1. ## euclidean geometry! help {solved}

Suppose that in a triangle ABC the angle ABC=angle ACB show that side AB=AC. In triangle ABC let X lie on BC in such a way that angle AXB is a right angle. Show X is the midpoint of BC.

ok its clearly isosceles so i thought mentioning Angle-side-angle then to show right angle i got confused where to go!

2. Originally Posted by skystar
Suppose that in a triangle ABC the angle ABC=angle ACB show that side AB=AC. In triangle ABC let X lie on BC in such a way that angle AXB is a right angle. Show X is the midpoint of BC.

ok its clearly isosceles so i thought mentioning Angle-side-angle then to show right angle i got confused where to go!
Since triangle ABC is isosceles, and $\overline{AB}\cong{\overline{AC}} \; \; and \; \; \angle {B} \cong{\angle{C}} \; \; then \; \; \triangle{ABX}\cong{\triangle{ACX}}$ by AAS (angle-angle-side) or HL (hypotenuse-leg) theorems.

$\overline{CX} \cong{\overline{BX}}$ by CPCTC (Corresponding Parts of Congruent Triangles are Congruent.

X is the midpoint by definition of midpoint.

3. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Since we knew that $\angle{B}\cong{\angle{C}} \; \; then \; \; we \; \;conclude \; \;that \; \; \overline{AB}\cong{\overline{AC}}$