Question

state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not
type your answer...
fill in the blank 4 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.
type your answer...
2.3 fill in the blank 4 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.
type your answer...

Explanation:

First Triangle Problem (Top):

Step1: Identify Right Angles

Triangles \( \triangle CBA \) and \( \triangle DFE \) are right - angled at \( B \) and \( F \) respectively (\( \angle B=\angle F = 90^{\circ}\)).

Step2: Identify Equal Sides and Angles

We can see that one pair of non - hypotenuse sides (let's say \( BA = FE \)) and one pair of angles (the marked acute angles) are equal. Also, the hypotenuses (or another pair of sides) can be considered. Since we have a right angle, a pair of equal angles, and a pair of equal sides, by the AAS (Angle - Angle - Side) congruence criterion, the triangles are congruent by AAS. But wait, if we consider the right angle, one leg and the hypotenuse, maybe HL? Wait, no, let's re - examine. Wait, the right angles, one pair of legs (if \( BA = FE \)) and the other pair of angles (the marked angles). Alternatively, if we assume that the marked sides (the legs) and the right angles and the other angles. Wait, maybe AAS. But let's check again. Wait, the first triangle: right angle at \( B \) and \( F \), \( \angle CAB=\angle EDF \) (marked angles), and \( AB = FE \) (the bases). So by AAS (two angles and a non - included side), the triangles are congruent by AAS. But wait, maybe HL? No, HL is for right triangles with hypotenuse and one leg. If we don't know the hypotenuse, AAS is more appropriate. Wait, maybe I made a mistake. Wait, the problem is to state the congruence. Let's assume that the triangles are right - angled, have one pair of equal legs and one pair of equal acute angles, so AAS. But maybe the correct answer is AAS. Wait, no, maybe HL? Wait, no, HL requires hypotenuse and one leg. If we have two right triangles, with one leg equal and the hypotenuse equal, then HL. But from the diagram, if the legs \( AB = FE \) and the hypotenuses \( AC = DE \), then HL. But the diagram shows marked angles. Maybe the correct answer is AAS. Wait, maybe I should re - think. Let's assume that the triangles are congruent by AAS. But maybe the first one is AAS. Wait, no, let's check the second problem.

Second Triangle Problem (Middle):

Step1: Analyze Angles and Sides

In triangles \( \triangle ABC \) and \( \triangle DFE \), we have angles at \( B \) and \( C \) marked, and angles at \( F \) and \( E \) marked. But the correspondence of angles is not clear. The angle at \( A \) and the angle at \( D \) are marked, but the sides are not clearly equal in a way that satisfies SSS, SAS, ASA, AAS, or HL. So the triangles are not congruent, so the answer is NOT.

Third Triangle Problem (Bottom):

Step1: Identify Vertical Angles and Equal Sides

In triangles \( \triangle OEW \) and \( \triangle AET \), we have vertical angles at \( E \) ( \( \angle OEW=\angle AET \) ). Also, the marked sides \( OE = AE \) and \( WE = TE \). So we have two sides and the included angle equal (SAS: Side - Angle - Side). The included angle is the vertical angle, and the two sides \( OE = AE \) and \( WE = TE \). So by SAS congruence criterion, the triangles are congruent by SAS.

First Triangle Answer:

Answer:

AAS (assuming the correct congruence)

Second Triangle Answer: