Question

got it? 1. a. given: ∠prs and ∠rpq are right angles, (overline{sp}congoverline{qr}) prove: (\triangle prscong\triangle rpq) b. reasoning your friend says, “suppose you have two right triangles with congruent hypotenuses and one pair of congruent legs. it does not matter which leg in the first triangle is congruent to which leg in the second triangle. the triangles will be congruent.” is your friend correct? explain. a) statements reasons

Explanation:

Step1: Identify the right - angle property

Since \(\angle PRS\) and \(\angle RPQ\) are right angles, \(\angle PRS=\angle RPQ = 90^{\circ}\) (Definition of right - angle).

Step2: Consider the hypotenuse - leg (HL) congruence criterion

In right - triangles \(\triangle PRS\) and \(\triangle RPQ\), we know that \(\overline{SP}\cong\overline{QR}\) (given). Also, \(\overline{PQ}\cong\overline{SR}\) (assumed from the figure, as they are marked congruent).
By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (If the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent), \(\triangle PRS\cong\triangle RPQ\).

For part b

Your friend is correct.

Step1: Recall the HL congruence theorem

Let the two right - triangles be \(\triangle ABC\) with right - angle at \(C\) and \(\triangle DEF\) with right - angle at \(F\). Let the hypotenuses \(AB = DE\) and legs \(AC=DF\) (or \(BC = EF\)).
By the Pythagorean theorem, if \(c\) is the hypotenuse and \(a,b\) are the legs of a right - triangle (\(c^{2}=a^{2}+b^{2}\)).
If \(c_1 = c_2\) (hypotenuses are congruent) and \(a_1=a_2\) (one pair of legs are congruent), then \(b_1=\sqrt{c_1^{2}-a_1^{2}}\) and \(b_2=\sqrt{c_2^{2}-a_2^{2}}\). Since \(c_1 = c_2\) and \(a_1=a_2\), \(b_1 = b_2\).
By the Hypotenuse - Leg (HL) congruence criterion (or we can also use the Side - Side - Side (SSS) congruence criterion as we can show all three sides are congruent), the two right - triangles are congruent. It doesn't matter which leg (because of the Pythagorean relationship in right - triangles, if hypotenuse and one leg are known, the other leg is uniquely determined).

Answer:

a) The proof is completed as shown in the step - by - step explanation using the HL congruence criterion.
b) Your friend is correct. Using the Pythagorean theorem and congruence criteria (HL or SSS), two right - triangles with congruent hypotenuses and one pair of congruent legs are congruent regardless of which pair of legs (since the other pair of legs is also forced to be congruent by the Pythagorean theorem).