Understanding exterior angles of a triangle is crucial for mastering geometry. This worksheet will guide you through the concept, providing examples, practice problems, and solutions to help you solidify your knowledge. We'll explore the properties of exterior angles and how they relate to the interior angles of a triangle.
What are Exterior Angles of a Triangle?
An exterior angle of a triangle is formed by extending one of the sides of the triangle. It's the angle formed outside the triangle, adjacent to an interior angle. Each vertex of a triangle has two exterior angles, one on each side of the extended line. We typically focus on the single exterior angle supplementary to the adjacent interior angle.
Key Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote) interior angles.
This is a fundamental theorem in geometry and is incredibly useful for solving various problems related to triangles.
Understanding the Relationship Between Interior and Exterior Angles
Let's consider a triangle ABC. If we extend side BC, we create an exterior angle at vertex C. This exterior angle is supplementary to interior angle C (meaning their sum is 180°). Importantly, the exterior angle's measure is also equal to the sum of angles A and B.
Example: If angle A = 50° and angle B = 60°, then the exterior angle at C = 50° + 60° = 110°. Angle C itself would be 180° - 110° = 70°.
Practice Problems
Here are some problems to test your understanding. Remember the key property: exterior angle = sum of remote interior angles.
Problem 1: In triangle XYZ, angle X = 45° and angle Y = 70°. Find the measure of the exterior angle at Z.
Problem 2: An exterior angle of a triangle measures 120°. If one of the remote interior angles is 40°, what is the measure of the other remote interior angle?
Problem 3: The exterior angle at vertex A of triangle ABC is 115°. Angle B is twice the measure of angle C. Find the measures of angles B and C.
Solutions to Practice Problems
Problem 1 Solution: The exterior angle at Z = angle X + angle Y = 45° + 70° = 115°
Problem 2 Solution: Let the remote interior angles be A and B. We know that A + B = 120° and A = 40°. Therefore, B = 120° - 40° = 80°.
Problem 3 Solution: Let angle B = 2x and angle C = x. The exterior angle at A is 115°, so angle B + angle C = 115°. Substituting, we get 2x + x = 115°, which simplifies to 3x = 115°. Solving for x, we find x = 38.33°. Therefore, angle C = 38.33° and angle B = 2 * 38.33° = 76.67°.
Frequently Asked Questions (FAQs)
Can an exterior angle of a triangle be greater than 180°?
No, an exterior angle is supplementary to its adjacent interior angle. Since interior angles are always less than 180°, the exterior angle will always be less than 180°.
How many exterior angles does a triangle have?
A triangle has six exterior angles – two at each vertex. However, typically, when we refer to the exterior angle at a vertex, we're referring to the single angle supplementary to the adjacent interior angle.
What is the relationship between the sum of the exterior angles of a triangle and the sum of its interior angles?
The sum of the exterior angles of a triangle (one at each vertex) is always 360°. The sum of the interior angles is always 180°.
This worksheet provides a solid foundation for understanding exterior angles of a triangle. Remember to practice regularly to reinforce your knowledge and improve your problem-solving skills. Further exploration into similar geometrical concepts will build upon this understanding.
