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

Question 1: Corresponding parts (2 points)

\(\triangle JKL\cong\triangle PQR\) as shown, with a right angle at K.

1. Find \(PR\). (1 point)
2. Find \(m\angle Q\). (1 point)

Question 2: Rotation on the plane (2 points)

Triangle ABC is rotated \(90^{\circ}\) counterclockwise about the origin, which sends \((x,y)\) to \((-y,x)\).

1. Write the coordinates of \(A^{\prime}\), \(B^{\prime}\), and \(C^{\prime}\). (1 point)
2. Find \(BC\) and \(B^{\prime}C^{\prime}\), and state why \(\triangle ABC\cong\triangle A^{\prime}B^{\prime}C^{\prime}\). (1 point)

Question 3: Choose the criterion (2 points)

In each case, name the criterion that proves \(\triangle ABC\cong\triangle DEF\), or write not enough information.

1. \(AB=DE=7\), \(BC=EF=10\), and \(\angle B\cong\angle E\). (1 point)
2. \(AB=DE\), \(AC=DF\), and \(\angle B\cong\angle E\), where B and E are not between the given sides. (1 point)

Question 4: Solve with congruent parts (2 points)

\(\triangle ABC\cong\triangle DEF\), \(AB=(4x-5)\text{ cm}\), \(DE=(2x+9)\text{ cm}\), \(m\angle C=(6y+2)^{\circ}\), and \(m\angle F=(8y-18)^{\circ}\).

1. Find \(x\) and \(AB\). (1 point)
2. Find \(y\) and \(m\angle C\). (1 point)

Question 5: Right triangles and HL (2 points)

In the figure, \(\overline{BD}\perp\overline{AC}\), \(AB=BC=17\text{ ft}\), and \(BD=15\text{ ft}\).

1. Find \(AD\). (1 point)
2. Name the criterion that proves \(\triangle ABD\cong\triangle CBD\) and state \(DC\). (1 point)

Question 6: Two-column proof (3 points)

Given: O is the midpoint of both \(\overline{PQ}\) and \(\overline{RS}\). Prove \(\triangle POR\cong\triangle QOS\). Write a two-column proof.

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

Triangles ABC and ABC′ share \(\angle A=30^{\circ}\) and \(AB=10\text{ in}\), and \(BC=BC^{\prime}=7\text{ in}\). Measuring gives AC of about 13.6 in and AC′ of about 3.8 in.

1. List the three pairs of congruent parts the triangles share. (1 point)
2. Are the triangles congruent? Explain. (1 point)

Question 8: Bisector and CPCTC (2 points)

In the figure, \(\overline{BD}\) bisects \(\angle ABC\) and \(\overline{BD}\perp\overline{AC}\). Also \(AB=(2x+1)\text{ in}\) and \(CB=(3x-5)\text{ in}\).

1. Name the criterion that proves \(\triangle ABD\cong\triangle CBD\). (1 point)
2. Find \(x\) and \(AB\). (1 point)

Question 9: Congruence on the plane (3 points)

\(\triangle ABC\) has \(A(0,0)\), \(B(3,4)\), and \(C(8,0)\). \(\triangle DEF\) has \(D(1,0)\), \(E(-3,3)\), and \(F(1,8)\).

1. Find the lengths of all three sides of each triangle. (2 points)
2. Prove \(\triangle ABC\cong\triangle DEF\) and name the criterion. (1 point)

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