Difference between revisions of "Base Angle Theorem"
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The '''Base Angle Theorem''' states that in an [[isosceles triangle]], the angles opposite the congruent sides are congruent. | The '''Base Angle Theorem''' states that in an [[isosceles triangle]], the angles opposite the congruent sides are congruent. | ||
− | ==Proof== | + | == Proofs == |
+ | |||
+ | == Proof 1 == | ||
+ | |||
+ | By the [[Law of Sines]], we have <math>\tfrac{b}{\sin(B)}=\tfrac{c}{\sin(C)}</math>. We know <math>b=c</math>, so <math>\sin(B)=\sin(C)</math>. Then either <math>B=C</math> or <math>B=180-C</math>, but the second case would imply <math>A=0^{\circ}</math>, so <math>B=C</math>. | ||
+ | |||
+ | == Proof 2 == | ||
+ | |||
+ | We know that <math>\overline{AB} \cong \overline{AC}</math> (given). By the reflexive property, we know that <math>\overline{BC} \cong \overline{CB}</math>. We know that <math>\overline{CA} \cong \overline{BA}</math> (given). By SSS, we conclude that <math>\Delta ABC \cong \Delta ACB</math>. By CPCTC, we conclude that <math>\angle ABC \cong \angle ACB</math>. | ||
+ | |||
+ | == Proof 3 == | ||
+ | |||
Since the triangle only has three sides, the two congruent sides must be adjacent. Let them meet at vertex <math>A</math>. | Since the triangle only has three sides, the two congruent sides must be adjacent. Let them meet at vertex <math>A</math>. | ||
− | Now we draw [[ | + | Now we draw [[altitude]] <math>AD</math> to <math>BC</math>. From the [[Pythagorean Theorem]], <math>BD=CD</math>, and thus <math>\triangle ABD</math> is congruent to <math>\triangle ACD</math>, and <math>\angle DBA=\angle DCA</math>. <asy> |
unitsize(5); defaultpen(fontsize(10)); | unitsize(5); defaultpen(fontsize(10)); | ||
pair A,B,C,D,E,F,G,H; | pair A,B,C,D,E,F,G,H; | ||
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[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
+ | {{stub}} |
Latest revision as of 17:41, 17 January 2025
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 . We know , so . Then either or , but the second case would imply , so .
Proof 2
We know that (given). By the reflexive property, we know that . We know that (given). By SSS, we conclude that . By CPCTC, we conclude that .
Proof 3
Since the triangle only has three sides, the two congruent sides must be adjacent. Let them meet at vertex .
Now we draw altitude to . From the Pythagorean Theorem, , and thus is congruent to , and . This article is a stub. Help us out by expanding it.