Question

ABC is an isosceles triangle with BA=BC

Answer 1

Explanation:

1. ∠BAC≅∠BCA, ∠ABD≅∠ADB; Reason: definition of isosceles triangles

2. ∠ABD +∠BAC +∠ADB = 180°; Reason: sum of internal angles is 180°

3. ∠BAC = 180° -2(∠ABD) = 36°; Reason: Subtraction and substitution properties of equality

4. ∠BAC +∠BCA +∠ABC = 180°; Reason: sum of internal angles is 180°

5. ∠BCA = 180° -2(∠BAC) = 108°; Reason: Subtraction and substitution properties of equality

6. ∠ABD +∠DBC = ∠ABC; Reason: Angle sum theorem

7. ∠DBC = ∠ABC -∠ABD = 108° -72° = 36°; Reason: Subtraction and substitution properties of equality

8. ∠BCA = ∠BAC = 36°; Reason: Substitution property of congruence

9. ΔBCD is isosceles; Reason: Base angles DBC and BCA are congruent.

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There may be extra steps involved if you separately use subtraction and substitution properties of equality, or if you separately claim congruence of angles and equality of their measures. We have assumed that the definition of "isosceles triangle" includes the fact of equal side lengths and equal base angles.