Congruent Triangles: SSS, SAS, ASA, AAS and HL: Geometry Math Test Answer Key – PDF Download

Question 1: Corresponding parts (2 points)

1. J matches P and L matches R, so \(PR=JL=10\text{ in}\).
2. K matches Q, so \(m\angle Q=m\angle K=90^{\circ}\).

Scoring: 1 point for each correct measure with the correct correspondence.

Question 2: Rotation on the plane (2 points)

1. \(A^{\prime}(-1,1)\), \(B^{\prime}(-1,4)\), \(C^{\prime}(-3,1)\).
2. \(BC=\sqrt{(4-1)^{2}+(1-3)^{2}}=\sqrt{13}\) and \(B^{\prime}C^{\prime}=\sqrt{(-1+3)^{2}+(4-1)^{2}}=\sqrt{13}\). A rotation is a rigid motion, so it preserves lengths and angles and the triangles are congruent.

Scoring: 1 point for all three image points. 1 point for both distances and the rigid-motion reason.

Question 3: Choose the criterion (2 points)

1. The angle at B is between sides \(AB\) and \(BC\): SAS.
2. \(AC\) is opposite \(\angle B\), so the angle is not between the sides. This is SSA: not enough information.

Scoring: 1 point per correct criterion; part (b) needs the conclusion that SSA does not work.

Question 4: Solve with congruent parts (2 points)

1. \(4x-5=2x+9\) gives \(2x=14\), so \(x=7\). \(AB=4(7)-5=23\) cm.
2. \(6y+2=8y-18\) gives \(20=2y\), so \(y=10\). \(m\angle C=6(10)+2=62^{\circ}\).

Scoring: 1 point for each part, awarded for the correct equation and the correct final value.

Question 5: Right triangles and HL (2 points)

1. \(AD^{2}=17^{2}-15^{2}=289-225=64\), so \(AD=8\text{ ft}\).
2. Hypotenuses \(AB\cong CB\) and the shared leg \(BD\) in right triangles: HL. By CPCTC, \(DC=AD=8\text{ ft}\).

Scoring: 1 point for the Pythagorean calculation and \(AD\). 1 point for HL and \(DC=8\).

Question 6: Two-column proof (3 points)

Scoring: 1 point for the two pairs of congruent sides with reasons. 1 point for the vertical angles with reason. 1 point for concluding the congruence with the criterion SAS.

Question 7: Two sides and an angle (2 points)

1. \(\overline{AB}\) (shared), \(\overline{BC}\cong\overline{BC^{\prime}}\) (7 in each), and \(\angle A\) (shared, \(30^{\circ}\)).
2. No. The angle is not between the two sides (SSA), and the third sides 13.6 in and 3.8 in differ. SSA does not prove congruence.

Scoring: 1 point for the three parts. 1 point for the answer no with the SSA reason.

Question 8: Bisector and CPCTC (2 points)

1. \(\angle ABD\cong\angle CBD\) (bisector), \(\overline{BD}\cong\overline{BD}\) (shared side, between the angles), and \(\angle ADB\cong\angle CDB\) (right angles): ASA.
2. By CPCTC, \(AB=CB\): \(2x+1=3x-5\), so \(x=6\) and \(AB=2(6)+1=13\text{ in}\).

Scoring: 1 point for ASA with the correct three parts. 1 point for using CPCTC to set the sides equal and finding x and AB.

Question 9: Congruence on the plane (3 points)

1. \(AB=\sqrt{3^{2}+4^{2}}=5\), \(AC=8\), \(BC=\sqrt{5^{2}+4^{2}}=\sqrt{41}\). \(DE=\sqrt{(-3-1)^{2}+3^{2}}=5\), \(DF=8\), \(EF=\sqrt{(1+3)^{2}+(8-3)^{2}}=\sqrt{41}\).
2. \(AB=DE\), \(AC=DF\), and \(BC=EF\): all three pairs of sides are congruent, so \(\triangle ABC\cong\triangle DEF\) by SSS.

Scoring: 1 point for the three lengths of \(\triangle ABC\), 1 point for the three lengths of \(\triangle DEF\), 1 point for SSS with all three pairs matched.

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