Prove that, if the bisector of ∠BAC of ∆ABC is perpendicular to si

1.	Prove that, if the bisector of ∠BAC of ∆ABC is perpendicular to side BC, then AABC is an isosceles triangle.
Answer» Given: Seg AD is the bisector of ∠BAC. seg AD ⊥ seg BC To prove: AABC is an isosceles triangle. Proof: In ∆ABD and ∆ACD, ∠BAD ≅ ∠CAD [seg AD is the bisector of ∠BAC] seg AD ≅ seg AD [Common side] ∠ADB ≅ ∠ADC [Each angle is of measure 90°] ∴ ∆ABD ≅ ∆ACD [ASA test] ∴ seg AB ≅ seg AC [c. s. c. t.] ∴ ∆ABC is an isosceles triangle.

Prove that, if the bisector of ∠BAC of ∆ABC is perpendicular to side BC, then AABC is an isosceles triangle.

Answer»

Given:

Seg AD is the bisector of ∠BAC. seg AD ⊥ seg BC

To prove: AABC is an isosceles triangle.

Proof:

In ∆ABD and ∆ACD,

∠BAD ≅ ∠CAD [seg AD is the bisector of ∠BAC]

seg AD ≅ seg AD [Common side]

∠ADB ≅ ∠ADC [Each angle is of measure 90°]

∴ ∆ABD ≅ ∆ACD [ASA test]

∴ seg AB ≅ seg AC [c. s. c. t.]

∴ ∆ABC is an isosceles triangle.