Difference between revisions of "Base Angle Theorem"

Revision as of 16:59, 31 August 2008

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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Difference between revisions of "Base Angle Theorem"

Revision as of 16:59, 31 August 2008

Proof

@@ Line 2: / Line 2: @@
 ==Proof==
-Since the triangle only has three sides, the two equal-measuring sides must be adjacent. Let them meet at vertex <math>A</math>. Now we draw [[height]] <math>AD</math> to <math>BC</math>. From the [[Pythagorean Theorem]], <math>BD=CD</math>, and thus <math>\triangle ABD</math> is similar to <math>\triangle ACD</math>, and <math>\angle DBA=\angle DCA</math>.
+Since the triangle only has three sides, the two equal-measuring sides must be adjacent. Let them meet at vertex <math>A</math>.
+Now we draw [[height]] <math>AD</math> to <math>BC</math>. From the [[Pythagorean Theorem]], <math>BD=CD</math>, and thus <math>\triangle ABD</math> is similar to <math>\triangle ACD</math>, and <math>\angle DBA=\angle DCA</math>. <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>
 [[Category:Theorems]]
 [[Category:Geometry]]