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4.7 Use Isosceles and Equilateral Triangles
Goal  Use theorems about isosceles and equilateral triangles.
Your Notes
VOCABULARY
Legs
The legs of an isosceles triangle are the two congruent sides.
Vertex angle
The vertex angle of an isosceles triangle is the angle formed by the legs.
Base
The base of an isosceles triangle is the side that is not a leg.
Base angles
The base angles of an isosceles triangle are the two angles adjacent to the base.
THEOREM 4.7: BASE ANGLES THEOREM
If two sides of a triangle are congruent, then the angles opposite them are congruent.
If AB  AC , then B  _C_.
THEOREM 4.8: CONVERSE OF BASE ANGLES THEOREM
If two angles of a triangle are congruent, then the sides opposite them are congruent.
If B  C, then AB  _____.
AC
Example 1
Apply the Base Angles Theorem
In FGH, FH  GH . Name two congruent angles.
Solution
FH  GH , so by the Base Angles Theorem, _F_  _G_.
Your Notes
COROLLARY TO THE BASE ANGLES THEOREM
If a triangle is equilateral, then it is _equiangular_.
The corollaries state
that a triangle is
equilateral if and
only if it is
equiangular.
COROLLARY TO THE CONVERSE OF BASE ANGLES THEOREM
If a triangle is equiangular, then it is _equilateral_.
Example 2
Find measures in a triangle
Find the measures of R, S, and T.
Solution
The diagram shows that RST is _equilateral_. Therefore, by the Corollary to the Base
Angles Theorem, RST is _equiangular_. So, mR = mS = mT.
3(mR) = _180_
Triangle Sum Theorem
mR = _60°_
Divide each side by 3.
The measures of R, S, and T are all _60_.
Example 3
Use isosceles and equilateral triangles
Find the values of x and y in the diagram.
Solution
Step 1
Find the value of x. Because JKL is _equiangular_, it is also _equilateral_ and
KL _____
JL . Therefore, x  _8_.
You cannot use J to
refer to LJM because
three angles have J as
their vertex.
Step 2
LJ and LMJ is isosceles. You
Find the value of y. Because JML  _LJM_, LM  ____,
know that LJ = _8_.
LM  _LJ_
2y = _8_
y  _4_
Definition of congruent segments
Substitute 2y for LM and _8_ for LJ.
Divide each side by 2.
Your Notes
Example 4
Solve a multi-step problem
Quilting The pattern at the right is present in a quilt.
a. Explain why ADC is equilateral.
b. Show that CBA  ADC.
Solution
a. By the Base Angles Theorem, DAC  _DCA_. So, ADC is _equiangular_. By the
_Corollary to the Converse of Base Angles Theorem_, ADC is equilateral.
b. By the Base Angles Theorem, ABC  _ACB_. So, CBA  ADC by the
_AAS_Congruence Theorem_.
Checkpoint Complete the following exercises.
1. Copy and complete the statement: If FH  FJ , then  _?_   _?_.
H; J
2. Copy and complete the statement: If FGK is equiangular and FG = 15, then
GK = _?_.
15
3. Use parts (a) and (b) in Example 4 to show that mBAD = 120°.
DCA is equiangular. So, mADC= mDCA = mCAD.
3(mCAD) = 180° Triangle Sum Theorem
mCAD = 60°
Divide each side by 3.
Because DCA is equiangular and CBA  ADC, you know that mBAC= 60°.
mBAD = mBAC+ mCAD
= 60° + 60°
= 120°
Homework
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