Understanding exterior angles of a triangle is crucial for mastering geometry. This worksheet will guide you through the concept, providing examples and exercises to solidify your understanding. We'll explore the properties of exterior angles and how they relate to the interior angles of a triangle. By the end, you'll be able to confidently solve problems involving exterior angles.
What is an Exterior Angle of a Triangle?
An exterior angle of a triangle is formed by extending one side of the triangle. It's the angle between the extended side and the adjacent side of the triangle. Each vertex of a triangle has two exterior angles, one formed by extending each of the two sides meeting at that vertex. We typically focus on just one exterior angle per vertex for calculations.
Key Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non-adjacent) interior angles.
Let's illustrate this with a triangle ABC:
- If we extend side AB, the exterior angle at vertex B is the angle formed outside the triangle between the extended AB and side BC.
- This exterior angle is equal to the sum of angles A and C (the remote interior angles).
Understanding the Relationship Between Interior and Exterior Angles
The sum of the interior angles of any triangle is always 180 degrees. This fact is fundamental to understanding exterior angles. Since an exterior angle and its adjacent interior angle form a straight line (180 degrees), we can easily derive the relationship mentioned above.
Example:
Imagine a triangle with angles:
- Angle A = 60°
- Angle B = 70°
- Angle C = 50°
The exterior angle at vertex C (let's call it angle X) will be:
- Angle X = Angle A + Angle B = 60° + 70° = 130°
Notice that Angle X + Angle C = 180° (they form a straight line).
Calculating Exterior Angles: Practice Problems
Let's put this knowledge into practice. Solve the following problems:
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 x° angle.
Problem 2:
One exterior angle of a triangle measures 110°. If one of the remote interior angles is 55°, what is the measure of the other remote interior angle?
Problem 3:
In a triangle XYZ, the exterior angle at X is 120°. If angle Y is twice the size of angle Z, find the measures of angles Y and Z.
Frequently Asked Questions (FAQs)
How many exterior angles does a triangle have?
A triangle has six exterior angles – two at each vertex. However, we usually focus on one exterior angle per vertex, as they are supplementary to the adjacent interior angles.
What is the relationship between the exterior angle and the adjacent interior angle?
An exterior angle and its adjacent interior angle are supplementary; their sum is 180 degrees.
Can an exterior angle of a triangle be greater than 180 degrees?
No. Exterior angles are formed by extending a side of the triangle, and thus they must be less than 180 degrees.
How are exterior angles used in real-world applications?
Exterior angles are used in many applications, including surveying, navigation, and construction. Understanding angles is essential for accurate measurements and planning.
Solutions to Practice Problems
Problem 1:
The sum of interior angles is 180°. Therefore, 45° + 65° + x° = 180°. Solving for x, we get x = 70°. The exterior angle at the vertex with the 70° angle is 180° - 70° = 110°.
Problem 2:
The exterior angle is equal to the sum of the two remote interior angles. Let y be the other remote interior angle. Then 110° = 55° + y. Solving for y, we get y = 55°.
Problem 3:
Let angle Z be represented by z. Then angle Y is 2z. The exterior angle at X is equal to the sum of angles Y and Z, so 120° = 2z + z. This simplifies to 3z = 120°, meaning z = 40°. Therefore, angle Z = 40° and angle Y = 80°.
This worksheet provides a comprehensive overview of exterior angles of a triangle. Remember to practice consistently to master this crucial geometrical concept. Further exploration of triangle properties will enhance your understanding of geometric relationships.
