Interior And Exterior Angles Of Triangles Worksheet With Answers Pdf

Interior and Exterior Angles of Triangles Worksheet with Answers: A Comprehensive Guide

This comprehensive guide delves into the world of triangles, specifically focusing on their interior and exterior angles. We'll explore the fundamental concepts, provide you with practice problems, and offer solutions to help solidify your understanding. This resource serves as a virtual worksheet, complete with answers, perfect for students of all levels seeking to master this crucial geometric topic.

Understanding Interior Angles

The interior angles of a triangle are the angles formed inside the triangle by its sides. A fundamental property of triangles is that the sum of their interior angles always equals 180 degrees. This is a cornerstone theorem in geometry, and understanding it is key to solving many triangle-related problems.

Theorem: Sum of Interior Angles

The sum of the interior angles of any triangle is always 180°. This holds true regardless of the triangle's type (equilateral, isosceles, scalene, acute, obtuse, or right-angled).

Types of Triangles Based on Angles

• Acute Triangle: All three interior angles are less than 90°.
• Right Triangle: One interior angle is exactly 90°.
• Obtuse Triangle: One interior angle is greater than 90°.

Practice Problems: Interior Angles

Problem 1: Find the missing angle (x) in a triangle with angles 70° and 60°.

Solution: Since the sum of interior angles is 180°, we have: 70° + 60° + x = 180°. Solving for x, we get x = 50°.

Problem 2: A triangle has angles of 45°, 45°, and x. What type of triangle is it, and what is the value of x?

Solution: 45° + 45° + x = 180°. This gives x = 90°. This is a right-angled isosceles triangle.

Problem 3: Two angles of a triangle are 30° and 110°. Is this possible? Why or why not?

Solution: No, this is not possible. 30° + 110° = 140°. The remaining angle would need to be 40° to reach 180°, which contradicts the given information about the two angles being 30° and 110°.

Understanding Exterior Angles

An exterior angle of a triangle is formed by extending one of the sides of the triangle. It is the angle between the extended side and the adjacent side of the triangle. Each vertex of a triangle has two exterior angles, but we typically focus on one at a time.

Theorem: Exterior Angle and Remote Interior Angles

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. The remote interior angles are the two angles inside the triangle that are not adjacent to the exterior angle.

Practice Problems: Exterior Angles

Problem 4: In a triangle, two interior angles are 55° and 75°. Find the measure of the exterior angle adjacent to the 55° angle.

Solution: The exterior angle is equal to the sum of the remote interior angles (75° + 55° = 130°).

Problem 5: An exterior angle of a triangle measures 110°. One of its remote interior angles is 60°. Find the other remote interior angle.

Solution: Let the other remote interior angle be x. Then 110° = 60° + x. Solving for x, we get x = 50°.

Problem 6: If an exterior angle of a triangle is 120° and one remote interior angle is twice the other, find the measures of the two remote interior angles.

Solution: Let the remote interior angles be x and 2x. Then 120° = x + 2x = 3x. Solving for x, we get x = 40°. Therefore, the remote interior angles are 40° and 80°.

Combining Interior and Exterior Angles

Many problems will require you to utilize both interior and exterior angle properties simultaneously.

Practice Problems: Combined Interior and Exterior Angles

Problem 7: In a triangle, one exterior angle is 100°, and one of the remote interior angles is twice the other. Find the measures of all three interior angles.

Solution: Let the remote interior angles be x and 2x. The exterior angle is 100° = x + 2x = 3x. Therefore, x = 100°/3 ≈ 33.33°. The remote interior angles are approximately 33.33° and 66.67°. The third interior angle is 180° - 33.33° - 66.67° ≈ 80°.

Problem 8: The angles of a triangle are in the ratio 2:3:4. Find the measures of the exterior angles.

Solution: Let the angles be 2x, 3x, and 4x. The sum of interior angles is 180°, so 2x + 3x + 4x = 180°. This gives 9x = 180°, so x = 20°. The angles are 40°, 60°, and 80°. The exterior angles are 180° - 40° = 140°, 180° - 60° = 120°, and 180° - 80° = 100°.

Problem 9: Two exterior angles of a triangle are 115° and 125°. Find all three interior angles.

Solution: The interior angles corresponding to the exterior angles are 180° - 115° = 65° and 180° - 125° = 55°. The sum of these two interior angles is 65° + 55° = 120°. The third interior angle is 180° - 120° = 60°.

Advanced Problems and Applications

The concepts of interior and exterior angles have broad applications in various fields, including:

• Surveying: Determining land boundaries and distances.
• Navigation: Calculating bearings and routes.
• Engineering: Designing structures and ensuring stability.
• Architecture: Planning building layouts and ensuring structural integrity.

Problem 10: A surveyor needs to determine the angle of a triangular plot of land. Two sides of the triangle meet at an angle of 70°, and the exterior angle formed by extending the other side is 110°. What are the other two angles of the triangle?

Solution: The interior angle opposite the exterior angle of 110° is 180° - 110° = 70°. Since one angle is 70°, and another is also 70°, the remaining angle is 180° - 70° - 70° = 40°.

Problem 11: A triangular traffic sign has exterior angles of 100°, 120°, and x. Find the value of x and the measures of the interior angles.

Solution: The sum of exterior angles of any polygon is always 360°. Therefore, 100° + 120° + x = 360°. This gives x = 140°. The interior angles are 180° - 100° = 80°, 180° - 120° = 60°, and 180° - 140° = 40°.

Conclusion

Mastering the concepts of interior and exterior angles of triangles is crucial for success in geometry and related fields. This guide, with its numerous practice problems and detailed solutions, provides a solid foundation for understanding these essential geometric principles. Remember to practice regularly and apply these concepts to various problem types to build a strong understanding. By consistently working through problems and applying the theorems, you'll develop the skills necessary to confidently tackle more complex geometric challenges. Remember to always check your work and ensure your solutions make logical sense within the context of the problem. Good luck!