Question

1. triangle jkl and triangle pqr are shown above. if ∠j is congruent to ∠p, which of the following must be true in order to prove that triangles jkl and pqr are congruent?

a. ∠l≅∠r and jl = pr
b. kl = qr and pr = jl
c. jk = pq and kl = qr
d. ∠k≅∠q and ∠l≅∠r

Explanation:

Step1: Recall triangle congruence criteria

The main triangle congruence criteria are \(ASA\) (Angle - Side - Angle), \(AAS\) (Angle - Angle - Side), \(SAS\) (Side - Angle - Side), \(SSS\) (Side - Side - Side).

Step2: Analyze option A

If \(\angle J\cong\angle P\), \(\angle L\cong\angle R\) and \(JL = PR\), then by \(ASA\) (the angle - side - angle criterion: we have two angles and the included side).
For \(ASA\), in \(\triangle JKL\) and \(\triangle PQR\), \(\angle J\) and \(\angle P\) are one pair of angles, \(JL\) and \(PR\) are the included sides between \(\angle J,\angle L\) and \(\angle P,\angle R\) respectively.

Step3: Analyze other options

• Option B: \(KL = QR\) and \(PR = JL\) gives two sides. But we only know one angle (\(\angle J\cong\angle P\)) which is not the included angle between the given sides. So, it does not satisfy any congruence criterion.
• Option C: \(JK = PQ\) and \(KL = QR\) gives two sides. But we only know one angle (\(\angle J\cong\angle P\)) which is not the included angle between the given sides. So, it does not satisfy any congruence criterion.
• Option D: \(\angle K\cong\angle Q\) and \(\angle L\cong\angle R\) gives three angles. But having three angles congruent (\(AAA\)) does not prove triangle congruence (it proves similarity).

Answer:

A. \(\angle L\cong\angle R\) and \(JL = PR\)