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Wyatt and Parker are comparing two triangles to determine if they are congruent. For triangle ABC, <B measures 90 degrees and for triangle PQR, <Q is a right angle. They also know that AB¯¯¯¯¯¯¯¯ ≅QR¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ ≅ PR¯¯¯¯¯¯¯¯

. Wyatt says the triangles are congruent by the HL Congruence Theorem and Parker thinks that they are congruent by SAS. Who is correct? (Hint: Draw a picture)(1 point)
Responses

Both are wrong, the triangles are congruent by SSA
Both are wrong, the triangles are congruent by SSA

Parker is correct
Parker is correct

Wyatt is correct
Wyatt is correct

Both are wrong, the triangles are not congruent
Both are wrong, the triangles are not congruent

1 answer

To determine the congruence type for triangles ABC and PQR given the provided information, let’s analyze the situation:

  1. Both triangles have a right angle:

    • Triangle ABC has ∠B = 90°.
    • Triangle PQR has ∠Q = 90°.

  2. The sides are given as:

    • AB ≅ QR
    • AC ≅ PR

Now let’s look at the two arguments made by Wyatt and Parker:

  • Wyatt's argument (HL Congruence Theorem): This theorem states that two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other triangle. However, we need to identify the hypotenuses first. The hypotenuse in each triangle is the side opposite to the right angle:
    • In triangle ABC, AC would be the hypotenuse.
    • In triangle PQR, PR would be the hypotenuse.

Since the information given indicates that AB ≅ QR and AC ≅ PR, it appears that if AC and PR are indeed the hypotenuses, then both triangles can be considered under HL, but we need to have information stating that AC and PR are indeed hypotenuses, which isn’t clear from the given data without assumptions.

  • Parker's argument (SAS): The Side-Angle-Side theorem states that two triangles are congruent if two sides and the angle between them in one triangle are congruent to the corresponding two sides and the angle in the other triangle. For Parker's argument to hold, he would need to show that two sides and the included angle are congruent.

Since both triangles have a right angle, we have an included right angle, and the two triangles have one side that corresponds to the angle between them (AB ≅ QR and AC ≅ PR are sides adjacent to the right angles). Thus, if the configuration supports this arrangement, Parker can be correct under SAS.

Based on the analysis:

  • Since both are right triangles with a leg and the included right angle, and the sides adjacent to the right angles are congruent, Parker is correct in stating that the triangles can be considered congruent by the SAS theorem.

Thus the correct response is: Parker is correct.