For the picture below: XY = 8, CD = 8. <X = 87 degrees, <Z=57 degrees, <D = 36 degrees and <B = 57 degrees.

Question

Part 1. Explain how you can use the Triangle Sum theorem and the AAS congruency theorem to prove the two triangles congruent. (2pts)

Part 2. Write a correct congruency statement. You may use the = instead of the congruency symbol. (2pts)

Answer 1

Part 1: To prove that the two triangles are congruent using the Triangle Sum theorem and the AAS (Angle-Angle-Side) congruency theorem, we can follow these steps:

1. Triangle Sum Theorem: The Triangle Sum theorem states that the sum of the interior angles of a triangle is always 180 degrees. We can apply this theorem to Triangle XYZ and Triangle DBC.

   • In Triangle XYZ:

     • We are given:
       • ∠X = 87 degrees.
       • ∠Z = 57 degrees.
       • To find ∠Y, we use the Triangle Sum theorem:
         • ∠Y = 180 - ∠X - ∠Z = 180 - 87 - 57 = 36 degrees.

   • In Triangle DBC:

     • We are given:
       • ∠D = 36 degrees.
       • ∠B = 57 degrees.
       • To find ∠C, we apply the Triangle Sum theorem:
         • ∠C = 180 - ∠D - ∠B = 180 - 36 - 57 = 87 degrees.

   Now we have the angles:

   • For Triangle XYZ: ∠X = 87 degrees, ∠Y = 36 degrees, ∠Z = 57 degrees.
   • For Triangle DBC: ∠D = 36 degrees, ∠B = 57 degrees, ∠C = 87 degrees.

2. AAS Congruency Theorem: The AAS theorem states that if two angles and a non-included side of one triangle are respectively equal to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

   • We have:
     • ∠X = ∠C (87 degrees)
     • ∠Z = ∠B (57 degrees)
     • Side XY = Side CD (both are equal to 8 units).

   Therefore, by the AAS theorem, we can conclude that Triangle XYZ is congruent to Triangle DBC.

Part 2: The correct congruency statement is: Triangle XYZ = Triangle DBC.