Lesson 2

Congruent Parts, Part 2

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Problem 1

Line \(SD\) is a line of symmetry for figure \(AXPDZHMS\). Noah says that \(AXPDS\) is congruent to \(HMZDS\) because sides \(AX\) and \(HM\) are corresponding.

  1. Why is Noah’s congruence statement incorrect?
  2. Write a correct congruence statement for the pentagons.
Figure A X P D Z H M S. 8 sides. Line of symmetry S D is drawn.

Solution

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Problem 2

FIgure \(MBJKGH\) is the image of figure \(AFEKJB\) after being rotated 90 degrees counterclockwise about point \(K\). Draw a segment in figure \(AFEKJB\) to create a quadrilateral. Draw the image of the segment when rotated 90 degrees counterclockwise about point \(K\). 

Write a congruence statement for the quadrilateral you created in figure \(AFEKJB\) and the image of the quadrilateral in figure \(MBJKGH\).

L shaped figure.

 

Solution

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Problem 3

Triangle \(HEF\) is the image of triangle \(FGH\) after a 180 degree rotation about point \(K\). Select all statements that must be true.

Triangle HEF is the image of triangle FGH after a 180 degree rotation about point K.
A:

Triangle \(FGH \) is congruent to triangle \(FEH\).

B:

Triangle \(EFH \) is congruent to triangle \(GFH\).

C:

Angle \(KHE\) is congruent to angle \(KFG\).

D:

Angle \(GHK\) is congruent to angle \(KHE\).

E:

Segment \(EH\) is congruent to segment \(FG\).

F:

Segment \(GH\) is congruent to segment \(EF\).

Solution

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Problem 4

When triangle \(ABC\) is reflected across line \(AB\), the image is triangle \(ABD\). Why are segment \(AD\) and segment \(AC\) congruent?

Triangle ACD with altitude AB.
A:

Congruent parts of congruent figures are corresponding.

B:

Corresponding parts of congruent figures are congruent.

C:

An isosceles triangle has a pair of congruent sides.

D:

Segment \(AB\) is a perpendicular bisector of segment \(DC\).

Solution

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(From Unit 2, Lesson 1.)

Problem 5

Elena needs to prove angles \(BED\) and \(BCA\) are congruent. Provide reasons to support each of her statements.

  1. Line \(m\) is parallel to line \(l\).
  2. Angles \(BED\) and \(BCA\) are congruent.
Triangle and two horizontal line segments.

Solution

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(From Unit 1, Lesson 20.)

Problem 6

Triangle \(FGH\) is the image of isosceles triangle \(FEH\) after a reflection across line \(HF\). Select all the statements that are a result of corresponding parts of congruent triangles being congruent.

Figure FGHE with diagonal HF and sides HE and FE congruent.
A:

\(EFGH\) is a rectangle.

B:

\(EFGH\) is a rhombus.

C:

Diagonal \(FH\) bisects angles \(EFG\) and \(EHG\).

D:

Diagonal \(FH\) is perpendicular to side \(FE\).

E:

Angle \(EHF\) is congruent to angle \(FGH\).

F:

Angle \(FEH\) is congruent to angle \(FGH\).

Solution

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(From Unit 2, Lesson 1.)

Problem 7

This design began from the construction of a regular hexagon.

  1. Draw 1 segment so the diagram has another hexagon that is congruent to hexagon \(ABCIHG\).

  2. Explain why the hexagons are congruent.

2 Hexagons and quadrilaterals.

Solution

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(From Unit 1, Lesson 22.)