Question

question 13 of 13 (1 point) | question attempt: 1 of unlimited
(a)
△ uvw ≅ △□ by the
select
(b)
○ not necessarily congruent
○ congruent:
△ ghi ≅ △□ by the
select
(c)
○ not necessarily congruent
○ congruent:
△ mno ≅ △□ by the
select
try again

Explanation:

Step1: Analyze Part (b) Triangles

In part (b), we have $\triangle GHI$ and $\triangle JKL$ (assuming the second triangle is $\triangle JKL$ with vertices $J, K, L$). Let's check the given markings:
- $\angle H$ and $\angle J$ are marked as equal (angle markings).
- One side in $\triangle GHI$ (say $HI$) and one side in $\triangle JKL$ (say $KL$) are marked as equal (side markings).
- $\angle I$ and $\angle L$ are marked as equal (angle markings).

So, by the **ASA (Angle - Side - Angle)** congruence criterion, if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. Wait, actually, let's re - check: $\angle H=\angle J$, $HI = KL$, $\angle I=\angle L$. So, the angles are at the ends of the equal side. So, it's **AAS (Angle - Angle - Side)**? Wait, no: AAS is two angles and a non - included side. But here, the side is between the two angles? Wait, no, in $\triangle GHI$, the angles at $H$ and $I$ with side $HI$ between them? Wait, no, $\angle H$ is at vertex $H$, $\angle I$ is at vertex $I$, and side $HI$ is between them. In $\triangle JKL$, $\angle J$ is at vertex $J$, $\angle L$ is at vertex $L$, and side $KL$ is between them? Wait, maybe the correct correspondence is $\triangle GHI\cong\triangle JKL$ by AAS? Wait, no, let's think again.

Wait, the triangles: $\triangle GHI$ has vertices $G, H, I$; $\triangle JKL$ has vertices $J, K, L$. The marked angles: $\angle H=\angle J$, $\angle I=\angle L$, and side $HI = KL$. So, in $\triangle GHI$, angles $\angle H$ and $\angle I$ with side $HI$ (the side opposite to $\angle G$? No, $HI$ is between $\angle H$ and $\angle I$? Wait, no, in a triangle, the side between two angles is the included side. So, if $\angle H$ and $\angle I$ are two angles, and $HI$ is the side between them, and in the other triangle, $\angle J$ and $\angle L$ are two angles, and $KL$ is the side between them, and $\angle H=\angle J$, $\angle I=\angle L$, $HI = KL$, then by ASA (Angle - Side - Angle), $\triangle GHI\cong\triangle JKL$.

Step2: Analyze Part (c) Triangles

In part (c), we have $\triangle MNO$ and $\triangle PQR$ (assuming the second triangle is $\triangle PQR$ with vertices $P, Q, R$). The markings:
- One side in $\triangle MNO$ (say $MO$) and one side in $\triangle PQR$ (say $QR$) are marked as equal (side markings).
- $\angle M$ and $\angle P$ are marked as equal (angle markings).
- $\angle N$ and $\angle Q$ are marked as equal (angle markings).

Wait, but the correspondence of sides and angles: Let's see, $\angle M=\angle P$, $\angle N=\angle Q$, and side $MO$ (in $\triangle MNO$) and side $QR$ (in $\triangle PQR$). Wait, maybe the triangles are not necessarily congruent? Wait, no, let's check the congruence criteria. If we have two angles equal, but the side is not the included side or the corresponding non - included side? Wait, the problem says "Not Necessarily Congruent" or "Congruent". Let's see: For $\triangle MNO$ and $\triangle PQR$, we have $\angle M=\angle P$, $\angle N=\angle Q$, but the equal side: is it the corresponding side? If the equal side is not the corresponding side (e.g., in $\triangle MNO$, side $MO$ and in $\triangle PQR$, side $QR$ which may not be the corresponding side), then the triangles are not necessarily congruent. But wait, maybe I made a mistake. Wait, the first triangle in (c) is $\triangle MNO$ with vertex $N$ at the top, $M$ and $O$ at the base (with $MO$ marked as equal to a side in the second triangle). The second triangle is $\triangle PQR$ with vertex $P$ at the bottom? Wait, no, the diagram shows $\triangle MNO$ with $\angle M$ and $\angle O$? Wait, maybe the correct answer for (b) is Congruent: $\triangle GHI\cong\triangle JKL$ by ASA (or AAS), and for (c) it's Not Necessarily Congruent? Wait, no, let's go back to the problem.

Wait, the problem is about triangle congruence, which is a topic in Geometry (subfield of Mathematics).

For part (b):
We have two triangles with two angles and the included side equal (or two angles and a non - included side, but in this case, the markings suggest that $\angle H=\angle J$, $HI = KL$, $\angle I=\angle L$. So, by AAS (since the side is not between the two angles? Wait, no, $\angle H$ is at $H$, $\angle I$ is at $I$, and $HI$ is the side between them. $\angle J$ is at $J$, $\angle L$ is at $L$, and $KL$ is the side between them. So, if $\angle H=\angle J$, $HI = KL$, $\angle I=\angle L$, then by ASA, $\triangle GHI\cong\triangle JKL$. So, the answer for (b) is Congruent: $\triangle GHI\cong\triangle JKL$ (assuming the second triangle is $\triangle JKL$) by AAS or ASA.

For part (c):
We have two triangles with two angles equal, but the equal side may not be the corresponding side. So, the triangles are Not Necessarily Congruent.

Answer:

(b) Congruent: $\triangle GHI\cong\triangle JKL$ (assuming the second triangle is $\triangle JKL$) by AAS (or ASA)
(c) Not Necessarily Congruent

(Note: The exact triangle names for the second triangle in each part depend on the diagram labels, but the key is the congruence analysis as above.)