Transformations, Transversals, and Proof
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 and angle are both 40 degrees? If I draw a line perpendicular to line through point , I get this triangle. Angle would be 50 degrees because . And because the angles of a triangle sum to 180 degrees, angle is 90 degrees. It's also a right angle!
Han: Oh! Then line and line 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 and 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?
4) Prove lines and are parallel.
5) What is the measure of angle ?
6) Lines and are perpendicular. The dashed rays bisect angles and . Explain why the measure of angle is 45 degrees.
7) Identify a figure that is not the image of quadrilateral after a sequence of transformations.
8) Explain how you know.
9) Quadrilateral is congruent to quadrilateral . Describe a sequence of rigid motions that takes to , to , to , and to .
10) Triangle is congruent to triangle . Describe a sequence of rigid motions that takes to , to , and to .
11) Identify any angles of rotation that create symmetry.
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 b) 60 c) 90 d) 120 e) 135 f) 150 g) 180
13) Three line segments form the letter N. Rotate the letter N clockwise around the midpoint of segment by 180 degrees. Describe the result.