Base Angle Theorem

The Hinge Theorem states that in an isosceles triangle, the measures of the angles opposite the equal-measuring sides are equal.

Proof

Since the triangle only has three sides, the two equal-measuring sides must be adjacent. Let them meet at vertex $A$ .

Now we draw height $AD$ to $BC$ . From the Pythagorean Theorem, $BD=CD$ , and thus $\triangle ABD$ is similar to $\triangle ACD$ , and $\angle DBA=\angle DCA$ . $[asy] unitsize(5); defaultpen(fontsize(10)); pair A,B,C,D,E,F,G,H; A=(0,10); B=(-5,0); C=(5,0); D=(0,0); E=(1,1); F=(-1,1); G=(-1,0); H=(1,0); draw(A--B); draw(B--C); draw(C--A); draw(A--D); draw(E--F); draw(E--H); draw(F--G); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,S);[/asy]$

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Base Angle Theorem

Proof