Lesson 3

Congruent Triangles, Part 1

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Problem 1

Triangle \(ABC\) is congruent to triangle \(EDF\). So, Kiran 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 \(F\).

B:

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

C:

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

D:

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

E:

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

Solution

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Problem 2

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 ray \(CA\) lines up with ray \(CE\).
  2. Explain why the image of \(A\) coincides with \(E\).
  3. Is triangle \(CBA\) congruent to triangle \(CDE\)? Explain your reasoning.

Solution

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Problem 3

The triangles are congruent. Which sequence of rigid motions will take triangle \(XYZ\) onto triangle \(BCA\)?

Triangle ABC and ZXY.
A:

Translate \(XYZ\) using directed line segment \(YC\). Rotate \(X’Y’Z’\) using \(C\) as the center so that \(X’\) coincides with \(B\). Reflect \(X’’Y’’Z’’\) across line \(CB\). 

B:

Translate \(XYZ\) using directed line segment \(YC\). Rotate \(X’Y’Z’\) using \(C\) as the center so that \(X’\) coincides with \(B\). Reflect \(X’’Y’’Z’’\) across line \(AC\).

C:

Translate \(XYZ\) using directed line segment \(YC\). Rotate \(X’Y’Z’\) using \(C\) as the center so that \(X’\) coincides with \(A\). Reflect \(X’’Y’’Z’’\) across line \(CB\).

D:

Translate \(XYZ\) using directed line segment \(YC\). Rotate \(X’Y’Z’\) using \(C\) as the center so that \(X’\) coincides with \(A\). Reflect \(X’’Y’’Z’’\) across line \(AC\).

Solution

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Problem 4

Triangle \(HEF\) is the image of triangle \(FGH\) after a 180 degree rotation around 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 \(HGF\) is congruent to triangle \(FEH\).

B:

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

C:

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

D:

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

E:

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

F:

Segment \(HG\) is congruent to segment \(FE\).

G:

Segment \(FH\) is congruent to segment \(HF\).

Solution

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(From Unit 2, Lesson 2.)

Problem 5

Line \(SD\) is a line of symmetry for figure \(ASMHZDPX\). Tyler says that \(ASDPX\) is congruent to \(SMDZH\) because sides \(AS\) and \(MS\) are corresponding.

Line SD is a line of symmetry for figure ASMHZDPX.
  1. Why is Tyler's congruence statement incorrect?
  2. Write a correct congruence statement for the pentagons.

Solution

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(From Unit 2, Lesson 2.)

Problem 6

Triangle \(ABC\) is congruent to triangle \(DEF\).  Select all the statements that are a result of corresponding parts of congruent triangles being congruent.

Two congruent triangles labeled A B C and D E F.
A:

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

B:

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

C:

Angle \(BAC\) is congruent to angle \(EDF\).

D:

Angle \(BCA\) is congruent to angle \(EDF\).

E:

Angle \(CBA\) is congruent to angle \(FED\).

Solution

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(From Unit 2, Lesson 1.)

Problem 7

When triangle \(ABC\) is reflected across line \(AB\), the image is triangle \(ABD\). Why is angle \(ACD\) congruent to angle \(ADB\)?

Triangle ACD with altitude AB.
A:

Corresponding parts of congruent figures are congruent.

B:

Congruent parts of congruent figures are corresponding.

C:

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

D:

An isosceles triangle has a pair of congruent angles.

Solution

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(From Unit 2, Lesson 1.)

Problem 8

Line \(DE\) is parallel to line \(BC\).

  1. What is the measure of angle \(EAC\)?
  2. What is the measure of angle \(DAB\)?
Line segment DE, with Point A towards the middle, is parallel to line segment BC. Line segment AB and AC create 3 triangles. Angle BAC is labeled Angle A. Angle ABC is labeled Angle b. Angle ACB is labeled Angle c.

Solution

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(From Unit 1, Lesson 21.)