically opposite to each other's supplements — actually they are the same angle formed by extending either of the two sides at that vertex, and both equal 180° minus the interior angle).

Exterior Angle Property of Triangle Formula

Exterior Angle Property:

Exterior Angle = Sum of two Interior Opposite Angles

If the exterior angle is d and the two remote interior angles are a and b:

d = a + b

Related results:

Exterior angle + Adjacent interior angle = 180° (linear pair)
If the interior angles of the triangle are a, b, and c, and d is the exterior angle adjacent to c, then: d = a + b and d + c = 180°.
Since a + b + c = 180° (angle sum property), we get a + b = 180° − c = d.

Units: Angles are measured in degrees (°).

Derivation and Proof

Proof of the Exterior Angle Property:

Consider triangle ABC. Extend side BC beyond C to a point D. The angle ACD is an exterior angle.

To prove: ∠ACD = ∠A + ∠B

Proof:

In triangle ABC, by the angle sum property:
∠A + ∠B + ∠ACB = 180° ... (1)
BCD is a straight line, so ∠ACB and ∠ACD form a linear pair:
∠ACB + ∠ACD = 180° ... (2)
From equation (1): ∠A + ∠B = 180° − ∠ACB
From equation (2): ∠ACD = 180° − ∠ACB
Comparing the right-hand sides:
∠ACD = ∠A + ∠B

Hence proved: The exterior angle of a triangle equals the sum of the two interior opposite angles.

Key insight:

This proof uses two facts: the angle sum property (interior angles add to 180°) and the linear pair axiom (angles on a straight line add to 180°).
The same method works at any vertex of the triangle.

Types and Properties

Types of problems using the Exterior Angle Property:

1. Finding the exterior angle:

Given two interior opposite angles, add them to find the exterior angle.
Example: If ∠A = 50° and ∠B = 70°, the exterior angle at C = 50° + 70° = 120°.

2. Finding a missing interior angle:

Given the exterior angle and one interior opposite angle, subtract to find the other.
Example: If exterior angle = 110° and ∠A = 45°, then ∠B = 110° − 45° = 65°.

3. Finding the adjacent interior angle:

Since the exterior angle and its adjacent interior angle form a linear pair, subtract from 180°.
Example: If exterior angle = 120°, adjacent interior angle = 180° − 120° = 60°.

4. Problems with algebraic expressions:

Angles are given as expressions (like 2x, 3x + 10). Set up an equation using the exterior angle property and solve.

5. Multiple triangles sharing sides:

The exterior angle of one triangle may be an interior angle of another triangle. Apply the property step by step.

Solved Examples

Example 1: Example 1: Finding the exterior angle

Problem: In triangle PQR, ∠P = 55° and ∠Q = 75°. Side QR is extended to point S. Find ∠PRS (the exterior angle at R).

Solution:

Given:

∠P = 55°
∠Q = 75°

Using the Exterior Angle Property:

∠PRS = ∠P + ∠Q
∠PRS = 55° + 75°
∠PRS = 130°

Answer: The exterior angle ∠PRS = 130°.

Example 2: Example 2: Finding a missing interior angle

Problem: In triangle ABC, side BC is extended to D. If ∠ACD = 115° and ∠A = 50°, find ∠B.

Solution:

Given:

Exterior angle ∠ACD = 115°
∠A = 50°

Using the Exterior Angle Property:

∠ACD = ∠A + ∠B
115° = 50° + ∠B
∠B = 115° − 50°
∠B = 65°

Answer: ∠B = 65°.

Example 3: Example 3: Finding all angles of a triangle

Problem: In triangle XYZ, side YZ is extended to W. If ∠XZW = 140° and ∠X = 60°, find all three interior angles.

Solution:

Given:

Exterior angle ∠XZW = 140°
∠X = 60°

Step 1: Find ∠Y using exterior angle property:

∠XZW = ∠X + ∠Y
140° = 60° + ∠Y
∠Y = 80°

Step 2: Find ∠XZY (interior angle at Z):

∠XZY + ∠XZW = 180° (linear pair)
∠XZY = 180° − 140° = 40°

Verification: 60° + 80° + 40° = 180° ✓

Answer: ∠X = 60°, ∠Y = 80°, ∠Z = 40°.

Example 4: Example 4: Algebraic problem

Problem: In a triangle, one interior angle is 40°, another is (2x + 10)°, and the exterior angle adjacent to the third angle is (3x + 20)°. Find x and all angles.

Solution:

Given:

∠A = 40°
∠B = (2x + 10)°
Exterior angle at C = (3x + 20)°

Using the Exterior Angle Property:

Exterior angle at C = ∠A + ∠B
3x + 20 = 40 + 2x + 10
3x + 20 = 2x + 50
3x − 2x = 50 − 20
x = 30

Finding the angles:

∠B = 2(30) + 10 = 70°
Exterior angle = 3(30) + 20 = 110°
∠C (interior) = 180° − 110° = 70°

Verification: 40° + 70° + 70° = 180° ✓ and 40° + 70° = 110° ✓

Answer: x = 30; angles are 40°, 70°, 70°.

