Question

1. decide whether enough information is given to prove that $\triangle lmp \cong \triangle npm$ using the sss congruence theorem (thm. 5.8). if so, write a proof. if not, explain why.
2. decide whether enough information is given to prove that $\triangle wxz \cong \triangle yzx$ using the hl congruence theorem (thm. 5.9). if so, write a proof. if not, explain why.

Explanation:

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Question 15 (SSS Proof)

Step1: List given congruent sides

$\overline{LM} \cong \overline{NP}$ (marked), $\overline{LP} \cong \overline{NM}$ (marked), $\overline{MP} \cong \overline{PM}$ (reflexive property)

Step2: Apply SSS Congruence

If 3 sides of $\triangle LMP$ are congruent to 3 sides of $\triangle NPM$, then $\triangle LMP \cong \triangle NPM$ by SSS Congruence Theorem.

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Question 16 (HL Proof)

Step1: Confirm right triangles

$\angle W$ and $\angle Y$ are right angles, so $\triangle WXZ$ and $\triangle YZX$ are right triangles.

Step2: List HL congruent parts

$\overline{WX} \cong \overline{YZ}$ (marked), $\overline{XZ} \cong \overline{ZX}$ (reflexive hypotenuse)

Step3: Apply HL Congruence

For right triangles, if hypotenuse and one leg are congruent, $\triangle WXZ \cong \triangle YZX$ by HL Congruence Theorem.

Answer:

1. For question 15: Yes, $\triangle LMP \cong \triangle NPM$ by SSS.
2. For question 16: Yes, $\triangle WXZ \cong \triangle YZX$ by HL.