# 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.

- Label the diagram based on Priya and Han's conversation.
- 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 \(ABE\)?

### 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.

### Problem 5

Identify a figure that is *not** * the image of quadrilateral \(ABCD\) after a sequence of transformations. Explain how you know.

### 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 \(D’\).

### 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 \(C’\).

### Problem 8

Identify any angles of rotation that create symmetry.

### Problem 9

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

45

60

90

120

135

150

180

### 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.