# Lesson 21

One Hundred and Eighty

- Let’s prove the Triangle Angle Sum Theorem.

### Problem 1

The triangles here are each obtained by applying rigid motions to triangle 1.

- Which triangles are translations of triangle 1? Explain how you know.
- Which triangles are not translations of triangle 1? Explain how you know.

### Problem 2

The quadrilateral is a parallelogram. Find the measure of angles 1, 2, and 3.

### Problem 3

In the figure shown, lines \(f\) and \(g\) are parallel. Select the angle that is congruent to angle 1.

Angle 2

Angle 6

Angle 7

Angle 8

### Problem 4

Angle \(BDE\) is congruent to angle \(BAC\). Name another pair of congruent angles. Explain how you know.

### Problem 5

- Describe a transformation that could be used to show that corresponding angles are congruent.
- Describe a transformation that could be used to show that alternate interior angles are congruent.

### Problem 6

Lines \(AD\) and \(EC\) meet at point \(B\).

Which of these *must* be true? Select **all** that apply.

A 180 degree clockwise rotation using center \(B\) takes \(D\) to \(A\).

The image of \(D\) after a 180 degree rotation using center \(B\) lies on ray \(BA\).

If a 180 degree rotation using center \(B\) takes \(C\) to \(E\) then it also takes \(E\) to \(C\).

Angle \(ABC\) is congruent to angle \(DBE\).

Angle \(ABE\) is congruent to angle \(ABC\).

### Problem 7

Points \(E\), \(B\), and \(C\) are collinear. Explain why points \(A\), \(B\), and \(D\) are collinear.

### Problem 8

- Draw the image of figure \(ACTS\) after a clockwise rotation around point \(C\) using angle \(CTS\) and then a translation by the directed line segment \(CT\).
- Describe another sequence of transformations that will result in the same image.

### Problem 9

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 \(C’\).