Question

Which triangles are congruent by AAS?

Answer 1

For two triangles to be congruent by AAS:
1- Two angles in the first triangle must be equal to two angles in the second triangle
2- A non included side in the first triangle is equal to a non included side in the second triangle

Now, let's check our options. We will find that:
For the two triangles UTV and ABC:
angle T = angle A
angle V = angle C
TU (non-included between angles T & V) = AB (non-included between angles A & C)

Therefore, we can conclude that:
Triangles ABC and UTV are congruent by AAS

Answer 2

The $\boxed{\Delta {\text{VTU and }}\Delta {\text{CAB}}}$ are congruent by AAS congruency criterion. Option (a) is correct.

Further Explanation:

The congruent triangles are those in which all the corresponding angles and the sides are equal.

There are many congruency rules and are as follows.

1. Angle Angle Side (AAS)

2. Angle Side Angle (ASA)

3. Side SideSide (SSS)

4. Side Angle Side (SAS)

Given:

The options are as follows,

(a). $\Delta {\text{VTU and }}\Delta {\text{CAB}}$

(b). $\Delta {\text{VTU and }}\Delta {\text{HFG}}$

(c). $\Delta{\text{ HFG and }}\Delta {\text{CAB}}$

(d). None of the above.

Explanation:

In $\Delta {\text{VTU and }}\Delta {\text{CAB}},$

$\begin{aligned}\angle {\text{UVT}}&= \angle {\text{BCA}}\\\angle {\text{VTU}}&= \angle {\text{CAB}}\\{\text{AB}}&= {\text{TU}}\\\Delta {\text{VTU}} \cong \Delta {\text{CAB}}\left( {{\text{AAS congruency}}} \right)\\\end{aligned}$

The $\boxed{\Delta {\text{VTU and }}\Delta {\text{CAB}}}$ are congruent by AAS congruency criterion. Option (a) is correct.

Option (b) is not correct.

Option (c) is not correct.

Option (d) is not correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Triangle

Keywords: congruent, angles, triangle, ASA, angle side angle, congruent sides, acute angle, side, corresponding angles, congruent triangle.