Exterior Angles of Triangles
Stretch one side of a triangle past a corner and you get an exterior angle. It always equals the two far-away inside angles added together - and the three exterior angles of any triangle add to one full turn.
Know these first
- Triangle Angle Sum 8.G.5
This unlocks
- More lessons that build on this are coming.
What you'll be able to do
- Find an exterior angle from the two remote interior angles, and work backward to a missing one.
- Solve for x when the angles are given as expressions.
- Use the fact that a triangle's three exterior angles add to 360°.
- Move between interior and exterior angles to solve multi-step problems.
The big idea
Extend one side of a triangle past a corner. The angle outside, between the extended side and the next side, is an exterior angle.
The exterior angle and the inside angle at that corner sit on a straight line, so they add to 180°. The three inside angles also add to 180°. So the exterior angle equals the sum of the two remote interior angles - the two far-away ones.
Walk around a triangle and turn at each corner: the three turns are the exterior angles, and together they bring you back to facing the way you started. The three exterior angles add to 360°.
Words you need
- Exterior angle
- The angle outside a triangle when one side is extended past a corner.
- Remote interior angles
- The two inside angles that are not at the exterior angle's corner.
- Adjacent interior angle
- The inside angle at the same corner. It adds to 180° with the exterior angle.
Worked examples
Example 1
The remote interior angles are 45° and 70°. Find the exterior angle.
- The exterior angle equals the two remote angles added together.
- 45° + 70° = 115°.
Answer: 115°
Example 2
An exterior angle is 125°. One remote interior angle is 38°. Find the other.
- Remote angles add to the exterior angle: 38° + ? = 125°.
- 125° − 38° = 87°.
Answer: 87°
Example 3
The remote angles are (x + 20)° and 2x°. The exterior angle is (4x − 10)°. Find x and the exterior angle.
- Set the sum of the remote angles equal to the exterior angle: (x + 20) + 2x = 4x − 10.
- Simplify: 3x + 20 = 4x − 10, so x = 30.
- Exterior angle = 4(30) − 10 = 110°.
Answer: x = 30; the exterior angle is 110°
Common mistakes
"The exterior angle equals the angle next to it." The angle next to it adds to 180° with the exterior angle. The exterior angle equals the two FAR angles added.
Using all three inside angles. Only the two remote angles add up to the exterior angle.
Adding exterior angles to 180°. One exterior angle at each corner - together they make a full turn, 360°.
Stopping at x. If the question asks for the angle, put x back into the expression.
Practice: 20 questions
13 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Find the exterior angle.
- Find the exterior angle.
- A triangular park has far corners of 52° and 63°. A sidewalk runs along one side and continues straight past the third corner. What exterior angle does it make there? Word problem
- A triangular pennant has angles of 24° and 78° at its far corners. The flagpole side extends past the third corner. Find the exterior angle there. Word problem
- A shelf bracket is a right triangle: 90° at the wall and 35° at the outer tip. Its diagonal arm extends past the bottom corner. What is the exterior angle there? Word problem
Tier 2 Intermediate One more step to think through
- The exterior angle is 130°. Find the missing remote angle.
- The exterior angle is 140°. Find the missing remote angle.
- A ramp support is a right triangle. At its base, the ground extended outward makes a 140° exterior angle with the ramp. The far right angle is 90°. Find the other remote angle. Word problem
- A skateboard ramp's side is a triangle. The exterior angle at the ground is 160°, and the remote angle at the top is 128°. Find the other remote angle. Word problem
- Outside one eave of a roof, the exterior angle is 145°. One remote angle is 35°. Find the other remote angle. Word problem
- A hiking loop forms a triangle. Where the trail continues past one corner, the exterior angle is 118°. One remote corner is 55°. Find the other. Word problem
- A triangle has exterior angles of 110° and 130° at two of its corners. Find the exterior angle at the third corner.
Tier 3 Advanced Several steps - find the hidden structure
- The remote angles are x° and (x + 10)°, and the exterior angle is (3x − 20)°. Find x and the exterior angle.
- A bridge truss triangle has remote angles of 3x° and (x + 20)°, and an exterior angle of (5x − 10)°. Find x and the exterior angle. Word problem
- A tent's front is a triangle. Outside the left corner, the ground and tent side form a 116° exterior angle. The two remote corners are equal. What does each measure? Word problem
- Mia jogs around a triangular path. At the first two corners she turns 125° and 140°. How far does she turn at the third corner to face her starting direction? Word problem
- Outside the eave of an isosceles roof, the exterior angle is 130°. The two remote angles are equal. Find each. Word problem
Tier 4 Challenge Stretch problems, some above grade level
- In triangle ABC, the exterior angle at A is 115° and the exterior angle at B is 140°. Find the interior angle at C.
- A triangular sail has remote angles of x° and (2x + 10)°, and an exterior angle of (4x − 20)°. Find x and the exterior angle. Word problem
- In a triangular garden, the exterior angle at one corner is 3 times the interior angle there, and the two remote angles are equal. Find each remote angle. Word problem
Answers
Tier 1 Foundations
- 115°. The exterior angle equals the two remote interior angles added: 50° + 65° = 115°.
- 122°. The exterior angle equals the two remote interior angles added: 32° + 90° = 122°.
- 115°. The exterior angle equals the two remote interior angles added: 52° + 63° = 115°.
- 102°. The exterior angle equals the two remote interior angles added: 24° + 78° = 102°.
- 125°. The exterior angle equals the two remote interior angles added: 90° + 35° = 125°.
Tier 2 Intermediate
- 82°. Exterior angle = sum of the remote interior angles, so the missing one is 130° − 48° = 82°.
- 69°. Exterior angle = sum of the remote interior angles, so the missing one is 140° − 71° = 69°.
- 50°. Exterior angle = sum of the remote interior angles, so the missing one is 140° − 90° = 50°.
- 32°. Exterior angle = sum of the remote interior angles, so the missing one is 160° − 128° = 32°.
- 110°. Exterior angle = sum of the remote interior angles, so the missing one is 145° − 35° = 110°.
- 63°. Exterior angle = sum of the remote interior angles, so the missing one is 118° − 55° = 63°.
- 120°. A triangle's three exterior angles add to 360°: 360° − 110° − 130° = 120°.
Tier 3 Advanced
- x = 30; the exterior angle is 70°. An exterior angle equals the sum of the two remote interior angles, so set them equal and solve.
- x = 30; the exterior angle is 140°. An exterior angle equals the sum of the two remote interior angles, so set them equal and solve.
- x = 58; the exterior angle is 116°. An exterior angle equals the sum of the two remote interior angles, so set them equal and solve.
- 95°. A triangle's three exterior angles add to 360°: 360° − 125° − 140° = 95°.
- x = 65; the exterior angle is 130°. An exterior angle equals the sum of the two remote interior angles, so set them equal and solve.
Tier 4 Challenge
- 75. Interior A = 180° − 115° = 65° and interior B = 180° − 140° = 40°, so C = 180° − 65° − 40° = 75°.
- x = 30; the exterior angle is 100°. An exterior angle equals the sum of the two remote interior angles, so set them equal and solve.
- 67.5. Interior + exterior = 180° and exterior = 3 × interior, so the interior is 45° and the exterior is 135°. The equal remote angles split 135°: 67.5° each.
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