# Transformations, Transversals, and Proof

ID: pihoh-hizod
Illustrative Math
Subject: Geometry

# Transformations, Transversals, and Proof

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##### 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?

True or false? Write below.

3) Will any 2 lines cut by a transversal with congruent alternate interior angles, be parallel?

True or false? Write below.
##### Problem 2

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

##### Problem 3

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

##### Problem 4

6) 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

7) Identify a figure that is not the image of quadrilateral ﻿$ABCD$﻿ after a sequence of transformations.

a) $\text{Quadrilateral } BHGF$b) $\text{Quadrilateral } DJMA$c) $\text{Quadrilateral } CBFE$d) $\text{Quadrilateral } JDKL$e) $\text{Quadrilateral } OCIN$

8) Explain how you know.

##### Problem 6

9) 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

10) 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

11) Identify any angles of rotation that create symmetry.

##### Problem 9

12) Select all the angles of rotation that produce symmetry for this flower. Write each corresponding letter in the answer box and separate letters with commas.

a) 45 ﻿$\quad\quad$﻿ b) 60 ﻿$\quad\quad$﻿ c) 90 ﻿$\quad\quad$﻿ d) 120 ﻿$\quad\quad$﻿ e) 135 ﻿$\quad\quad$﻿ f) 150 ﻿$\quad\quad$﻿ g) 180

##### Problem 10

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