Download 5.3 The Isosceles Triangle Theorems

IsoscelesTriangleTheorems20052006.nb
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5.3 The Isosceles Triangle Theorems
As we discussed in a previous section isosceles triangles are triangles with at least two
sides congruent. The two congruent sides are called legs and the third side is called the base.
Since an equilateral triangle, by definition, is also isosceles then any two sides can be referred
to as the legs and the third side the base. Look at the figure below.
vertex angle
leg
leg
base angles
base
Note that the definition of an isosceles triangle tells us only which triangles are isosceles. We will now look at two theorems and several corollaries which will tell us some of the
properties of isosceles triangles.
The Isosceles Triangle Theorem
If two sides of one triangle are congruent, then the angles opposite those sides are
congruent.
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Given: AB @ AC
Prove: —B @ —C
IsoscelesTriangleTheorems20052006.nb
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A
B
D
C
Plan for proof: You can show that —B and —C are corresponding parts of congruent triangles if
you draw an auxiliary line that will give you such triangles. For example, draw the bisector of
êêêêê
êêêêê
—A. Since AD @ AD by the reflexive property DADB and DADC are congruent by the Side
Angle Side Postulate.
There are some corollaries that follow from the Isosceles Triangle Theorem and they are listed
below. See if you can determine how these corollaries follow from the theorem.
Corollary 1
An equilateral triangle is also equiangular.
Corollary 2
An equilateral triangle has three 60° angles.
Corollary 3
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its
midpoint.
The converse of the Isosceles Triangle Theorem can also be proven and is stated below.
IsoscelesTriangleTheorems20052006.nb
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Converse of the Isosceles Triangle Theroem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Given : —B @ —C
êêêêê
êêêêê
Prove: AB @ AC
A
B
D
C
êêêêê
êêêêê
Plan for proof: You can show that AB and AC are corresponding parts of congruent triangles.
Draw the bisector of —A as your auxiliary line, show that —ADB @ —ADC (remember that if
two angles of one triangle are congruent to two angles of another triangle then the third angles
are congruent), and then use the Angle Side Angle Postulate.
The Converse of the Isosceles Triangle Theroem also has a corollary which is stated below.
Corollary
An equiangular triangle is also equilateral.
Summary:
The definition of an isosceles triangle tells us which triangles can be classified as isosceles. The Isosceles Triangle Theorem tells us if a triangle is isosceles then its base angles are
congruent. The corollaries of the Isosceles Triangle Theorem tell us that: 1) an equilateral
triangle is also equiangular, 2) an equilateral triangle has three 60° angles, and 3) the bisector
of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
The Converse of the Isosceles Triangle Theorem tells us that if two angles of a trian-
IsoscelesTriangleTheorems20052006.nb
gle are congruent then the sides opposite those angles are congruent. The corollary of this
theorem states that an equiangular triangle is also equilateral.
Ç
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