# Lesson 19

Evidence, Angles, and Proof

### Problem 1

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

### Problem 2

Select **all** true statements about the figure.

$c + b = d + c$

$d + b = 180$

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

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

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

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.

- Rotate \(ABCD\) clockwise by angle \(ADC\) using point \(D\) as the center.
- Translate the image by the directed line segment \(DE\).

### 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 \(D’\).

### 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 \(C’\).

### Problem 7

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

- Based on the line of symmetry, explain why the diagonals \(AC\) and \(BD\) are perpendicular.
- Based on the line of symmetry, explain why angles \(ACB\) and \(ACD\) have the same measure.

### Problem 8

Here are 2 polygons:

Select **all** sequences of translations, rotations, and reflections below that would take polygon \(P\) to polygon \(Q\).

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

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

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

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

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