Unit 1, Lesson 9

Transformations, Transversals, and Proof

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Integrated Math 2

9.1

Warm-up

Angle Relationships

Lines and are parallel. Find the value of in each figure.

<p>Parallel lines l and m cut by transversal line p, creating two congruent angles above l and m and to the right to the right of the transversal. Upper angle is 40 degrees and lower angle is x degrees.</p> — Figure A

<p>Parallel lines l and m cut by transversal line p, creating two alternate interior angles marked congruent. Upper left angle is 61 degrees and lower right angle is x degrees.</p> — Figure B

<p>Parallel lines l and m cut by transversal line p, creating two same-side inside angles marked congruent. Upper angle is 98 degrees and lower angle is x degrees.</p> — Figure C

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9.2

Activity

Make a Mark? Give a Reason

Here are intersecting lines and :

Translate lines and by the directed line segment from to . Label the images of as .
What is true about lines and ? Explain your reasoning.
Take turns with your partner to identify congruent angles.
1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

9.3

Activity

An Alternate Explanation

Here are intersecting lines and :

Rotate line by 180 degrees around point . Label the images of as .
What is true about lines and ? Explain your reasoning.
Take turns with your partner to identify congruent angles.
1. For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Student Lesson Summary

There are often several different ways to explain why statements are true. Comparing the different ways can lead to new insights or more flexible understanding. Consider the angles formed when 2 parallel lines and are cut by a transversal:

<p>Parallel lines l and m cut by transversal EF at points B and C and having midpoint M. Point A is to the left of the transversal on l and point G is to the right of the transversal on m.</p>

Suppose we want to explain why angle is congruent to angle . Label the midpoint of as . Rotating 180 degrees around takes angle to angle . Why? Well, and are equidistant from , so the rotation takes to . Also, it takes the transversal to itself, so it takes the ray to the ray . Finally, the rotation takes line onto line because 180-degree rotations take lines onto parallel lines, and is the only line parallel to that also goes through .

A different explanation can prove the same fact using a translation and the idea that vertical angles are congruent. Try thinking of that explanation yourself.

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Transformations, Transversals, and Proof

Warm-up

Angle Relationships

Are You Ready for More?

Activity

Make a Mark? Give a Reason

Activity

An Alternate Explanation

Student Lesson Summary

Glossary

Are You Ready for More?

Are You Ready for More?