Lesson 19

Evidence, Angles, and Proof

  • Let’s make convincing explanations.

Problem 1

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

Line segment AD is intersected at Point B by line segment EC. Measurement of Angle CBD is 40 degrees.

Problem 2

Select all true statements about the figure.

Line segment AD intersects line segment EC at point B. Angle EBD is labeled Angle a, Angle DBC is labeled Angle b, Angle ABC is labeled Angle C, and Angle ABE is labeled Angle d.
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 \(BA\).

Line segment AD is intersected at Point B by line segment EC.

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 \(DE\). 
Three quadrilaterals.
(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 \(D’\).

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

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

 

(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.

Quadrilateral B A D C, upside down kite. Starting at the bottom vertex and moving clockwise, A, D, C, B.
  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:

5-sided polygons, P and Q, on an isometric grid. Q is a transformation of P. P has vertices A, B, C, D, and E. Q has vertices J, A, F, G, and H.

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.)