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If △PQR∼△JKL, which statements are correct?(1 point) Responses ∠P≅∠J and QR¯¯¯¯¯¯¯¯ corresponds to JK¯¯¯¯¯¯¯¯. angle upper P congruent to angle upper J and Modifying above upper Q upper R with bar corresponds to Modifying above upper J upper K with bar . ∠Q≅∠K and PR¯¯¯¯¯¯¯¯ corresponds to JL¯¯¯¯¯¯¯. angle upper Q congruent to angle upper K and Modifying above upper P upper R with bar corresponds to Modifying above upper J upper L with bar . ∠Q≅∠L and PQ¯¯¯¯¯¯¯¯ corresponds to JK¯¯¯¯¯¯¯¯. angle upper Q congruent to angle upper L and Modifying above upper P upper Q with bar corresponds to Modifying above upper J upper K with bar . ∠P≅∠J and PQ¯¯¯¯¯¯¯¯ corresponds to JL¯¯¯¯¯¯¯. angle upper P congruent to angle upper J and Modifying above upper P upper Q with bar corresponds to Modifying above upper J upper L with bar .

1 answer

Given that triangles \( \triangle PQR \) is similar to \( \triangle JKL \) (denoted as \( \triangle PQR \sim \triangle JKL \)), it follows that corresponding angles are congruent and corresponding sides are in proportion.

Based on the properties of triangle similarity, we can derive the following pairs of congruent angles and corresponding sides:

  • \( \angle P \cong \angle J \)
  • \( \angle Q \cong \angle K \)
  • \( \angle R \cong \angle L \)

And the corresponding sides based on their angle pairs:

  • \( \overline{PQ} \) corresponds to \( \overline{JK} \)
  • \( \overline{QR} \) corresponds to \( \overline{KL} \)
  • \( \overline{PR} \) corresponds to \( \overline{JL} \)

Now let's analyze the statements provided:

  1. \( \angle P \cong \angle J \) and \( \overline{QR} \) corresponds to \( \overline{JK} \):

    • The angle statement is correct, but the corresponding sides are incorrect; \( \overline{QR} \) should correspond to \( \overline{KL} \).

  2. \( \angle Q \cong \angle K \) and \( \overline{PR} \) corresponds to \( \overline{JL} \):

    • This statement is correct; both the angle and side correspondences are accurate.

  3. \( \angle Q \cong \angle L \) and \( \overline{PQ} \) corresponds to \( \overline{JK} \):

    • The angle statement is incorrect; \( \angle Q \) corresponds to \( \angle K\), not \( \angle L\). The side correspondence is also misplaced.

  4. \( \angle P \cong \angle J \) and \( \overline{PQ} \) corresponds to \( \overline{JL} \):

    • The angle statement is correct, but the corresponding sides do not match; \( \overline{PQ} \) should correspond to \( \overline{JK} \).

Therefore, the only completely correct statement is:

2. \( \angle Q \cong \angle K \) and \( \overline{PR} \) corresponds to \( \overline{JL} \).