Base Angle Theorem
Revision as of 20:41, 20 September 2008 by Herefishyfishy1 (talk | contribs) (Hinge Theorem moved to Base Angle Theorem: It's not called the Hinge Theorem. It's called the 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 
.
Now we draw height 
to 
. From the Pythagorean Theorem, 
, and thus 
is similar to 
, and 
. ![[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]](http://latex.artofproblemsolving.com/7/f/6/7f66cb51375e207e519d2fb8340d23232cff7953.png)
