Question
Given: Line A R bisects ∠BAC; AB = AC
Triangle A B C is shown. Point R is at the middle of the triangle. Lines are drawn from point R to each of the points of the triangle. Angles B A R and R A C are congruent. Sides A B and A C are congruent.
Which congruence theorem can be used to prove ΔABR ≅ ΔACR?
AAS
SSS
ASA
SAS
Triangle A B C is shown. Point R is at the middle of the triangle. Lines are drawn from point R to each of the points of the triangle. Angles B A R and R A C are congruent. Sides A B and A C are congruent.
Which congruence theorem can be used to prove ΔABR ≅ ΔACR?
AAS
SSS
ASA
SAS
Answers
GPT-4o mini
To prove that triangles ΔABR and ΔACR are congruent, we analyze the information given:
1. **Angle Bisector Information**: Since line AR bisects ∠BAC, we have that angle BAR is congruent to angle CAR (∠BAR ≅ ∠CAR).
2. **Side Information**: We know that sides AB and AC are congruent (AB = AC).
3. **Common Side**: The side AR is common to both triangles ΔABR and ΔACR.
Now we can summarize:
- We have the two angles (∠BAR and ∠CAR) that are congruent.
- We have the side AB that is congruent to side AC.
- We have the side AR that is common to both triangles.
Putting this together, we can apply the **ASA (Angle-Side-Angle)** congruence theorem:
1. Angle ∠BAR ≅ ∠CAR (from the angle bisector),
2. Side AR is common to both ΔABR and ΔACR,
3. The sides AB and AC are congruent.
Thus, we can conclude that:
**ΔABR ≅ ΔACR by ASA (Angle-Side-Angle) congruence theorem.**
1. **Angle Bisector Information**: Since line AR bisects ∠BAC, we have that angle BAR is congruent to angle CAR (∠BAR ≅ ∠CAR).
2. **Side Information**: We know that sides AB and AC are congruent (AB = AC).
3. **Common Side**: The side AR is common to both triangles ΔABR and ΔACR.
Now we can summarize:
- We have the two angles (∠BAR and ∠CAR) that are congruent.
- We have the side AB that is congruent to side AC.
- We have the side AR that is common to both triangles.
Putting this together, we can apply the **ASA (Angle-Side-Angle)** congruence theorem:
1. Angle ∠BAR ≅ ∠CAR (from the angle bisector),
2. Side AR is common to both ΔABR and ΔACR,
3. The sides AB and AC are congruent.
Thus, we can conclude that:
**ΔABR ≅ ΔACR by ASA (Angle-Side-Angle) congruence theorem.**