# Thread: isosceles triangle proof II

1. ## isosceles triangle proof II

Prove that a triangle is isosceles if and only if two altitudes are congruent.

2. If you covered the Pythagoras theorem, you could prove that the small triangles having altitudes as catheti and the common side as a hypotenuse are equal.

3. Originally Posted by MATNTRNG
Prove that a triangle is isosceles if and only if two altitudes are congruent.
A triangle is isosceles if "at least" two sides have equal lengths.

Triangle area is $A=\frac{1}{2}base(perpendicular\ height)$

If two sides are equal, then the altitudes of the 3rd vertex above them must also be equal.

If no two altitudes are equal, no two sides can be equal.

if you begin with "at least 2 acute angles are equal", you can use the Sine Rule

$\frac{SinA}{a}=\frac{SinB}{b}\Rightarrow\frac{SinA }{a}=\frac{SinA}{b}\Rightarrow\ a=b$