Example 5: Example 5: Isosceles triangle with exterior angle

Problem: In an isosceles triangle ABC, AB = AC and ∠A = 50°. Side BC is extended to D. Find ∠ACD.

Solution:

Given:

AB = AC (isosceles triangle)
∠A = 50°

Step 1: Find base angles.

Since AB = AC, ∠B = ∠C (base angles of isosceles triangle are equal).
∠A + ∠B + ∠C = 180°
50° + 2∠B = 180°
2∠B = 130°
∠B = 65°

Step 2: Find the exterior angle ∠ACD:

∠ACD = ∠A + ∠B = 50° + 65° = 115°

Answer: ∠ACD = 115°.

Example 6: Example 6: Two exterior angles

Problem: In triangle PQR, ∠P = 70°. Side QR is extended to S, and side QP is extended to T. Find ∠PRS and ∠RQT if ∠Q = 60°.

Solution:

Given:

∠P = 70°, ∠Q = 60°

Step 1: Find ∠R:

∠R = 180° − 70° − 60° = 50°

Step 2: Exterior angle at R (∠PRS):

∠PRS = ∠P + ∠Q = 70° + 60° = 130°

Step 3: Exterior angle at Q (∠RQT):

∠RQT = ∠P + ∠R = 70° + 50° = 120°

Answer: ∠PRS = 130°, ∠RQT = 120°.

Example 7: Example 7: Right triangle

Problem: In a right triangle, one acute angle is 35°. Find the exterior angle at the vertex of the right angle.

Solution:

Given:

∠C = 90° (right angle)
∠A = 35°

Step 1: Find ∠B:

∠B = 180° − 90° − 35° = 55°

Step 2: Exterior angle at C:

Exterior angle at C = ∠A + ∠B = 35° + 55° = 90°

Note: The exterior angle at the right-angle vertex is always 90° in a right triangle. This makes sense because the adjacent interior angle is 90°, and 180° − 90° = 90°.

Answer: The exterior angle at the right-angle vertex = 90°.

Example 8: Example 8: Finding angles using two exterior angles

Problem: In triangle ABC, the exterior angle at B is 125° and the exterior angle at C is 130°. Find all interior angles.

Solution:

Given:

Exterior angle at B = 125°
Exterior angle at C = 130°

Step 1: Find interior angles at B and C:

∠B = 180° − 125° = 55° (linear pair)
∠C = 180° − 130° = 50° (linear pair)

Step 2: Find ∠A:

∠A = 180° − 55° − 50° = 75°

Verification using exterior angle property:

Exterior at B should equal ∠A + ∠C = 75° + 50° = 125° ✓
Exterior at C should equal ∠A + ∠B = 75° + 55° = 130° ✓

Answer: ∠A = 75°, ∠B = 55°, ∠C = 50°.

Example 9: Example 9: Exterior angle is twice an interior angle

Problem: In triangle ABC, the exterior angle at C is twice ∠A. If ∠B = 50°, find ∠A and the exterior angle.

Solution:

Given:

Exterior angle at C = 2 × ∠A
∠B = 50°

Using the Exterior Angle Property:

Exterior angle at C = ∠A + ∠B
2∠A = ∠A + 50°
2∠A − ∠A = 50°
∠A = 50°

Finding the exterior angle:

Exterior angle = 2 × 50° = 100°

Finding ∠C:

∠C = 180° − 100° = 80°

Verification: 50° + 50° + 80° = 180° ✓

Answer: ∠A = 50°, Exterior angle = 100°.

Example 10: Example 10: Equilateral triangle exterior angle

Problem: Find the exterior angle of an equilateral triangle.

Solution:

Given:

In an equilateral triangle, all interior angles are equal.
Each interior angle = 180° ÷ 3 = 60°

Finding the exterior angle:

Exterior angle = sum of two remote interior angles = 60° + 60° = 120°

Alternatively:

Exterior angle = 180° − interior angle = 180° − 60° = 120° (linear pair)

Answer: Each exterior angle of an equilateral triangle is 120°.

Real-World Applications

Real-world uses of the Exterior Angle Property:

Navigation and surveying: Surveyors use exterior angles to calculate directions and bearings when mapping land. If two angles of a triangle formed by landmarks are known, the exterior angle gives the turning angle.
Architecture: Roof trusses form triangles. The exterior angle helps determine the slope and pitch of the roof at the eaves.
Engineering: In bridge design, triangular supports are used. Knowing exterior angles helps in calculating forces acting on the structure.
Astronomy: The concept of exterior angles is used when calculating the angles of observation of celestial objects from different locations on Earth.
Art and design: Understanding exterior angles helps in creating symmetric designs and patterns using triangles.
Robotics and gaming: When a robot or game character turns at the corner of a triangular path, the exterior angle determines the turning angle.

