Question

According to the SAS theorem, which of the following sets of relationships can be used to show that the triangles pictured above are congruent?

1) mo pr, mn pq, no qr
2) mn pq, m p, no qr
3) mn pq, n r, no qr
4) mn pq, n q, no qr

Answer 1

Option 2) mn pq, m p, no qr correctly applies the SAS theorem for congruent triangles by comparing two sides and the included angle of respective triangles.

Step-by-step explanation:

The Side-Angle-Side (SAS) theorem states that two triangles are congruent if they have two sides of the same length and the angle between those two sides with the same measure. When checking for congruence using the SAS theorem, you need to compare two sides and the included angle between those sides for both triangles.

Based on your options to demonstrate that the triangles are congruent using SAS, option 2) mn pq, m p, no qr satisfies the SAS theorem requirements. This means that triangles have two sides mn and no equal in length to pq and qr respectively, and the included angle m is equal to angle p. Hence, the triangles are congruent by SAS, provided that angle m and angle p are the angles included between sides mn and no, and sides pq and qr, respectively.

Answer 2

Set 2) mn ≅ pq, m ≅ p, and no ≅ qr is the correct set of relationships to prove that two triangles are congruent according to the SAS theorem, as it provides two sides and the included angle between them for each triangle.

Step-by-step explanation:

The SAS (Side-Angle-Side) theorem states that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle. Given the relationships amongst line segments and angles mentioned: 1) mo ≅ pr, mn ≅ pq, and no ≅ qr; 2) mn ≅ pq, m ≅ p, and no ≅ qr; 3) mn ≅ pq, n ≅ r, and no ≅ qr; 4) mn ≅ pq, n ≅ q, and no ≅ qr; the correct set of relationships to prove two triangles are congruent using the SAS theorem is the set that provides two sides and the angle between them.

After examining the relationships, the only set that fits the criteria of the SAS theorem is set 2) mn ≅ pq, m ≅ p, and no ≅ qr. This set shows that two sides (mn and no) and the included angle (angle m or p) of one triangle are equal to the corresponding two sides (pq and qr) and the included angle (angle p) of another triangle, proving congruence between the two triangles under the SAS theorem.