Question

given that △pqr≅△xyz by asa, which of the following pairs of angles are congruent? a. ∠p and ∠y b. ∠q and ∠x c. ∠p and ∠z d. ∠r and ∠z what is the benefit of using the asa postulate in geometry? a. it verifies similarity b. it proves all three sides are equal c. it is easier to use than any other congruence rule d. it allows for a simple proof of congruence with only two angles and one side given that △xyz≅△abc by asa, which pair of sides must be congruent? a. xy and ab b. xz and bc c. xz and ac d. yz and ab

Explanation:

Step1: Recall ASA congruence

ASA (Angle - Side - Angle) means two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. Corresponding angles of congruent triangles are congruent.

Step2: Solve first question

In \(\triangle PQR\cong\triangle XYZ\) by ASA, the corresponding angles are congruent. \(\angle P\) corresponds to \(\angle X\), \(\angle Q\) corresponds to \(\angle Y\), and \(\angle R\) corresponds to \(\angle Z\). So \(\angle R\) and \(\angle Z\) are congruent.

Step3: Solve second question

The ASA postulate allows for a simple proof of congruence with only two angles and one side. It doesn't verify similarity (AA is for similarity), doesn't prove all three - sides are equal (SSS does that), and there's no basis to say it's easier than other rules.

Step4: Solve third question

In \(\triangle XYZ\cong\triangle ABC\) by ASA, the included sides between the congruent angles are congruent. If the congruent angles are arranged such that the included side for \(\triangle XYZ\) is \(XY\) and for \(\triangle ABC\) is \(AB\), then \(XY\) and \(AB\) are the congruent sides.

Answer:

d. \(\angle R\) and \(\angle Z\)
d. It allows for a simple proof of congruence with only two angles and one side
a. \(XY\) and \(AB\)