# Lesson 3

Congruent Triangles, Part 1

### 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 \(ED\).

### Solution

For access, consult one of our IM Certified Partners.

### 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.

### Solution

For access, consult one of our IM Certified Partners.

### 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 \(AC\).

### Solution

For access, consult one of our IM Certified Partners.

### 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 \(HF\).

### Solution

For access, consult one of our IM Certified Partners.

(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.

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

### Solution

For access, consult one of our IM Certified Partners.

(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.

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 \(FED\).

### Solution

For access, consult one of our IM Certified Partners.

(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 \(ADB\)?

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.

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 2, Lesson 1.)### Problem 8

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

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

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 1, Lesson 21.)