[tex]\boxed{\bf AAS congruence theorem}[/tex] proves that these two triangles are congruent.
Further explanation:
AAS congruence rule:
If two angles and the non-included side of one triangle are equal to two angles and the non-included side of other triangle, then both the triangles are congruent from the Angle-Angle-Side- congruence theorem.
There are two triangles [tex]\triangle\text{MNR}[/tex] and [tex]\triangle\text{PNQ}[/tex] as shown in attached Figure 1.
Now, assume that [tex]\angle\text{RMN}[/tex] and [tex]\angle\text{QPN}[/tex] are equal and the line MP bisects the line RQ, therefore the side RN of [tex]\triangle\text{MNR}[/tex] and the side NQ of [tex]\triangle\text{PNQ}[/tex] is equal as shown in Figure 1.
The angle [tex]\angle\text{MNR}[/tex] and [tex]\angle\text{QNP}[/tex] are equal because they are vertically opposite angles.
If all the angles of triangle are equal to corresponding angles of another triangle, then it is not necessary that the triangles are congruent.
Thus, AAS congruence theorem can be used to prove that the triangles are congruent.
The [tex]\boxed{\bf option\ b}[/tex] i.e., AAS is correct option.
Learn more:
1. Learn more about triangles https://brainly.com/question/7437053
2. Learn more about slope of a line https://brainly.com/question/1473992
Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Triangles
Keywords: Congruence, two angles, Triangles, one side, AAS, theorem, ASA, SSS, SAS, HL, prove, non-included side, congruency, theorem, mathematics.
