The angle outside
An exterior angle and the interior angle beside it make 180°.
What is left of the straight line?
Extend the side BC beyond C. The new piece of line is CD. Now there are two angles at C that touch the line: the corner of the triangle, and the angle between CD and CA.
The corner of the triangle at C is 70°. What do you think the other angle at C is?
What this lesson covers
The idea
The angle formed between the extension of a side of a triangle and the other side is an exterior angle; it forms a straight angle with the adjacent interior angle, so the two add up to 180°.
What is left of the straight line?
Extend the side BC beyond C. The new piece of line is CD. Now there are two angles at C that touch the line: the corner of the triangle, and the angle between CD and CA.
The corner of the triangle at C is 70°. What do you think the other angle at C is?
- Also 70°
- 110°
- We cannot know without measuring
Extend a side at every corner
Extend a side past a corner. Look at the two angles next to the straight line. Try every corner.
The angle outside
When a side of a triangle is extended beyond a corner, the angle between the extension and the other side is an exterior angle.
The interior angle and the exterior angle next to it sit on one straight line. So they always add up to 180°.
An exterior angle is formed between an extended side of a triangle and the other side at that corner. It sits on a straight line with the interior angle beside it, so the two add up to 180°.
- Angle at the corner | Is
- ∠ACB | the interior angle
- ∠ACD | the exterior angle, between CD and CA
- ∠ACB + ∠ACD | a straight angle, 180°
Notes
An exterior angle is formed between an extended side of a triangle and the other side at that corner. It sits on a straight line with the interior angle beside it, so the two add up to 180°.
Check yourself
The interior angle at a corner of a triangle is 65°. How big is the exterior angle at that corner?
Answer: 115
The interior and exterior angles add up to 180°. So the exterior angle is 180° − 65° = 115°.
In triangle ABC, ∠A = 50° and ∠B = 60°. The side BC is extended beyond C to D. How big is the exterior angle ∠ACD?
Answer: 110
∠ACB = 180° − 50° − 60° = 70°. The exterior angle is on a straight line with it: ∠ACD = 180° − 70° = 110°.
An exterior angle of a triangle is 110°. How big is the interior angle beside it?
Which pair of angles is an interior angle and the exterior angle beside it?
- 110°. The interior angle beside it is not equal to it. The two make a straight line, so they add up to 180°.
- 70° — correct. Yes! The two are on a straight line: 110° + 70° = 180°.
- 250°. 110° + 250° = 360°, a full turn. The two angles are on a straight line, so they add up to 180°.
- The corners at B and at C of the triangle. Both of these are interior angles. An exterior angle needs an extended side.
- The three corners of the triangle together. The three interior angles together make 180°, but that is not an interior and exterior pair.
- The corner of the triangle at C and the angle between CD and CA, where D is on BC extended — correct. Yes! They share the side CA and sit on the straight line BCD.