Understanding exterior angles of a triangle is crucial for mastering geometry. This worksheet will guide you through the concepts, providing examples and exercises to solidify your understanding. We'll explore the relationship between interior and exterior angles, and how to solve problems involving them.
What are Exterior Angles of a Triangle?
An exterior angle of a triangle is the angle formed by one side of the triangle and the extension of an adjacent side. Each vertex of a triangle has two exterior angles, but we typically focus on one at a time. Notice that an exterior angle and its adjacent interior angle form a linear pair (they add up to 180°).
Key Properties of Exterior Angles of a Triangle
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Exterior Angle Theorem: 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.
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Sum of Exterior Angles: The sum of the measures of the three exterior angles of a triangle (one at each vertex) is always 360°.
Example Problem: Applying the Exterior Angle Theorem
Let's say we have a triangle with interior angles measuring 60°, 70°, and 50°. What is the measure of the exterior angle at the vertex with the 70° interior angle?
Solution: According to the Exterior Angle Theorem, the exterior angle is equal to the sum of the two remote interior angles. In this case, it's 60° + 50° = 110°.
Frequently Asked Questions (FAQs) about Exterior Angles
Here are some common questions students have regarding exterior angles of a triangle, addressed in detail:
How do I find the measure of an exterior angle given two interior angles?
To find the measure of an exterior angle, simply add the measures of the two remote interior angles (the angles that are not adjacent to the exterior angle). This directly applies the Exterior Angle Theorem.
What is the relationship between interior and exterior angles of a triangle?
An interior angle and its adjacent exterior angle are supplementary, meaning they add up to 180°. The exterior angle is also equal to the sum of the two remote interior angles.
How do I find the missing angle in a triangle given an exterior angle?
If you know an exterior angle and one remote interior angle, you can subtract the remote interior angle from the exterior angle to find the other remote interior angle. Alternatively, you can find the adjacent interior angle (180° - exterior angle) and then use the fact that the sum of interior angles in a triangle is 180° to find the missing angle.
Can the exterior angle of a triangle be obtuse?
Yes, an exterior angle can be obtuse. If the sum of the two remote interior angles is greater than 90°, the exterior angle will be obtuse.
What is the sum of the exterior angles of a triangle (one at each vertex)?
The sum of the exterior angles of a triangle (taking one exterior angle at each vertex) is always 360°.
Practice Problems: Exterior Angles of a Triangle Worksheet
Now let's put your knowledge to the test with some practice problems. Try solving these problems using the concepts and theorems we've discussed:
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Problem 1: A triangle has interior angles of 45°, 65°, and x°. Find the value of x and the measure of the exterior angle at the vertex with the 45° interior angle.
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Problem 2: An exterior angle of a triangle measures 120°. One of the remote interior angles measures 50°. What is the measure of the other remote interior angle?
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Problem 3: A triangle has exterior angles of a, b, and c. What is a + b + c?
Solutions: (These will be provided in a separate document or section to allow for independent problem-solving).
This worksheet provides a solid foundation for understanding exterior angles of a triangle. Remember to practice consistently to build confidence and proficiency in solving geometry problems. By understanding the relationships between interior and exterior angles, you'll unlock the ability to tackle more complex geometric challenges.