Key Points to Remember

An exterior angle is formed when one side of a triangle is extended beyond a vertex.
The Exterior Angle Property: An exterior angle of a triangle equals the sum of the two interior opposite angles.
If the exterior angle is d and the remote interior angles are a and b, then d = a + b.
The exterior angle is always greater than either of the two remote interior angles.
The exterior angle and its adjacent interior angle form a linear pair (sum = 180°).
A triangle has 6 exterior angles (two at each vertex), but only 3 are distinct (180° minus each interior angle).
The sum of the three distinct exterior angles of any triangle is 360°.
In a right triangle, the exterior angle at the right-angle vertex is always 90°.
In an equilateral triangle, each exterior angle is 120°.
This property is derived from the angle sum property and the linear pair axiom.

Practice Problems

In triangle ABC, ∠A = 45° and ∠B = 65°. Side BC is extended to D. Find ∠ACD.
The exterior angle of a triangle is 128°. One of the interior opposite angles is 53°. Find the other interior opposite angle.
In triangle PQR, the exterior angle at R is 135°. If ∠P = ∠Q, find ∠P and ∠Q.
In a triangle, the angles are in the ratio 2:3:5. Find all three exterior angles.
In triangle ABC, ∠A = (x + 20)°, ∠B = (2x)° and the exterior angle at C is (4x − 10)°. Find x and all angles.
The exterior angles at two vertices of a triangle are 110° and 120°. Find all interior angles.
In an isosceles triangle, the exterior angle at the vertex angle is 80°. Find the base angles.
Prove that the exterior angle of a triangle can never be less than 90° if the triangle has a right angle.

Frequently Asked Questions

Q1. What is the exterior angle property of a triangle?

The exterior angle property states that an exterior angle of a triangle is equal to the sum of the two interior opposite angles (also called remote interior angles). If the exterior angle is d and the two remote interior angles are a and b, then d = a + b.

Q2. How many exterior angles does a triangle have?

A triangle has 6 exterior angles — two at each vertex. However, the two exterior angles at the same vertex are vertically related (both equal 180° minus the interior angle), so there are only 3 distinct exterior angle values.

Q3. Is the exterior angle always greater than the interior angles?

The exterior angle is always greater than either of the two remote interior angles individually. Since d = a + b (where a and b are both positive), d is necessarily greater than a alone and greater than b alone.

Q4. What is the sum of all exterior angles of a triangle?

The sum of the three distinct exterior angles of any triangle is 360°. Each exterior angle = 180° minus the corresponding interior angle. So the sum = 3 × 180° − (sum of interior angles) = 540° − 180° = 360°.

Q5. What is the exterior angle of an equilateral triangle?

In an equilateral triangle, each interior angle is 60°. Each exterior angle = 180° − 60° = 120°. Alternatively, each exterior angle = 60° + 60° = 120° (sum of the two remote interior angles).

Q6. How is the exterior angle property different from the angle sum property?

The angle sum property says the three interior angles add up to 180°. The exterior angle property says each exterior angle equals the sum of the two non-adjacent interior angles. The exterior angle property is actually derived from the angle sum property.

Q7. Can the exterior angle be 90°?

Yes. In a right triangle, the exterior angle at the right-angle vertex is exactly 90°. The two remote interior angles (the two acute angles) add up to 90°.

Q8. What are remote interior angles?

Remote interior angles (also called non-adjacent interior angles or interior opposite angles) are the two angles of the triangle that are NOT adjacent to the exterior angle. They are the angles at the other two vertices of the triangle.

Exterior Angle Property of Triangle

What is Exterior Angle Property of Triangle?

Exterior Angle Property of Triangle Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Example 1: Finding the exterior angle

Example 2: Example 2: Finding a missing interior angle

Example 3: Example 3: Finding all angles of a triangle

Example 4: Example 4: Algebraic problem

Example 5: Example 5: Isosceles triangle with exterior angle

Example 6: Example 6: Two exterior angles

Example 7: Example 7: Right triangle

Example 8: Example 8: Finding angles using two exterior angles

Example 9: Example 9: Exterior angle is twice an interior angle

Example 10: Example 10: Equilateral triangle exterior angle

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the exterior angle property of a triangle?

Q2. How many exterior angles does a triangle have?

Q3. Is the exterior angle always greater than the interior angles?

Q4. What is the sum of all exterior angles of a triangle?

Q5. What is the exterior angle of an equilateral triangle?

Q6. How is the exterior angle property different from the angle sum property?

Q7. Can the exterior angle be 90°?

Q8. What are remote interior angles?

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