Lesson 22

Now What Can You Build?

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Problem 1

This design began from the construction of a regular hexagon. Name 2 pairs of congruent figures.

Hexagon with line segments.

Solution

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Problem 2

This design began from the construction of a regular hexagon. Describe a rigid motion that will take the figure to itself.

Regular hexagon with center I and vertices A B C D E F. Segment F C contains points H I G. Segment A E passes through H. Segment B D passes through G. Segment B E passes through I. Triangles I H E and I G B are highlighted.

Solution

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Problem 3

Noah starts with triangle \(ABC\) and makes 2 new triangles by translating \(B\) to \(A\) and by translating \(B\) to \(C\). Noah thinks that triangle \(DCA\)  is congruent to triangle \(BAC\). Do you agree with Noah? Explain your reasoning.

Large triangle BFE has small triangle ADC at its midpoints.

 

Solution

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(From Unit 1, Lesson 21.)

Problem 4

In the image, triangle \(ABC\) is congruent to triangle \(BAD\) and triangle \(CEA\). What are the measures of the 3 angles in triangle \( CEA\)? Show or explain your reasoning.

Trapezoid DECB base BC and point A midpoint of DE making triangle ABC. Angle D is 45 degrees and angle BAC is 75 degrees.

Solution

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(From Unit 1, Lesson 21.)

Problem 5

In the figure shown, angle 3 is congrent to angle 6. Select all statements that must be true.

Parallel lines f and g cut by a transversal, creating angles 1, 2, 3 and 4 at intersection of lines g and transversal, and creating angles 5, 6, 7 and 8 at intersection of lines f and transversal.
A:

Lines \(f\) and \(g\) are parallel. 

B:

Angle 2 is congruent to angle 6

C:

Angle 2 and angle 5 are supplementary

D:

Angle 1 is congruent to angle 7

E:

Angle 4 is congruent to angle 6

Solution

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(From Unit 1, Lesson 20.)

Problem 6

In this diagram, point \(M\) is the midpoint of segment \(AC\) and \(B’\) is the image of \(B\) by a rotation of \(180^\circ\) around \(M\).

  1. Explain why rotating \(180^\circ\) using center \(M\) takes \(A\) to \(C\).
  2. Explain why angles \(BAC\) and \(B’CA\) have the same measure.
Point M is the midpoint of segment AC and B Prime  is the image of B by a rotation of 180 degrees around Point M.

Solution

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(From Unit 1, Lesson 20.)

Problem 7

Lines \(AB\) and \(BC\) are perpendicular. The dashed rays bisect angles \(ABD\) and \(CBD\).

Select all statements that must be true: 

Lines AB and BC are perpendicular and meet at point B. Dashed rays BF and BE bisect angles ABD and CBD.
A:

Angle \(CBF\) is congruent to angle \(DBF\)

B:

Angle \(CBE\) is obtuse

C:

Angle \(ABC\) is congruent to angle \(EBF\)

D:

Angle \(DBC\) is congruent to angle \(EBF\)

E:

Angle \(EBF\) is 45 degrees

Solution

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(From Unit 1, Lesson 19.)

Problem 8

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

Give an example of a rotation using an angle greater than 0 degrees and less than 360 degrees, that takes both lines to themselves. Explain why your rotation works.

Lines AD and EC intersect at point B.

Solution

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(From Unit 1, Lesson 19.)

Problem 9

Draw the image of triangle \(ABC\) after this sequence of rigid transformations.

  1. Reflect across line segment \(AB\).
  2. Translate by directed line segment \(u\).
Triangle ABC with directed line segment u formed by ray BA.

Solution

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(From Unit 1, Lesson 18.)

Problem 10

  1. Draw the image of figure \(CAST\) after a clockwise rotation around point \(T\) using angle \(CAS\) and then a translation by directed line segment \(AS\).
  2. Describe another sequence of transformations that will result in the same image.
A figure CAST as a rectangle with missing side CT on the bottom.

Solution

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(From Unit 1, Lesson 18.)