Lesson 20

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.

Diagram of 4 lines.
  1. Label the diagram based on Priya and Han's conversation.
  2. Is there something special about 40 degrees? Will any 2 lines cut by a transversal with congruent alternate interior angles, be parallel?

Solution

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Problem 2

Prove lines \(AI\) and \(GJ\) are parallel.

Diagram of 4 lines.

 

Solution

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Problem 3

What is the measure of angle \(ABE\)?

Lines AD and EC intersect at point B. Angle CBD is 50 degrees.

Solution

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(From Unit 1, Lesson 19.)

Problem 4

Lines \(AB\) and \(BC\) are perpendicular. The dashed rays bisect angles \(ABD\) and \(CBD\). Explain why the measure of angle \(EBF\) is 45 degrees. 

Lines AB and BC are perpendicular and meet at point B. Dashed rays BF and BE bisect angles ABD and CBD.

 

Solution

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(From Unit 1, Lesson 19.)

Problem 5

Identify a figure that is not the image of quadrilateral \(ABCD\) after a sequence of transformations. Explain how you know.

6 quadrilaterals.

Solution

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(From Unit 1, Lesson 18.)

Problem 6

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 \(D’\).

Two congruent quadrilaterals labeled ABCD and A’B’C’D’.

Solution

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(From Unit 1, Lesson 17.)

Problem 7

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 \(C’\).

Two congruent triangles labeled A prime B prime C prime and ABC.

Solution

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(From Unit 1, Lesson 17.)

Problem 8

Identify any angles of rotation that create symmetry.

Flower on polar grid. Grid is made up of 3 circles with the same center. Flower is made up of 3 congruent petals, equally spaced. 2 petals on top half of grid, 1 petals on bottom. Petal length reaches the third outermost circle on the grid.

Solution

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(From Unit 1, Lesson 16.)

Problem 9

Select all the angles of rotation that produce symmetry for this flower.

Flower on polar grid.
A:

45

B:

60

C:

90

D:

120

E:

135

F:

150

G:

180

Solution

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(From Unit 1, Lesson 16.)

Problem 10

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

The letter N, formed by 3 line segments. 4 points on the endpoints of the segments. Starting at the bottom left, A. Moving upward, B. Slanting down and to the right, C. Moving upward, D.

Solution

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(From Unit 1, Lesson 14.)