Pregunta
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III. Direction: Determine the congruent parts of \( \triangle \mathrm{PQR} \cong \) \( \triangle \mathrm{XYZ} \).

Ask by Salinas Ford. in the Philippines
Feb 21,2025

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The congruent parts of \( \triangle PQR \) and \( \triangle XYZ \) are: - \( PQ \cong XY \) - \( QR \cong YZ \) - \( RP \cong ZX \) - \( \angle P \cong \angle X \) - \( \angle Q \cong \angle Y \) - \( \angle R \cong \angle Z \)

Solución

To determine the congruent parts of the triangles \( \triangle PQR \) and \( \triangle XYZ \), we need to identify the corresponding sides and angles that are congruent between the two triangles. 1. **Congruent Triangles**: When two triangles are congruent, it means that all their corresponding sides and angles are equal. 2. **Corresponding Parts**: The notation \( \triangle PQR \cong \triangle XYZ \) indicates that: - Side \( PQ \) is congruent to side \( XY \) - Side \( QR \) is congruent to side \( YZ \) - Side \( RP \) is congruent to side \( ZX \) - Angle \( P \) is congruent to angle \( X \) - Angle \( Q \) is congruent to angle \( Y \) - Angle \( R \) is congruent to angle \( Z \) Thus, the congruent parts of the triangles are: - \( PQ \cong XY \) - \( QR \cong YZ \) - \( RP \cong ZX \) - \( \angle P \cong \angle X \) - \( \angle Q \cong \angle Y \) - \( \angle R \cong \angle Z \) This establishes the complete correspondence between the two triangles.

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Beyond the Answer

In congruent triangles, corresponding parts are equal. For \( \triangle PQR \cong \triangle XYZ \), it means that the angles and sides of both triangles match up perfectly. This implies that \( \angle P \) is congruent to \( \angle X \), \( \angle Q \) to \( \angle Y \), and \( \angle R \) to \( \angle Z \). Additionally, the side lengths must also correspond: \( PQ \cong XY \), \( QR \cong YZ \), and \( RP \cong ZX \). This concept is super useful in real-world applications such as architecture and engineering, where the stability and symmetry of structures are designed by ensuring that parts are congruent. For instance, in designing a bridge, engineers often need to calculate angles and dimensions that ensure each side is equal and contributes to the overall stability of the structure.

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