2.4

How Can I Use It?Pg. 14

Angles In a Triangle

2.4 – How Can I Use It?_____________Angles In a Triangle

So far in this chapter, you have investigated the angle relationships created when two lines intersect, forming vertical angles. You have also investigated the relationships created when a transversal intersects two parallel lines. Today you will study the angle relationships that result when three non-parallel lines intersect, forming a triangle.

2.22 – ANGLE RELATIONSHIPS

Marcos decided to study the angle relationships in triangles by making a tiling. Find the pattern below.

a. Color in one of the angles with a pen or pencil. Then use the same color to shade every angle on the pattern that is equal to the shaded angle.

b. Repeat this process for the other two angles of the triangle, using a different color for each angle in the triangle. When you are done, every angle in your tiling should be shaded with one of the three colors.

c. Now examine your colored tiling. What relationship can you find between the three different-colored angles? You may want to focus on the angles that form a straight angle. What does this tell you about the angles in a triangle?

"If a polygon is a triangle, then the sum

of the interior angles is ___________"180°

d. Let us see if this works for any triangle. Each team member will cut out a different type of triangle: isosceles, scalene, right, or obtuse. Rip off the angles of the triangle and put them together to form a straight line. Do these three angles all add to 180°?

e. How can you convince yourself that your conjecture is true for all triangles? Match the reasons to the proof below to show that the sum of the interior angles of any triangle adds to 180°.

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

a d

c e 180d b e

180a b c

Alternate interior Alternate interiorSupplementarysubstitution

2.23 – TRIANGLE ANGLE RELATIONSHIPS

Use your proof about the angles in a triangle to find x in each diagram below.

x + 40 + 80 = 180

x + 120 = 180

x = 60°

2x + x + 12 + 96 = 180

3x + 108 = 180

x = 24°3x = 72

2.24 – TRIANGLE SUM THEOREMWhat can the Triangle Angle Sum Theorem help you learn about special triangles?

a. Find the measure of each angle in an equilateral triangle. Justify your conclusion.

180°3

60° 60°

60°

b. Consider the isosceles right triangle at right. Find the measures of all the angles.

180 – 902

45°

45°

c. What if you only know one angle of an isosceles triangle? For example, if what are the measures of the other two angles?

34 ,m A

34°180 – 34

2

73° 73°

d. Find the measure of each indicated angle in a regular octagon. Start by finding the central angle, then remember that each triangle inside is equal.

x°

y° z°

3608

x° = = 45°

y° = 180 – 452

67.5°y& z=

2.25 – TEAM REASONING CHALLENGE

How much can you figure out about the figure using your knowledge of angle relationships? Work with your team to find the measures of all the labeled angles in the diagram.

123°123° 57°

57°123° 57°

42°

81°99°81° 99°

2.26 – ANGLE AND LINE RELATIONSHIPS Use your knowledge of angle relationships to answer the questions below.

a. In the diagram at right, what is the sum of the angles x and y?

180°

b. While looking at the diagram below, Rianna exclaimed, "I think something is wrong with this diagram." What do you think she is referring to? Be prepared to share your ideas.

c. Maria is still not convinced that the lines in must be parallel. She decides to assume that they are not parallel and draws the diagram below. Where do lines and intersect in Maria's diagram?

E

0°

Yes,

112° + 68° = 180°

d. Write a conjecture based on your conclusion to this problem.

"If the measures of same-side interior angles are __________________, then the lines are _____________."

supplementaryparallel

e. Complete the conjectures below based on this information.

"If the measures of corresponding angles are __________, then the lines are _____________."

congruentparallel

e. Complete the conjectures below based on this information.

"If the measures of alternate interior angles are __________, then the lines are _____________."

congruentparallel