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Questions and Answers
What is the definition of an exterior angle of a triangle?
What is the relationship between the measure of an exterior angle and the measures of the two non-adjacent interior angles?
What is the exterior angle theorem?
How do you find the measure of an exterior angle of a triangle?
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What is the relationship between an exterior angle and its adjacent interior angle?
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What is the purpose of using diagrams to visualize exterior angles?
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What is the measure of the exterior angle of the triangle shown below? 
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In the triangle shown below, what is the measure of the exterior angle at vertex A? 
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What is the measure of the exterior angle of the triangle shown below, if the measure of one interior angle is 30° and another interior angle is 60°? 
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In the triangle shown below, what is the measure of the exterior angle at vertex B? 
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What is the measure of the exterior angle of the triangle shown below, if the measure of one interior angle is 45° and another interior angle is 90°? 
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Study Notes
Exterior Angles of Triangles
Definition
- An exterior angle of a triangle is an angle formed by extending one side of the triangle.
- It is supplementary to the adjacent interior angle.
Properties
- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
- The exterior angle theorem: m∠ACD = m∠A + m∠B, where ∠ACD is the exterior angle and ∠A and ∠B are the non-adjacent interior angles.
Diagrams
- To find an exterior angle, draw an extension of one side of the triangle.
- Identify the adjacent interior angle and the two non-adjacent interior angles.
- Use the exterior angle theorem to find the measure of the exterior angle.
Example
- In the diagram below, find the measure of ∠ACD:
∠A = 30° ∠B = 40°
m∠ACD = m∠A + m∠B m∠ACD = 30° + 40° m∠ACD = 70°
Key Takeaways
- Exterior angles are supplementary to adjacent interior angles.
- The measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles.
- Diagrams can be used to visualize and find exterior angles.
Exterior Angles of Triangles
- An exterior angle is formed by extending one side of a triangle and is supplementary to the adjacent interior angle.
- The measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles.
Exterior Angle Theorem
- The exterior angle theorem states that m∠ACD = m∠A + m∠B, where ∠ACD is the exterior angle and ∠A and ∠B are the non-adjacent interior angles.
Finding Exterior Angles
- To find an exterior angle, draw an extension of one side of the triangle.
- Identify the adjacent interior angle and the two non-adjacent interior angles.
- Use the exterior angle theorem to find the measure of the exterior angle.
Example
- In the diagram, ∠A = 30° and ∠B = 40°, so m∠ACD = 30° + 40° = 70°.
Key Takeaways
- Exterior angles are supplementary to adjacent interior angles.
- The measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles.
- Diagrams can be used to visualize and find exterior angles.
Finding Angles of Triangles
Types of Angles
- Exterior angle: formed by one side of the triangle and an extension of another side
- Interior angle: formed by two sides of the triangle
- Vertical angles: equal angles formed by two intersecting lines
- Alternate interior angles: equal angles formed by a transversal and two parallel lines
- Corresponding angles: equal angles formed by a transversal and two parallel lines
- Adjacent angles: sharing a common vertex and a common side
Angle Properties
- Sum of interior angles: 180° for any triangle
- Exterior angle theorem: measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles
- Linear pair theorem: sum of measures of adjacent angles is 180°
Calculating Angle Measures
- Using trigonometric ratios: sine, cosine, and tangent
- Using angle properties and theorems to set up and solve equations
Solving Triangles
- Right triangles: use trigonometric ratios to find missing side lengths and angles
- Oblique triangles: use law of sines and law of cosines to find missing side lengths and angles
Some examples of diagram-based problems:
📊 Example 1
A
/ \
/ \
/ \
C-----B
- Find ∠A if ∠C = 40° and ∠B = 60°
- Use angle properties and theorems to set up and solve an equation
📊 Example 2
5
/ \
/ \
/ \
A-----C
3 / \
/ \
7-----------B
- Find the length of side AB using trigonometry
- Use the law of cosines to find the length of the hypotenuse
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Description
Learn about exterior angles of triangles, including their definition, properties, and the exterior angle theorem.
