To solve this problem, we will analyze the given information and apply the relevant theorems step by step:
Part 1: Explanation Using Triangle Sum Theorem and AAS Congruency Theorem
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Triangle Sum Theorem: The sum of the angles in a triangle is always 180 degrees. We need to establish that the remaining angles in the identified triangles are equal or can be derived using the Triangle Sum theorem.
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For Triangle XZY:
- We know that \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- We can find \( \angle Y \) using the Triangle Sum Theorem: \[ \angle Y = 180^\circ - (87^\circ + 57^\circ) = 180^\circ - 144^\circ = 36^\circ \]
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For Triangle BCD:
- We know that \( \angle D = 36^\circ \) and \( \angle B = 57^\circ \).
- We can find \( \angle C \) using the Triangle Sum Theorem: \[ \angle C = 180^\circ - (36^\circ + 57^\circ) = 180^\circ - 93^\circ = 87^\circ \]
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AAS Congruency Theorem: This theorem states that if two angles and the non-included side of one triangle are equal to two angles and the non-included side of another triangle, then the two triangles are congruent.
- We have shown that in Triangle XZY, \( \angle X = 87^\circ \), \( \angle Z = 57^\circ \), and \( Y = 36^\circ \) (which corresponds to CD) with side \( XY = 8 \).
- In Triangle BCD, we have \( \angle B = 57^\circ \), \( \angle D = 36^\circ \), and \( C = 87^\circ \) (which corresponds to XY) with side \( CD = 8 \).
Therefore, using the AAS theorem:
- Angle \( \angle X = \angle C \)
- Angle \( \angle Z = \angle B \)
- The side \( XY = CD \)
Thus, by the AAS theorem, Triangle XZY is congruent to Triangle BCD.
Part 2: Congruency Statement
Based on the information and analysis above, a correct congruency statement can be written as:
\( \triangle XZY = \triangle BCD \)
This statement indicates that the triangles are congruent based on their corresponding angles and sides.
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