Interior and Exterior Angles of Triangles

Different Angles of Triangles

Key Concepts

• Relate interior angle measures in a triangle
• Find the exterior angle measures
• Find the unknown angle measures using algebra

Interior and Exterior Angles of Triangles

• A triangle is a three-sided polygon that consists of three edges and three vertices. 

• If xx and yy are two parallel lines, a line that intersects two or more lines at different points is called a transversal. (Say tt) 

• We know the corresponding angles are congruent. 

So, ∠1=∠5, ∠2=∠6, ∠3=∠7 and ∠4=∠8

• The alternate interior angles are congruent. 

So, ∠4=∠6and ∠3=∠5

• The same-side interior angles are supplementary. 

So, ∠3+∠6=180° and ∠4+∠5=180°

Interior and Exterior Angles of Triangles 

Consider a triangle △ABC as shown 

Relate interior angle measures in a triangles 

Let us rotate the copies of △ABC and place them in order to bring all the angles together. 

∠A ∠B and ∠C appear to form a straight line. 

 A straight line has an angle of 180°

∴∠A+∠B+∠C=180°

Hence, the sum of the measures of interior angles of a triangle is 180°

Find exterior angle measures 

If we extend any side of a triangle, the angle is called an Exterior angle. 

In △PQR if we extend QR towards R, ∠PRS is the exterior angle. 

For ∠PRS, ∠QRP is the interior adjacent angle and ∠PQR and ∠RPQ are the interior opposite angles. 

Let us add the measures of ∠P and ∠Q and compare it to the measure of the exterior angle. 

∴∠PRS = ∠PQR+∠RPQ

Hence, the measure of an exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles. 

1.6.3: Use algebra to find unknown angle measures 

Example 1: Find the measure of x and y

Solution:  

Step 1: Find the measure of 𝒙

In the given triangle, x and 120° form a linear pair. 

A linear pair of angles must add up to 180°

⇒120°+x=180°

⇒x=180°−120°

⇒x=60°

∴ The measure of 𝒙 is 𝟔𝟎°

Step 2: Find the measure of𝒚

We know, x, y and 70° are the angles of the triangle. 

The sum of the interior angles of a triangle is 180°

⇒x+y+70°=180°

⇒60°+y+70°=180°

⇒ y=180°−70°−60°

⇒ y=50°

∴The measure of 𝒚 is 𝟓𝟎°

Exercise

1. Find m∠1 and m∠2.

2. In the figure, m ∠1=(8x+7)°, m∠2=(4x+14)°, and m∠4=(13x+12)°. Your friend incorrectly says that m∠4=51°. What is m∠4? What mistake might your friend have made?

3. In ∆ABC, what is m∠C?

4. The measure of ∠F is 110°. The measure of ∠E is 100°. What is the measure of ∠D?

Concept Map: 

What we have learned:

1. Sum of the measures of interior angles of a triangle is 180°.
2. The measure of an exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.