Lesson 4

Congruent Triangles, Part 2

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Problem 1

Match each statement using only the information shown in the pairs of congruent triangles.

A:

In the 2 triangles there are 3 pairs of congruent sides.

B:

The 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle.

C:

The 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle.

1:
Two adjacent triangles.
2:
Two adjacent triangles.
3:
Two adjacent congruent angles. In each triangle, one side has one tick mark, the second side has 2 tick marks, and the 3rd side overlaps the other triangle.  

Solution

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Problem 2

Sketch the unique triangles that can be made with angle measures \(40^{\circ}\) and \(100^{\circ}\) and side length 3. How do you know you have sketched all possibilities?

Solution

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Problem 3

What is the least amount of information that you need to construct a triangle congruent to this one?

Triangle JKL.

Solution

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Problem 4

Triangle \(ABC\) is congruent to triangle \(EDF\). So, Mai knows that there is a sequence of rigid motions that takes \(ABC\) to \(EDF\).  

Congruent triangles ABC and EFG.

Select all true statements after the transformations:

A:

Angle \(A\) coincides with angle \(E\).

B:

Angle \(B\) coincides with angle \(F\).

C:

Segment \(AB\) coincides with segment \(EF\).

D:

Segment \(BC\) coincides with segment \(DF\).

E:

Segment \(AC\) coincides with segment \(ED\).

Solution

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(From Unit 2, Lesson 3.)

Problem 5

A rotation by angle \(ACE\) using point \(C\) as the center takes triangle \(CBA\) onto triangle \(CDE\).

Triangle CBA and CDE.
  1. Explain why the image of segment \(CB\) lines up with segment \(CD\).
  2. Explain why the image of \(B\) coincides with \(D\).
  3. Is triangle \(ABC\) congruent to triangle \(EDC\)? Explain your reasoning.

Solution

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(From Unit 2, Lesson 3.)

Problem 6

Line \(EF\) is a line of symmetry for figure \(ABECDF\). Clare says that \(ABEF\) is congruent to \(CDFE\) because sides \(AB\) and \(CD\) are corresponding.

Line EF is a line of symmetry for figure ABECDF.
  1. Why is Clare's congruence statement incorrect?
  2. Write a correct congruence statement for the quadrilaterals.

Solution

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(From Unit 2, Lesson 2.)

Problem 7

Triangle \(HEF\) is the image of triangle \(HGF\) after a reflection across line \(FH\). Select all statements that must be true.

Triangle  H E F is the image of triangle  H G F.
A:

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

B:

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

C:

Angle \(HFE\) is congruent to angle \(FHG\).

D:

Angle \(EFG\) is congruent to angle \(EHG\).

E:

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

F:

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

Solution

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(From Unit 2, Lesson 2.)

Problem 8

When rectangle \(ABCD\) is reflected across line \(EF\), the image is \(BADC\). How do you know that segment \(AD\) is congruent to segment \(BC\)?

Rectangle ABCD with line EF through lines AB and DC at midpoints E and F.
A:

A rectangle has 2 pairs of parallel sides.

B:

Any 2 sides of a rectangle are congruent.

C:

Corresponding parts of congruent figures are congruent.

D:

Congruent parts of congruent figures are corresponding.

Solution

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(From Unit 2, Lesson 1.)

Problem 9

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

Hexagon with line segments.

Solution

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(From Unit 1, Lesson 22.)