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Lesson 22 Understand Angle Relationships in Triangles
Lesson 22Understand Angle Relationships in Triangles
You have seen that angles formed by parallel lines and a transversal have important relationships. Angles in triangles also have similar relationships.
Later in this lesson you will show that the sum of the three angle measures in any triangle is 180°.
You can use this angle relationship to help you solve problems about triangles and other polygons. For example, you could find the measure of each angle in an equilateral triangle.
An exterior angle is the angle formed by extending one side of a polygon past a vertex.
InteriorAngle
ExteriorAngle
The fact that the sum of the measures of the three interior angles is also 180° will help you find a relationship between the exterior angle and the other interior angles. You will reason that, in any triangle, the measure of an exterior angle (such as w) equals the sum of the measures of the two non-adjacent interior angles (y and z).
Think The measure of an exterior angle of a triangle plus the measure of the adjacent interior angle 5 180°.
What special angle relationships are in triangles?
Lesson 22 Understand Angle Relationships in Triangles
Angles in a Triangle
2 Explain how you know that m/1 1 m/2 1 m/3 5 180°.
Lines m and n are parallel. What is the relationship that tells you that each statement is true?
3 m/1 5 m/4
4 m/3 5 m/5
5 Explain how you can use congruent angles from problems 3 and 4 with the equation in problem 2 to show that the sum of the measures of the angles inside the triangle is 180°.
Now try this problem.
6 Use the fact about triangle angle sums and similar triangles to solve this problem. Triangle PQR and triangle STU are similar triangles. In triangle PQR, m/QPR 5 60° and /PQR is a right angle.
Explain how you can solve for x.
m
n
1 32
4 5
S
T
R
Q60°
P
U
x
Let’s Explore the Idea Let’s explore triangles in relationship to angles formed by intersecting lines.
Let’s Talk About It Now let’s explore the relationship between the angles in two triangles.
You demonstrated on the previous page that the sum of the measures of the angles in any triangle is 180°. Now consider these two triangles: /a > /d and /b > /e. Complete the reasoning to show that the third pair of angles must also be congruent.
7 How do you know that c 5 180 2 a 2 b and f 5 180 2 d 2 e?
8 Remember that a 5 d and b 5 e. Why can you say that c 5 180 2 d 2 e?
9 Remember that f 5 180 2 d 2 e. Why can you say that c 5 f?
10 Would this reasoning still be true if the triangles were right triangles or isosceles triangles? Explain.
Try It Another Way
11 Draw nRST similar to nABC using a scale factor of 1 ·· 2 .
Use a protractor to check that corresponding angles
are congruent.
12 Write the proportions you would use to confirm that the two triangles are similar to each other.
Lesson 22 Understand Angle Relationships in Triangles
Angles in a Triangle
Talk through these problems as a class, then write your answers below.
13 Prove Complete the logic to show that the sum of the measures of the two non-adjacent interior angles y and z equals the measure of the exterior angle w.
• Complete the two equations to show the angle relationships you know.
x 1 y 1 z 5
w 1 x 5
• Solve the first equation for the sum y 1 z and solve for the second equation for w.
y 1 z 5
w 5
• How do you know that the measure of the exterior angle w equals the sum of the measures of the two non-adjacent interior angles y and z?
14 Explain In the diagram, /ABC > /DEC. Explain how you know that nABC and nDEC are similar triangles.
15 Apply This diagram shows the same similar triangles as in Problem 14. Each segment except ··· AB is a road. Between points A and B is a pond. Explain how you can find the distance across the pond by measuring other distances.