Exterior Angle Theorem practice worksheet with problems requiring calculation of unknown angles in triangles.
Worksheet with geometry problems involving the Exterior Angle Theorem, showing triangles with labeled angles and variables to solve for.
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Step-by-step solution for: 4 The Exterior Angle Theorem (PDF) - Kuta Software
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Show Answer Key & Explanations
Step-by-step solution for: 4 The Exterior Angle Theorem (PDF) - Kuta Software
This image is a worksheet from Kuta Software on the "Exterior Angle Theorem" in geometry. The task is to find the measure of each indicated angle, and for problem 9, to solve for the variable 'x'.
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (the two angles inside the triangle that are not adjacent to the exterior angle).
I will solve each problem step-by-step.
---
- Given: In triangle UTV, the interior angles at U and T are 7° and 50° respectively. The exterior angle at V is 120°.
- Find: The measure of the indicated angle at vertex V (which appears to be an interior angle).
- Solution:
- The exterior angle at V is 120°.
- The two remote interior angles are at U (7°) and T (50°).
- According to the theorem, the exterior angle should equal the sum of the remote interior angles: 7° + 50° = 57°.
- However, the given exterior angle is 120°, which contradicts the theorem. This suggests there might be a typo in the worksheet, or the 120° is meant to be the interior angle.
- If we assume the 120° is the *interior* angle at V, then the exterior angle would be 180° - 120° = 60°.
- The sum of the remote interior angles is 7° + 50° = 57°, which still does not match 60°.
- Given the inconsistency, I will calculate the missing interior angle assuming the triangle's angles must sum to 180°.
- Interior angle at V = 180° - 7° - 50° = 123°.
- Therefore, the exterior angle at V would be 180° - 123° = 57°.
- Since the problem asks for the measure of the indicated angle, and the 120° is shown as the exterior angle, it is likely a misprint. The correct exterior angle should be 57°.
---
- Given: In triangle VUT, the interior angle at V is 50°, and the exterior angle at T is 115°.
- Find: The measure of the indicated interior angle at U.
- Solution:
- The exterior angle at T is 115°.
- The two remote interior angles are at V (50°) and U (unknown).
- According to the theorem: Exterior angle = Sum of remote interior angles.
- So, 115° = 50° + angle U.
- Therefore, angle U = 115° - 50° = 65°.
---
- Given: In triangle STU, the interior angle at S is 70°, and the interior angle at T is 50°. The exterior angle at U is indicated.
- Find: The measure of the exterior angle at U.
- Solution:
- The two remote interior angles are at S (70°) and T (50°).
- According to the theorem: Exterior angle at U = 70° + 50° = 120°.
---
- Given: In triangle STR, the interior angle at S is 52°, and the interior angle at T is 7°. The exterior angle at R is indicated.
- Find: The measure of the exterior angle at R.
- Solution:
- The two remote interior angles are at S (52°) and T (7°).
- According to the theorem: Exterior angle at R = 52° + 7° = 59°.
---
- Given: In triangle EDC, the interior angle at D is 45°, and the exterior angle at C is 140°.
- Find: The measure of the indicated interior angle at E.
- Solution:
- The exterior angle at C is 140°.
- The two remote interior angles are at D (45°) and E (unknown).
- According to the theorem: 140° = 45° + angle E.
- Therefore, angle E = 140° - 45° = 95°.
---
- Given: In triangle TUV, the interior angle at T is 80°, and the interior angle at V is 180°? (This seems incorrect as an interior angle cannot be 180° in a triangle). Looking closely, it appears to be 18°.
- Assumption: The angle at V is 18°, not 180°, as 180° is impossible for an interior angle of a triangle.
- Find: The measure of the exterior angle at U.
- Solution:
- The two remote interior angles are at T (80°) and V (18°).
- According to the theorem: Exterior angle at U = 80° + 18° = 98°.
---
- Given: In triangle EFG, the interior angle at E is 26°, and the interior angle at F is 38°. The exterior angle at G is indicated.
- Find: The measure of the exterior angle at G.
- Solution:
- The two remote interior angles are at E (26°) and F (38°).
- According to the theorem: Exterior angle at G = 26° + 38° = 64°.
---
- Given: In triangle HKQ, the interior angle at H is 35°, and the interior angle at Q is 95°. The exterior angle at K is indicated.
- Find: The measure of the exterior angle at K.
- Solution:
- The two remote interior angles are at H (35°) and Q (95°).
- According to the theorem: Exterior angle at K = 35° + 95° = 130°.
---
- Given: In triangle PQR, the interior angle at P is 20°, the interior angle at R is 4x + 2, and the exterior angle at Q is 8 + 6x.
- Find: Solve for x.
- Solution:
- The exterior angle at Q is 8 + 6x.
- The two remote interior angles are at P (20°) and R (4x + 2).
- According to the theorem: Exterior angle = Sum of remote interior angles.
- So, 8 + 6x = 20 + (4x + 2)
- Simplify the right side: 8 + 6x = 22 + 4x
- Subtract 4x from both sides: 8 + 2x = 22
- Subtract 8 from both sides: 2x = 14
- Divide by 2: x = 7
---
1. 57° (assuming the 120° is a misprint for the exterior angle)
2. 65°
3. 120°
4. 59°
5. 95°
6. 98° (assuming the angle at V is 18°)
7. 64°
8. 130°
9. x = 7
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (the two angles inside the triangle that are not adjacent to the exterior angle).
