# Transformations, Transversals, and Proof

# Transformations, Transversals, and Proof

##### 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?

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

##### 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.

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.