This geometry congruent triangles test answer key gives the full solution and the scoring guide for each of the 9 test questions on congruent triangles: sss, sas, asa, aas and hl.
Use this key to score the congruent triangles test out of 20 points. Each solution ends with a scoring guide. Give partial credit only as described, and accept any correct proof that names a valid criterion for the same triangles.
Geometry Congruent Triangles Test Answer Key: questions 1–5
Question 1: Corresponding parts (2 points)
- J matches P and L matches R, so \(PR=JL=10\text{ in}\).
- 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)
- \(A^{\prime}(-1,1)\), \(B^{\prime}(-1,4)\), \(C^{\prime}(-3,1)\).
- \(BC=\sqrt{(4-1)^{2}+(1-3)^{2}}=\sqrt{13}\) 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)
- The angle at B is between sides \(AB\) and \(BC\): SAS.
- \(AC\) 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)
- \(4x-5=2x+9\) gives \(2x=14\), so \(x=7\). \(AB=4(7)-5=23\) cm.
- \(6y+2=8y-18\) 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)
- \(AD^{2}=17^{2}-15^{2}=289-225=64\), so \(AD=8\text{ ft}\).
- 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 \(DC=8\).
Geometry Congruent Triangles Test Answer Key: questions 6–9
Question 6: Two-column proof (3 points)
| Statements | Reasons |
|---|---|
| \(O\) is the midpoint of \(\overline{PQ}\) | Given |
| \(\overline{PO}\cong\overline{QO}\) | Definition of midpoint |
| \(O\) is the midpoint of \(\overline{RS}\) | Given |
| \(\overline{RO}\cong\overline{SO}\) | Definition of midpoint |
| \(\angle POR\cong\angle QOS\) | Vertical angles are congruent |
| \(\triangle POR\cong\triangle QOS\) | SAS |
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)
- \(\overline{AB}\) (shared), \(\overline{BC}\cong\overline{BC^{\prime}}\) (7 in each), and \(\angle A\) (shared, \(30^{\circ}\)).
- 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)
- \(\angle ABD\cong\angle CBD\) (bisector), \(\overline{BD}\cong\overline{BD}\) (shared side, between the angles), and \(\angle ADB\cong\angle CDB\) (right angles): ASA.
- 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)
- \(AB=\sqrt{3^{2}+4^{2}}=5\), \(AC=8\), \(BC=\sqrt{5^{2}+4^{2}}=\sqrt{41}\). \(DE=\sqrt{(-3-1)^{2}+3^{2}}=5\), \(DF=8\), \(EF=\sqrt{(1+3)^{2}+(8-3)^{2}}=\sqrt{41}\).
- \(AB=DE\), \(AC=DF\), 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.
Scoring summary
20 points: add the points of every question of this geometry congruent triangles test answer key.
How to use this geometry congruent triangles test answer key
Use this geometry congruent triangles test answer key to score the test fairly and to see which skills need more practice.
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Geometry Congruent Triangles: SSS, SAS, ASA, AAS and HL test answer key
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