Draw the triangles that are to be shown congruent...

Draw the triangles that are to be shown congruent separately. Then complete the proof. Given: $\quad \frac{\angle M Q P}{M Q} \cong \frac{\text { and }}{N P} \angle N P Q$ are rt. $\angle \mathrm{s}$ Prove: $\quad \overline{M P} \cong \overline{N Q}$ (HINT: Show $\triangle M Q P \cong \triangle N P Q$.)

Draw the triangles that are to be shown congruent separately. Then complete the proof.
Given: $\quad \frac{\angle M Q P}{M Q} \cong \frac{\text { and }}{N P} \angle N P Q$ are rt. $\angle \mathrm{s}$
Prove: $\quad \overline{M P} \cong \overline{N Q}$
(HINT: Show $\triangle M Q P \cong \triangle N P Q$.)

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Key Concepts

Corresponding Parts of Congruent Triangles (CPCTC)

CPCTC is a logical consequence that follows once two triangles are proven congruent. It holds that all corresponding angles and sides of congruent triangles are equal. This is a crucial step in many geometric proofs because it allows the inference of the equality of other parts of the geometric figures once triangle congruence has been established.

Triangle Congruence

Triangle congruence is the principle that two triangles are identical in shape and size if all corresponding sides and angles are equal. This concept is fundamental in geometry, because once two triangles are shown to be congruent, one can infer that every corresponding element (side or angle) in one triangle is equal to its counterpart in the other.

Hypotenuse-Leg Theorem

The Hypotenuse-Leg (HL) Theorem is a specific criterion used to prove the congruence of right triangles. It states that if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. This theorem simplifies proofs when dealing with right triangles by reducing the need to check all three sides.

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