# Lesson 19

Evidence, Angles, and Proof

• Let’s make convincing explanations.

### Problem 1

What is the measure of angle $$ABE$$?

### Problem 2

Select all true statements about the figure.

A:

$$c + b = d + c$$

B:

$$d + b = 180$$

C:

Rotate clockwise by angle $$ABC$$ using center $$B$$. Then angle $$CBD$$ is the image of angle $$ABE$$.

D:

Rotate 180 degrees using center $$B$$. Then angle $$CBD$$ is the image of angle $$EBA$$.

E:

Reflect across the angle bisector of angle $$ABC$$. Then angle $$CBD$$ is the image of angle $$ABE$$.

F:

Reflect across line $$CE$$. Then angle $$CBD$$ is the image of angle $$EBA$$

### Problem 3

Point $$D$$ is rotated 180 degrees using $$B$$ as the center. Explain why the image of $$D$$ must lie on the ray $$BA$$.

### Problem 4

Draw the result of this sequence of transformations.

1. Rotate $$ABCD$$ clockwise by angle $$ADC$$ using point $$D$$ as the center.
2. Translate the image by the directed line segment $$DE$$
(From Unit 1, Lesson 18.)

### Problem 5

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 $$D’$$.

(From Unit 1, Lesson 17.)

### Problem 6

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 $$C’$$.

(From Unit 1, Lesson 17.)

### Problem 7

In quadrilateral $$BADC$$, $$AB=AD$$ and $$BC=DC$$. The line $$AC$$ is a line of symmetry for this quadrilateral.

1. Based on the line of symmetry, explain why the diagonals $$AC$$ and $$BD$$ are perpendicular.
2. Based on the line of symmetry, explain why angles $$ACB$$ and $$ACD$$ have the same measure.
(From Unit 1, Lesson 15.)

### Problem 8

Here are 2 polygons:

Select all sequences of translations, rotations, and reflections below that would take polygon $$P$$ to polygon $$Q$$.

A:

Reflect over line $$BA$$ and then translate by directed line segment $$CB$$.

B:

Translate by directed line segment $$BA$$ then reflect over line $$BA$$.

C:

Rotate $$60^\circ$$ clockwise around point $$B$$ and then translate by directed line segment $$CB$$.

D:

Translate so that $$E$$ is taken to $$H$$. Then rotate $$120^\circ$$ clockwise around point $$H$$.

E:

Translate so that $$A$$ is taken to $$J$$. Then reflect over line $$BA$$.

(From Unit 1, Lesson 13.)