Let's break down the problem into the two parts requested:
Part 1: Explanation of Congruence
To prove that the two triangles are congruent using the Triangle Sum Theorem and the AAS (Angle-Angle-Side) Congruency Theorem, follow these steps:
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Triangle Sum Theorem: This theorem states that the sum of the angles in a triangle is always 180 degrees. We can use the known angles to find any missing angles in the triangles.
- For one triangle, we know two angles, \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- We can find the third angle \( \angle Y \) using the Triangle Sum Theorem: \[ \angle Y = 180^\circ - \angle X - \angle Z = 180^\circ - 87^\circ - 57^\circ = 36^\circ \]
- The third angle \( \angle Y \) is found to be \( 36^\circ \).
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Using AAS Congruency Theorem: We now have enough information to apply the AAS Congruency Theorem. This theorem states that if two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then the two triangles are congruent.
- In Triangle XYZ: \( \angle X = 87^\circ \), \( \angle Z = 57^\circ \), and side \( YZ = 8 \) (since we established CD = 8).
- In Triangle CDZ: \( \angle D = 36^\circ \) (which we found), \( \angle B = 57^\circ\), and side \( CD = 8 \).
Since we have established:
- \( \angle X = \angle D = 87^\circ \)
- \( \angle Z = \angle B = 57^\circ \)
- Side \( YZ = CD = 8 \)
With two angles and the side included being equal, we conclude that the triangles are congruent by AAS.
Part 2: Congruency Statement
From the triangles we analyzed, we can write the congruency statement as follows:
\[ \triangle XYZ = \triangle DBC \]
Both parts of the answer highlight the importance of angle and side relationships in proving triangle congruency using geometric theorems.