Question

are the following triangles congruent? if so, how do you know? yes, sss yes, asa no, not enough information given yes, sas yes, aas

Explanation:

Step1: Identify Vertical Angles

Vertical angles \( \angle UNF \) and \( \angle ANM \) are equal.

Step2: Identify Given Angles and Sides

We have \( \angle F \cong \angle M \) (marked right angles), \( \angle UNF \cong \angle ANM \) (vertical angles), and \( FN \cong MN \) (marked segments). Wait, no, actually, looking at the congruence criteria: SAS (Side - Angle - Side) would require two sides and the included angle. Wait, let's re - examine. The triangles are \( \triangle UNF \) and \( \triangle ANM \). We know that \( \angle F=\angle M = 90^{\circ}\) (assuming the marked angles are right angles), \( \angle UNF=\angle ANM\) (vertical angles), and \( FN = MN\)? Wait, no, maybe the sides: Wait, the correct approach: Let's check the congruence criteria. The vertical angles are equal (\( \angle UNF=\angle ANM\)), \( \angle F=\angle M\) (right angles), and \( FN = MN\)? Wait, no, actually, if we consider the sides around the vertical angles: Wait, maybe it's AAS? No, wait, let's look at the options. The correct congruence here: We have two angles and a non - included side? Wait, no, let's see the marked parts. The triangles have a vertical angle, a right angle, and a side. Wait, the correct answer is "Yes, AAS"? No, wait, let's re - evaluate. Wait, the vertical angles are equal, one pair of angles (the right angles) are equal, and a side. Wait, AAS (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. But wait, maybe it's SAS? Wait, no, let's check the options. Wait, the correct answer is "Yes, AAS"? No, wait, the vertical angles are equal, \( \angle F=\angle M\), and \( FN = MN\)? Wait, no, maybe the sides: Wait, the line segments \( FN\) and \( MN\) are equal (marked), \( \angle F=\angle M\), and \( \angle UNF=\angle ANM\). So two angles and a side. If the side is not included between the two angles, it's AAS. But wait, let's check the options. Wait, the correct answer is "Yes, AAS"? No, wait, maybe I made a mistake. Wait, the other option: Let's see the triangles. The vertical angles are equal, \( \angle F=\angle M\), and \( FN = MN\). So angle - angle - side (AAS) because the side is not between the two angles. But wait, another way: If we have \( \angle F=\angle M\), \( \angle UNF=\angle ANM\), and \( FN = MN\), then by AAS, the triangles are congruent. But wait, the option "Yes, AAS" is there. Wait, no, wait, maybe it's SAS? Wait, no, SAS requires the side to be between the two angles. The included angle for SAS would be the angle between the two sides. In this case, if we have \( FN = MN\), \( \angle F=\angle M\), and \( \angle UNF=\angle ANM\), it's AAS. But wait, let's check the options again. The options are: Yes, SSS; Yes, ASA; No, not enough; Yes, SAS; Yes, AAS.

Wait, let's re - analyze the diagram. The triangles share the vertical angle at \( N\), so \( \angle UNF\cong\angle ANM\) (vertical angles). We have \( \angle F\cong\angle M\) (both right angles, marked with the same symbol). And we have a side: Let's say \( FN\cong MN\) (marked with the same symbol). So in \( \triangle UNF\) and \( \triangle ANM\), we have \( \angle F\cong\angle M\), \( \angle UNF\cong\angle ANM\), and \( FN\cong MN\). This is AAS (Angle - Angle - Side) because the side \( FN\) (or \( MN\)) is not included between the two angles \( \angle F\) and \( \angle UNF\) (or \( \angle M\) and \( \angle ANM\)). So the triangles are congruent by AAS.

Answer:

Yes, AAS