Question

Two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle. Which congruence theorem can be used to prove that the triangles are congruent?
a.SSS
b.AAS
c.SAS
d.HL

Answer 1

The correct option is $\boxed{\bf option B}$ i.e., AAS.

Further explanation:

Given that two angles of one triangle and the non-included side are congruent to the corresponding parts of another triangle.

ASA congruence rule:

If two angles and the included side of one triangle are congruent to two angles and the included side of other triangle, then both the triangles are congruent from the Angle-Side-Angle axiom of congruence.

Consider two triangles as $\triangle\text{ABC}$ and $\triangle\text{ACD}$ as shown in Figure 1.

Now, assume that $\angle\text{ABD}$ and $\angle\text{ADB}$ are equal and the side AC bisects $\angle\text{BAD}$ as shown in Figure 1.

The side AC is common to both the triangles.

The sum of three angles of a triangle is equal to $180^{\circ}$.

The $\angle\text{ACB}$ is calculated as  follows:

$\angle\text{ACB}=180-(\angle\text{CAB}+\angle\text{ABC})$

The $\angle\text{ACD}$ is calculated as  follows:

$\angle\text{ACD}=180-(\angle\text{CAD}+\angle\text{ADC})$

The $\angle\text{CAB}$ is equal to $\angle\text{CAD}$ and $\angle\text{ABC}$ is equal to $\angle\text{ADC}$.

Then the $\angle\text{ACB}$ is  calculted as follows:

$\begin{aligned}\angle\text{ACB}&=180-(\angle\text{CAB}+\angle\text{ABC})\\&=180-(\angle\text{CAD}+\angle\text{ADC})\\&=\angle\text{ACD}\end{aligned}$

Therefore, all the three angles of the $\triangle\text{ABC}$ are equal to corresponding angles of $\triangle\text{ADB}$.

If all the angles of triangle are equal to corresponding angles of another triangle, then it is not necessary that the triangles are congruent.  

So for triangles to be congruent from three equal parts there should be one side.

Thus, AAS congruence theorem can be used to prove that the triangles are congruent.

Therefore, the correct option is $\boxed{\bf option B}$ i.e., AAS.

Learn more:

1. Problem about triangles brainly.com/question/7437053

2. The product of a binomial and a trinomial is x 3 + 3 x 2 − x + 2 x 2 + 6 x − 3. Which expression is equivalent to this product after it has been fully simplified? brainly.com/question/1394854

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Triangles

Keywords: Triangles, congruence, two angles, one side, AAS, theorem, ASA, SSS, SAS, HL, prove, non-included side.

Answer 2

Answer:-B )" AAS congruence theorem" an be used to prove that the triangles are congruent .

AAS congruence theorem tells that if two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle then the triangles are congruent.

Therefore option B is the correct answer. "AAS congruence" theorem an be used to prove that the triangles are congruent .