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)

(4 points)

Answer 1

To determine the congruency of the two triangles based on the information provided, we will follow the steps outlined in your question.

Part 1: Using the Triangle Sum Theorem and AAS Congruency Theorem

Triangle Sum Theorem: The sum of the angles in a triangle is always 180 degrees.

1. For triangle XYZ:

   • We know that ∠X = 87 degrees and ∠Z = 57 degrees.
   • To find ∠Y, we can apply the Triangle Sum Theorem: \[ ∠Y = 180 - (∠X + ∠Z) = 180 - (87 + 57) = 180 - 144 = 36 \text{ degrees} \]

2. For triangle CDB:

   • We know that ∠D = 36 degrees and ∠B = 57 degrees.
   • Similarly, we can find ∠C using the Triangle Sum Theorem: \[ ∠C = 180 - (∠D + ∠B) = 180 - (36 + 57) = 180 - 93 = 87 \text{ degrees} \]

Now we have the angles for both triangles:

• In triangle XYZ: ∠X = 87 degrees, ∠Y = 36 degrees, ∠Z = 57 degrees.
• In triangle CDB: ∠C = 87 degrees, ∠D = 36 degrees, ∠B = 57 degrees.

AAS (Angle-Angle-Side) Congruency Theorem: This theorem states that if in two triangles, two angles and the non-included side of one triangle are equal to two angles and the non-included side of the other triangle, then the triangles are congruent.

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

Since we have two angles and the side included between them equal in both triangles, we can conclude that triangle XYZ is congruent to triangle CDB by the AAS Congruency Theorem.

Part 2: Congruency Statement

From the above analysis, we can write the congruency statement:

XYZ = CDB

This indicates that triangle XYZ is congruent to triangle CDB, or simply that their respective corresponding sides and angles are equal.

Answer 2

can you write the same thing again but use symbols insted of letters

Answer 3

Sure! Here’s the response using symbols instead of letters.

Part 1: Using the Triangle Sum Theorem and AAS Congruency Theorem

Triangle Sum Theorem: The sum of the angles in a triangle is always 180 degrees.

1. For triangle \( \triangle XYZ \):

   • Given: \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
   • To find \( \angle Y \): \[ \angle Y = 180^\circ - (\angle X + \angle Z) = 180^\circ - (87^\circ + 57^\circ) = 180^\circ - 144^\circ = 36^\circ \]

2. For triangle \( \triangle CDB \):

   • Given: \( \angle D = 36^\circ \) and \( \angle B = 57^\circ \).
   • To find \( \angle C \): \[ \angle C = 180^\circ - (\angle D + \angle B) = 180^\circ - (36^\circ + 57^\circ) = 180^\circ - 93^\circ = 87^\circ \]

Now we have the angles for both triangles:

• In triangle \( \triangle XYZ \): \( \angle X = 87^\circ \), \( \angle Y = 36^\circ \), \( \angle Z = 57^\circ \).
• In triangle \( \triangle CDB \): \( \angle C = 87^\circ \), \( \angle D = 36^\circ \), \( \angle B = 57^\circ \).

AAS (Angle-Angle-Side) Congruency Theorem: This theorem states that if in two triangles, two angles and the non-included side of one triangle are equal to two angles and the non-included side of the other triangle, then the triangles are congruent.

• We have:
  • \( \angle X = \angle C \) (i.e., \( 87^\circ = 87^\circ \))
  • \( \angle Z = \angle B \) (i.e., \( 57^\circ = 57^\circ \))
  • \( XY = CD \) (i.e., \( 8 = 8 \))

Since we have two angles and the side included between them equal in both triangles, we can conclude that \( \triangle XYZ \cong \triangle CDB \) by the AAS Congruency Theorem.

Part 2: Congruency Statement

The congruency statement is:

\( \triangle XYZ \cong \triangle CDB \)

This indicates that triangle \( XYZ \) is congruent to triangle \( CDB \), meaning their corresponding sides and angles are equal.