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Euclidean Geometry: Triangles and Circles · Ages 15–16

Triangle Congruence and Proofs Practice

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For 10th-grade geometry students. Identify the congruence theorem for each problem, or write the missing statement and reason.

  1. Triangle ABC and Triangle DEF have AB = DE, BC = EF, and AC = DF. State the theorem that proves them congruent.

  2. In Triangle JKL and Triangle MNO, JK = MN, angle K = angle N, and KL = NO. State the theorem that proves them congruent.

  3. Triangle PQR and Triangle STU share angle P = angle S, angle Q = angle T, and segment PQ = segment ST. State the congruence theorem.

  4. Two right triangles share congruent hypotenuses and one pair of congruent legs. Which specific congruence criterion proves them congruent?

  5. Segments WY and XZ bisect each other at point M. To prove Triangle WMX is congruent to Triangle YMZ, you identify WM = YM and XM = ZM. What is the third pair of parts and the reason?

  6. You have proved Triangle ABC is congruent to Triangle XYZ. What is the formal reason to conclude that angle C is congruent to angle Z?

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Triangle Congruence and Proofs Practice

Answer key

  1. Triangle ABC and Triangle DEF have AB = DE, BC = EF, and AC = DF. State the theorem that proves them congruent.

    Side-Side-Side (SSS) Congruence Theorem.

  2. In Triangle JKL and Triangle MNO, JK = MN, angle K = angle N, and KL = NO. State the theorem that proves them congruent.

    Side-Angle-Side (SAS) Congruence Theorem.

  3. Triangle PQR and Triangle STU share angle P = angle S, angle Q = angle T, and segment PQ = segment ST. State the congruence theorem.

    Angle-Side-Angle (ASA) Congruence Theorem.

  4. Two right triangles share congruent hypotenuses and one pair of congruent legs. Which specific congruence criterion proves them congruent?

    Hypotenuse-Leg (HL) Theorem, which is an instance of the Side-Side-Angle opposite the longer side criterion.

  5. Segments WY and XZ bisect each other at point M. To prove Triangle WMX is congruent to Triangle YMZ, you identify WM = YM and XM = ZM. What is the third pair of parts and the reason?

    Angle WMX = Angle YMZ because vertical angles are congruent. The triangles are then congruent by SAS.

  6. You have proved Triangle ABC is congruent to Triangle XYZ. What is the formal reason to conclude that angle C is congruent to angle Z?

    CPCTC (Corresponding Parts of Congruent Triangles are Congruent).