Transformations, Transversals, and Proof

ID: pihoh-hizod
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

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 ABCABC and angle BCJBCJ are both 40 degrees? If I draw a line perpendicular to line AIAI through point BB, I get this triangle. Angle CBXCBX would be 50 degrees because 40+50=9040+50=90. And because the angles of a triangle sum to 180 degrees, angle CXBCXB is 90 degrees. It's also a right angle!

Han: Oh! Then line AIAI and line GJGJ 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 AIAI and GJGJ must be parallel.

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1) Label the diagram based on Priya and Han's conversation.

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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 AIAI and GJGJ are parallel.

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Problem 3

5) What is the measure of angle ABEABE?

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Problem 4

6) Lines ABAB and BCBC are perpendicular. The dashed rays bisect angles ABDABD and CBDCBD. Explain why the measure of angle EBFEBF is 45 degrees.

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Problem 5

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

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

8) Explain how you know.

Problem 6

9) Quadrilateral ABCDABCD is congruent to quadrilateral ABCDA'B'C'D'. Describe a sequence of rigid motions that takes AA to AA', BB to BB', CC to CC', and DD to DD'.

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Problem 7

10) Triangle ABCABC is congruent to triangle ABCA'B'C'. Describe a sequence of rigid motions that takes AA to AA', BB to BB', and CC to CC'.

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Problem 8

11) Identify any angles of rotation that create symmetry.

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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

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Problem 10

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

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