I will solve each problem step-by-step.
---
Problem 1
- Given: In triangle UTV, the interior angles at U and T are 7° and 50° respectively. The exterior angle at V is 120°.
- Find: The measure of the indicated angle at vertex V (which appears to be an interior angle).
- Solution:
- The exterior angle at V is 120°.
- The two remote interior angles are at U (7°) and T (50°).
- According to the theorem, the exterior angle should equal the sum of the remote interior angles: 7° + 50° = 57°.
- However, the given exterior angle is 120°, which contradicts the theorem. This suggests there might be a typo in the worksheet, or the 120° is meant to be the interior angle.
- If we assume the 120° is the *interior* angle at V, then the exterior angle would be 180° - 120° = 60°.
- The sum of the remote interior angles is 7° + 50° = 57°, which still does not match 60°.
- Given the inconsistency, I will calculate the missing interior angle assuming the triangle's angles must sum to 180°.
- Interior angle at V = 180° - 7° - 50° = 123°.
- Therefore, the exterior angle at V would be 180° - 123° = 57°.
- Since the problem asks for the measure of the indicated angle, and the 120° is shown as the exterior angle, it is likely a misprint. The correct exterior angle should be 57°.
---
Problem 2
- Given: In triangle VUT, the interior angle at V is 50°, and the exterior angle at T is 115°.
- Find: The measure of the indicated interior angle at U.
- Solution:
- The exterior angle at T is 115°.
- The two remote interior angles are at V (50°) and U (unknown).
- According to the theorem: Exterior angle = Sum of remote interior angles.
- So, 115° = 50° + angle U.
- Therefore, angle U = 115° - 50° = 65°.
---
Problem 3
- Given: In triangle STU, the interior angle at S is 70°, and the interior angle at T is 50°. The exterior angle at U is indicated.
- Find: The measure of the exterior angle at U.
- Solution:
- The two remote interior angles are at S (70°) and T (50°).
- According to the theorem: Exterior angle at U = 70° + 50° = 120°.
---
Problem 4
- Given: In triangle STR, the interior angle at S is 52°, and the interior angle at T is 7°. The exterior angle at R is indicated.
- Find: The measure of the exterior angle at R.
- Solution:
- The two remote interior angles are at S (52°) and T (7°).
- According to the theorem: Exterior angle at R = 52° + 7° = 59°.
---
Problem 5
- Given: In triangle EDC, the interior angle at D is 45°, and the exterior angle at C is 140°.
- Find: The measure of the indicated interior angle at E.
- Solution:
- The exterior angle at C is 140°.
- The two remote interior angles are at D (45°) and E (unknown).
- According to the theorem: 140° = 45° + angle E.
- Therefore, angle E = 140° - 45° = 95°.
---
Problem 6
- Given: In triangle TUV, the interior angle at T is 80°, and the interior angle at V is 180°? (This seems incorrect as an interior angle cannot be 180° in a triangle). Looking closely, it appears to be 18°.
- Assumption: The angle at V is 18°, not 180°, as 180° is impossible for an interior angle of a triangle.
- Find: The measure of the exterior angle at U.
- Solution:
- The two remote interior angles are at T (80°) and V (18°).
- According to the theorem: Exterior angle at U = 80° + 18° = 98°.
---
Problem 7
- Given: In triangle EFG, the interior angle at E is 26°, and the interior angle at F is 38°. The exterior angle at G is indicated.
- Find: The measure of the exterior angle at G.
- Solution:
- The two remote interior angles are at E (26°) and F (38°).
- According to the theorem: Exterior angle at G = 26° + 38° = 64°.
---
Problem 8
- Given: In triangle HKQ, the interior angle at H is 35°, and the interior angle at Q is 95°. The exterior angle at K is indicated.
- Find: The measure of the exterior angle at K.
- Solution:
- The two remote interior angles are at H (35°) and Q (95°).
- According to the theorem: Exterior angle at K = 35° + 95° = 130°.
---
Problem 9
- Given: In triangle PQR, the interior angle at P is 20°, the interior angle at R is 4x + 2, and the exterior angle at Q is 8 + 6x.
- Find: Solve for x.
- Solution:
- The exterior angle at Q is 8 + 6x.
- The two remote interior angles are at P (20°) and R (4x + 2).
- According to the theorem: Exterior angle = Sum of remote interior angles.
- So, 8 + 6x = 20 + (4x + 2)
- Simplify the right side: 8 + 6x = 22 + 4x
- Subtract 4x from both sides: 8 + 2x = 22
- Subtract 8 from both sides: 2x = 14
- Divide by 2: x = 7
---
Final Answers:
1. 57° (assuming the 120° is a misprint for the exterior angle)
2. 65°
3. 120°
4. 59°
5. 95°
6. 98° (assuming the angle at V is 18°)
7. 64°
8. 130°
9. x = 7
Parent Tip: Review the logic above to help your child master the concept of exterior angle theorem worksheet with answer key.
