Given: Isosceles △ABC

Question

with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯


Prove: ∠A≅∠C


Statements Reasons
1. Isosceles △ABC
with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
1. Given
2. BD←→
the angle bisector of ∠B


2. Construction
3. ∠ABD≅∠CBD
3. Definition of angle bisector
4. BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯
4. Reflexive property of congruence
5. ∠ABD≅∠CBD
5. SAS Congruence Theorem
6. ∠A≅∠C
6. CPCTC Theorem

Answer 1

The proof is correct. Steps 1, 2, and 3 establish that angle ABD is congruent to angle CBD because BD is the angle bisector of angle B in isosceles triangle ABC. Step 4 uses the reflexive property of congruence to show that BD is congruent to itself. Step 5 applies the SAS Congruence Theorem to triangles ABD and CBD, concluding that angle A is congruent to angle C. Finally, step 6 uses the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem to state that angle A is congruent to angle C. Thus, the proof is valid.

Answer 2

the Triangle Inequality Theorem
the Triangle Inequality Theorem

the Base Angles Theorem
the Base Angles Theorem

the Triangle Sum Theorem
the Triangle Sum Theorem

the Isosceles Triangle Theorem
these are the answers provided

Answer 3

I apologize for the confusion. The correct answer is as follows:

Given: Isosceles triangle ABC with AB ≅ BC

Prove: ∠A ≅ ∠C

Statements Reasons

1. ABC is an isosceles triangle Given

2. AB ≅ BC Given

3. Definition of Isosceles Triangle AB ≅ BC

4. BD is the angle bisector of ∠B Construction

5. ∠ABD ≅ ∠CBD Definition of angle bisector

6. ∠ABD ≅ ∠CBD Reflexive property of congruence

7. ΔABD ≅ ΔCBD SAS Congruence Theorem

8. ∠A ≅ ∠C Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem

Therefore, ∠A ≅ ∠C is proved.