# Lesson 20

Transformations, Transversals, and Proof

• Let’s prove statements about parallel lines.

### Problem 1

Priya: I bet if the alternate interior angles are congruent, then the lines will have to be parallel.

Han: Really? We know if the lines are parallel then the alternate interior angles are congruent, but I didn't know that it works both ways.

Priya: Well, I think so. What if angle $$ABC$$ and angle $$BCJ$$ are both 40 degrees? If I draw a line perpendicular to line $$AI$$ through point $$B$$, I get this triangle. Angle $$CBX$$ would be 50 degrees because $$40+50=90$$. And because the angles of a triangle sum to 180 degrees, angle $$CXB$$ is 90 degrees. It's also a right angle!

Han: Oh! Then line $$AI$$ and line $$GJ$$ are both perpendicular to the same line. That's how we constructed parallel lines, by making them both perpendicular to the same line. So lines $$AI$$ and $$GJ$$ must be parallel.

1. Label the diagram based on Priya and Han's conversation.
2. Is there something special about 40 degrees? Will any 2 lines cut by a transversal with congruent alternate interior angles, be parallel?

### Problem 2

Prove lines $$AI$$ and $$GJ$$ are parallel.

### Problem 3

What is the measure of angle $$ABE$$?

(From Unit 1, Lesson 19.)

### Problem 4

Lines $$AB$$ and $$BC$$ are perpendicular. The dashed rays bisect angles $$ABD$$ and $$CBD$$. Explain why the measure of angle $$EBF$$ is 45 degrees.

(From Unit 1, Lesson 19.)

### Problem 5

Identify a figure that is not the image of quadrilateral $$ABCD$$ after a sequence of transformations. Explain how you know.

(From Unit 1, Lesson 18.)

### Problem 6

Quadrilateral $$ABCD$$ is congruent to quadrilateral $$A’B’C’D’$$. Describe a sequence of rigid motions that takes $$A$$ to $$A’$$, $$B$$ to $$B’$$, $$C$$ to $$C’$$, and $$D$$ to $$D’$$.

(From Unit 1, Lesson 17.)

### Problem 7

Triangle $$ABC$$ is congruent to triangle $$A’B’C’$$. Describe a sequence of rigid motions that takes $$A$$ to $$A’$$, $$B$$ to $$B’$$, and $$C$$ to $$C’$$.

(From Unit 1, Lesson 17.)

### Problem 8

Identify any angles of rotation that create symmetry.

(From Unit 1, Lesson 16.)

### Problem 9

Select all the angles of rotation that produce symmetry for this flower.

A:

45

B:

60

C:

90

D:

120

E:

135

F:

150

G:

180

(From Unit 1, Lesson 16.)

### Problem 10

Three line segments form the letter N. Rotate the letter N clockwise around the midpoint of segment $$BC$$ by 180 degrees. Describe the result.

(From Unit 1, Lesson 14.)