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4.6 Isosceles 4.6 Isosceles Triangles Triangles
4.6 Isosceles Triangles4.6 Isosceles Triangles
ObjectivesObjectives
Use properties of isosceles trianglesUse properties of isosceles triangles
Use properties of equilateral trianglesUse properties of equilateral triangles
Properties of Isosceles TrianglesProperties of Isosceles Triangles
The The formed by the ≅ sides is formed by the ≅ sides is called the called the vertex anglevertex angle. .
The two ≅ sides are called The two ≅ sides are called legslegs. The third side is called . The third side is called the the basebase..
The two The two s formed by the base s formed by the base and the legs are called theand the legs are called thebase anglesbase angles. .
leg leg
base
vertex
Isosceles Triangle TheoremIsosceles Triangle Theorem
Theorem 4.9Theorem 4.9If two sides of a If two sides of a ∆ are ∆ are ≅, then the ≅, then the s s opposite those sides are ≅ (if AC ≅ AB, opposite those sides are ≅ (if AC ≅ AB, then then B ≅ B ≅ C).C). A
B C
The Converse of Isosceles The Converse of Isosceles Triangle TheoremTriangle Theorem
Theorem 4.10Theorem 4.10
If two If two ss of a of a ∆ are ∆ are ≅, then the sides ≅, then the sides opposite those opposite those s are ≅ (if s are ≅ (if B ≅ B ≅ C, C, then AC ≅ AB).then AC ≅ AB).
Answer:
Name two congruent angles (not indicated).
Example 2:Example 2:
Answer:
Name two congruent segments (not indicated).
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,
Example 2:Example 2:
a. Name two congruent angles.
Answer:
Answer:
b. Name two congruent segments.
Your Turn:Your Turn:
Properties of Equilateral Properties of Equilateral ∆s∆s
Corollary 4.3Corollary 4.3A A ∆ is equilateral if it is equiangular.∆ is equilateral if it is equiangular.
Corollary 4.4Corollary 4.4Each Each of an equilateral of an equilateral ∆∆ measures measures 6060°.°.
Since the angle was bisected,Each angle of an equilateral triangle measures 60°.
EFG is equilateral, and bisects bisectsFind and
Example 3a:Example 3a:
Answer:
Add.
Exterior Angle Theorem
Substitution
is an exterior angle of EGJ.
Example 3a:Example 3a:
Subtract 75 from each side.
Linear pairs are supplementary.
Substitution
Answer: 105
EFG is equilateral, and bisects bisectsFind
Example 3b:Example 3b:
a. Find x.
b.
Answer: 90
Answer: 30
ABC is an equilateral triangle. bisects
Your Turn:Your Turn:
