Geometry Congruent Triangles and Rigid Motions Test Solutions and Scoring (PDF)

Geometry math test solutions: Congruent Triangles and Rigid Motions – Maths PDF

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This congruence test answers for Geometry covers congruent triangles and rigid motions (Common Core G-CO.B.7, G-CO.B.8), with a free PDF to print.

Use this answer key and scoring guide for the congruent triangles test. Each solution shows the full work, and the scoring note tells how the points are split between method and final answer. Partial credit is listed so that grading stays consistent from one paper to the next.

Geometry · Common Core State Standards: G-CO.B.7, G-CO.B.8

Congruence test answers: worked solutions

Question 1 – Corresponding parts [2 pts]

\(3x + 4 = 5x – 20\)

\(24 = 2x\)

\(x = 12\)

\(m\angle D = 5(12) – 20 = 40^{\circ}\)

x = 12, m∠D = 40°

Scoring: 1 point for the equation, 1 point for x and the angle.

Question 2 – Which test? [3 pts]

SSS
SAS (the angle A is between AB and AC)
Not enough: this is SSA, which does not prove congruence

(a) SSS; (b) SAS; (c) cannot be proved

Scoring: 1 point per part.

Question 3 – Congruence by coordinates [2 pts]

E(12, 1) and F(8, 4).

The rule is \((x, y) \text{ maps to } (x + 6, y + 1)\), so E(12, 1) and F(8, 4).

A translation is a rigid motion, so it preserves distances and angles, and DEF is congruent to ABC.

Scoring: 1 point for E and F, 1 point for the explanation.

Question 4 – Proof with a shared side [3 pts]

The diagonal makes two congruent triangles.

AB = AD and CB = CD (given).
AC = AC (reflexive property, shared side).
Triangle ABC is congruent to triangle ADC (SSS).
\(\angle B = \angle D\) (CPCTC).

Scoring: 1 point for the shared side, 1 point for naming SSS, 1 point for CPCTC.

Question 5 – Hypotenuse-leg [2 pts]

Hypotenuse and a leg match, so they are congruent by HL.

Other leg: \(\sqrt{17^{2} – 8^{2}} = \sqrt{225} = 15\). Area: \(\frac{1}{2} \cdot 8 \cdot 15 = 60\).

HL; area 60 each

Scoring: 1 point for HL, 1 point for the area.

Question 6 – SSA counterexample [2 pts]

The side of length 5 can meet the base at two places, forming an acute-angled and an obtuse-angled triangle with the same given parts.

Because two different triangles fit the information, SSA is not a congruence test.

Scoring: 1 point for naming SSA, 1 point for the two-triangle explanation.

Question 7 – Third angle [2 pts]

\(m\angle C = 180^{\circ} – 40^{\circ} – 75^{\circ} = 65^{\circ}\), so \(\angle C\) and \(\angle F\) are both 65°.

AB is opposite C and DE is opposite F, so the congruent side is not between the angles: AAS.

Yes, by AAS

Scoring: 1 point for the third angle, 1 point for AAS.

Question 8 – Surveying [2 pts]

Both triangles have angles 52° and 61° with the 24 yard side between them.

By ASA the triangles are congruent, so by CPCTC the matching distances to the rock are equal.

Yes, by ASA

Scoring: 1 point for ASA, 1 point for the conclusion.

Question 9 – Perimeter [2 pts]

\(2n + 1 = 3n – 4\)

\(n = 5\)

PQ = 11, so the sides are 11, 14 and 9.

\(11 + 14 + 9 = 34\)

n = 5, perimeter 34

Scoring: 1 point for n, 1 point for the perimeter.

Total score

Total

20 points

Key facts behind this congruence test answers

Key facts

Congruent triangles have three pairs of congruent sides and three pairs of congruent angles.
Valid tests: SSS, SAS, ASA, AAS and, for right triangles, HL.
SSA and AAA do not prove congruence.
CPCTC: after proving triangles congruent, corresponding parts are congruent.
A rigid motion maps a figure onto a congruent figure.

How to use this congruence test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

Download the congruence test answers as a PDF with the button below and keep it with your notes.

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