Question

unit 04: triangle congruence benchmark test

given \\(\delta pqr \cong \delta lmn\\) with \\(\overline{pq} \cong \overline{lm}\\), which additional congruence criteria will always prove \\(\delta pqr \cong \delta lmn\\)?

select one:
a. \\(\overline{pr} \cong \overline{ln}\\) and \\(\angle q \cong \angle m\\)
b. \\(\overline{pr} \cong \overline{ln}\\) and \\(\angle p \cong \angle l\\)
c. \\(\overline{pr} \cong \overline{ln}\\) and \\(\angle r \cong \angle m\\)
d. \\(\overline{pr} \cong \overline{ln}\\) and \\(\angle r \cong \angle n\\)

Explanation:

⚡ Using what you learned: congruence criteria for triangles (sss, sas, asa, aas, hl)

Step 1: Analyze the given information

We are given:

• One pair of congruent corresponding sides: \(\overline{PQ} \cong \overline{LM}\) (Side)

We need to find which additional pair of conditions will always prove \(\Delta PQR \cong \Delta LMN\).

All options include the side congruence:

$$ \overline{PR} \cong \overline{LN} \quad \text{(Side)} $$

If we use this, we have two pairs of congruent sides:

1. \(\overline{PQ} \cong \overline{LM}\)
2. \(\overline{PR} \cong \overline{LN}\)

Step 2: Apply triangle congruence criteria

To prove congruence with two known sides, we can use the SAS (Side-Angle-Side) congruence criterion.

The angle must be the included angle (the angle formed between the two known sides):

• For \(\Delta PQR\), the angle between \(\overline{PQ}\) and \(\overline{PR}\) is \(\angle P\).
• For \(\Delta LMN\), the angle between \(\overline{LM}\) and \(\overline{LN}\) is \(\angle L\).

Therefore, the required angle congruence is:

$$ \angle P \cong \angle L $$

This matches option b.

Answer:

b. \(\overline{PR} \cong \overline{LN}\) and \(\angle P \cong \angle L\)