# 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.

\(c + b = d + c\)

\(d + b = 180\)

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 \(BA\).