Given: Isosceles △ABC
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
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.
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
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.
To prove that ∠A≅∠C in isosceles triangle △ABC with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯, we can use the following statements and reasons:
1. Given: Isosceles △ABC with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯
Reason: This is the given information.
2. Construction: Draw BD←→, the angle bisector of ∠B
Reason: We construct the angle bisector BD to help us prove the congruent angles.
3. Definition of angle bisector: ∠ABD≅∠CBD
Reason: By definition, the angle bisector BD divides ∠B into two congruent angles, ∠ABD and ∠CBD.
4. Reflexive property of congruence: BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯
Reason: The reflexive property states that any segment is congruent to itself.
5. SAS Congruence Theorem: ∠ABD≅∠CBD
Reason: By the SAS (Side-Angle-Side) Congruence Theorem, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
6. CPCTC Theorem: ∠A≅∠C
Reason: Using the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Theorem, we can conclude that if ∠ABD≅∠CBD, then ∠A≅∠C.
Therefore, by proving that ∠ABD≅∠CBD using the angle bisector BD and applying the congruence theorems, we have shown that ∠A≅∠C in isosceles triangle △ABC with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯.