# Lesson 3

Congruent Triangles, Part 1

- Let’s use transformations to be sure that two triangles are congruent.

### Problem 1

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

Select **all** true statements after the transformations:

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

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

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

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

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

### Problem 2

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

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

### Problem 3

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

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 \(CB\).

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 \(AC\).

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 \(CB\).

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 \(AC\).

### 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 \(HGF\) is congruent to triangle \(FEH\).

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

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

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

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

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

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

### 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.

- Why is Tyler's congruence statement incorrect?
- Write a correct congruence statement for the pentagons.

### 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.

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

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

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

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

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

### Problem 7

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

Corresponding parts of congruent figures are congruent.

Congruent parts of congruent figures are corresponding.

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

An isosceles triangle has a pair of congruent angles.

### Problem 8

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

- What is the measure of angle \(EAC\)?
- What is the measure of angle \(DAB\)?