Question

homework 5-4 triangle proofs (asa, aas, sss) question 12 of 13 (1 point) | question attempt: 1 of unlimited (a) image of two triangles m, q, p with markings options: not necessarily congruent; congruent: δ mno ≅ δ□ by the select (b) image of two triangles a, b, c and d, f, e with markings options: not necessarily congruent; congruent: δ abc ≅ δ□ by the select (c) image of two triangles v, u, w and x, i, z with markings options: not necessarily congruent; congruent: δ uvw ≅ δ□ by the select try one last time

Explanation:

Step1: Analyze Part (a)

In the first triangle pair, we look for congruent triangles. The triangles have vertical angles (equal) and two sides marked equal. Wait, actually, let's check the markings. The first triangle (MNO) and the other triangle (let's say PQO? Wait, the labels: M, N, O and P, Q, O? Wait, the diagram has two triangles with a common side? Wait, no, the first diagram: triangle MNO and triangle PQO? Wait, the angles: angle at N and angle at P? Wait, no, the markings: two sides are equal (the segments with ticks) and the included angle? Wait, no, vertical angles. Wait, actually, let's re-examine. The first triangle: M, N, O. The second: P, Q, O. The sides: MN and PQ? Wait, no, the ticks: one side in MNO and one in PQO, and the common side? Wait, maybe it's ASA or AAS? Wait, no, the correct approach: in part (a), the triangles have two sides equal (marked) and the included angle? Wait, no, vertical angles are equal. Wait, maybe it's AAS? Wait, no, let's see: if we have two angles and a side? Wait, no, the first triangle: angle at N, side NO, and angle at O? Wait, no, maybe I mislabel. Wait, the first triangle: M, N, O. The second: P, Q, O. The sides: MN and PQ? No, the ticks: one side in MNO (say, NO) and one in PQO (PO) with two ticks? Wait, the diagram shows two triangles with a common vertex, and two sides marked equal (each with two ticks) and the vertical angle. So that's SAS: two sides and included angle. So triangle MNO congruent to triangle PQO? Wait, no, the labels: M, N, O and P, Q, O. Wait, maybe the other triangle is PQO? Wait, no, the correct triangle is probably PQO? Wait, no, let's check the congruence criteria. If two sides and the included angle are equal, SAS. So in part (a), the triangles are congruent by SAS. So ΔMNO ≅ ΔPQO by SAS? Wait, maybe the triangle is PQO? Wait, no, the labels: M, N, O and P, Q, O. Wait, maybe the second triangle is PQO, so ΔMNO ≅ ΔPQO by SAS.

Step2: Analyze Part (b)

In part (b), triangle ABC and triangle DEF? Wait, no, the labels are A, B, C and D, F, E. The angles: angle at A and angle at D are equal (marked), angle at B and angle at F are equal (marked), and one side (AB and DF? Wait, no, the sides: AB has one tick, DE? Wait, no, the triangles: ABC has AB and BC with one tick? Wait, no, ABC: AB and AC? Wait, no, ABC: angle at A, angle at B, and side AB. DEF: angle at D, angle at F, and side DE? Wait, no, the markings: ABC has AB with one tick, AC? No, ABC: angle at A (marked), angle at B (marked), and side AB. DEF: angle at D (marked), angle at F (marked), and side DE? Wait, no, the sides: ABC has AB with one tick, DEF has DE with one tick? No, the problem says "Not Necessarily Congruent" is selected? Wait, no, in part (b), the option "Not Necessarily Congruent" is selected? Wait, no, the user's diagram: in part (b), the radio button "Not Necessarily Congruent" is selected? Wait, no, the original problem: in part (b), the triangles: ABC has angle at A, angle at B, and side AB. DEF has angle at D, angle at F, and side DE. But the angles: angle at A and angle at D, angle at B and angle at F. But the side: AB and DF? Wait, no, the sides: ABC has AB with one tick, DEF has DE with one tick? No, the congruence criteria: ASA requires two angles and included side. Here, the angles are not included? Wait, angle at A, side AB, angle at B: that's ASA. But in the other triangle, angle at D, side DE, angle at F: but angle at F is not adjacent to DE. So maybe it's AAS, but the sides are not corresponding. So they are not necessarily congruent. So part (b) is Not Necessarily Congruent.

Step3: Analyze Part (c)

In part (c), triangle UVW and triangle XYZ? Wait, the diagram: UVW has two sides marked (one tick each), and XYZ has two sides marked (one tick and two ticks? No, XYZ has two sides with two ticks? Wait, no, the diagram: UVW is a small triangle with two sides marked (one tick), and XYZ is a triangle with two sides marked (two ticks) and one side with one tick? Wait, no, the markings: UVW has two sides with one tick, XYZ has two sides with two ticks? No, maybe it's SSS? Wait, no, the angles: XYZ has an angle marked. Wait, UVW: two sides and included angle? No, the problem says "Try one last time". Wait, maybe the triangles are congruent by SAS? Wait, no, let's see: UVW has two sides marked (one tick), XYZ has two sides marked (two ticks? No, maybe the correct congruence is SAS or SSS. Wait, maybe UVW ≅ XYZ by SAS? Wait, no, the labels: UVW and XYZ. Wait, the diagram: UVW is a triangle with vertex V, and XYZ with vertex Z. The sides: UVW has two sides with one tick, XYZ has two sides with two ticks? No, maybe it's SSS, but the markings don't match. Wait, maybe the correct answer for part (c) is congruent by SAS or SSS, but I need to check.

Wait, maybe I made a mistake. Let's re-express:

Part (a)

Triangles: MNO and PQO (assuming). The markings: two sides equal (SAS) and included angle (vertical angles). So ΔMNO ≅ ΔPQO by SAS.

Part (b)

Triangles ABC and DEF. The angles: angle at A, angle at B, and side AB. The other triangle: angle at D, angle at F, and side DE. The angles are not included, so not necessarily congruent.

Part (c)

Triangles UVW and XYZ. The markings: UVW has two sides, XYZ has two sides and an angle. Maybe SSS? No, maybe SAS. Wait, the correct answer for part (c) is congruent by SAS or SSS, but I need to check the diagram.

But the user's problem is to solve these triangle congruence problems. Let's focus on part (a) first.

In part (a), the triangles have two sides equal (marked) and the included angle (vertical angles) equal. So by SAS, ΔMNO ≅ ΔPQO (assuming the other triangle is PQO). So the answer for part (a) is Congruent: ΔMNO ≅ ΔPQO by SAS.

For part (b), the triangles do not have corresponding congruent parts (angles and sides) in the correct order, so Not Necessarily Congruent.

For part (c), the triangles: UVW has two sides marked, XYZ has two sides marked and an angle. Wait, maybe UVW ≅ XYZ by SAS? Or SSS? Wait, the diagram shows UVW with two sides (one tick) and XYZ with two sides (two ticks? No, maybe the markings are different. Wait, the correct answer for part (c) is Congruent: ΔUVW ≅ ΔXYZ by SAS (or SSS, depending on markings).

But since the user's problem is a bit unclear, but let's proceed with part (a) as an example.

Answer:

(a) Congruent: $\triangle MNO \cong \triangle PQO$ by SAS (assuming the other triangle is PQO)
(b) Not Necessarily Congruent
(c) Congruent: $\triangle UVW \cong \triangle XYZ$ by SAS (or other criteria, depending on markings)

(Note: The exact triangle names and criteria depend on the diagram's labeling, but the above is a general solution based on typical triangle congruence problems.)