Base Angle Theorem

The Base Angle Theorem states that in an isosceles triangle, the angles opposite the congruent sides are congruent.

Proofs

Proof 1

By the Law of Sines, we have $\tfrac{b}{\sin(B)}=\tfrac{c}{\sin(C)}$ . We know $b=c$ , so $\sin(B)=\sin(C)$ . Then either $B=C$ or $B=180-C$ , but the second case would imply $A=0^{\circ}$ , so $B=C$ .

Proof 2

We know that $\overline{AB} \cong \overline{AC}$ (given). By the reflexive property, we know that $\overline{BC} \cong \overline{CB}$ . We know that $\overline{CA} \cong \overline{BA}$ (given). By SSS, we conclude that $\Delta ABC \cong \Delta ACB$ . By CPCTC, we conclude that $\angle ABC \cong \angle ACB$ .

Proof 3

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

Now we draw altitude $AD$ to $BC$ . From the Pythagorean Theorem, $BD=CD$ , and thus $\triangle ABD$ is congruent 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$",D,S);[/asy]$ This article is a stub. Help us out by expanding it.

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Base Angle Theorem

Contents

Proofs

Proof 1

Proof 2

Proof 3