Isosceles and Equilateral Triangles in Geometry
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SECTION 4.5 ISOSCELES
SECTION 4.5 ISOSCELES
& EQUILATERAL
& EQUILATERAL
TRIANGLES
TRIANGLES
Geometry
Geometry
■
Legs are congruent
Legs are congruent
Isosceles Triangle
Isosceles Triangle
Vertex Angle
Vertex Angle
Base Angles
Base Angles
Legs
Legs
Theorem 4-3 (Isosceles Triangle Thm):
Theorem 4-3 (Isosceles Triangle Thm):
If two sides of a triangle
If two sides of a triangle
are congruent, then the angles opposite those sides are
are congruent, then the angles opposite those sides are
congruent.
congruent.
a
c
b
Theorem 4-4 (Converse of Isosceles Triangle Thm):
Theorem 4-4 (Converse of Isosceles Triangle Thm):
If two angles of a triangle are congruent, then the
If two angles of a triangle are congruent, then the
sides opposite the angles are congruent.
sides opposite the angles are congruent.
■
The bisector of the vertex angle of an isosceles
The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
triangle is the perpendicular bisector of the base.
Theorem 4-5
Theorem 4-5
A
B
C
■
Find the values of x and y.
Find the values of x and y.
Ex. 2)
Ex. 2)
M
L
N
63°
y°
x°
Corollary:
Corollary:
a statement that follows immediately from a theorem
a statement that follows immediately from a theorem
Corollary to Thm 4-3:
Corollary to Thm 4-3:
If a triangle is equilateral, then it is equiangular.
If a triangle is equilateral, then it is equiangular.
Corollary to Thm 4-4:
Corollary to Thm 4-4:
If a triangle is equiangular, then it is equilateral.
If a triangle is equiangular, then it is equilateral.
Ex. 3)
Ex. 3)
A
B
C
M
If
find
Ex. 4)
Ex. 4)
A
B
C
M
If
find
Ex. 5)
Ex. 5)
A
B
C
M
If
and
find x.
Ex. 6) Find the missing angles and sides.
Ex. 6) Find the missing angles and sides.
Exploring the properties and theorems related to isosceles and equilateral triangles in geometry. Discover concepts like the congruence of legs and angles, the relationship between sides and angles, and the bisector of the vertex angle in an isosceles triangle. Explore examples with detailed solutions to enhance your understanding of these triangle types.
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Presentation Transcript
SECTION 4.5 ISOSCELES SECTION 4.5 ISOSCELES & EQUILATERAL & EQUILATERAL TRIANGLES TRIANGLES Geometry Geometry
Isosceles Triangle Isosceles Triangle Legs are congruent Legs are congruent Vertex Angle Vertex Angle Legs Legs Base Angles Base Angles
Theorem 4-3 (Isosceles Triangle Thm): If two sides of a triangle are congruent, then the angles opposite those sides are congruent. c b a
Theorem 4 Theorem 4- -4 (Converse of Isosceles Triangle 4 (Converse of Isosceles Triangle Thm If two angles of a triangle are congruent, then the sides opposite the angles are congruent. Thm): ):
Theorem 4 Theorem 4- -5 5 The bisector of the vertex angle of an isosceles The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. triangle is the perpendicular bisector of the base. A A B B C C
Ex. 2) Ex. 2) Find the values of x and y. Find the values of x and y. M y x 63 L N
Corollary: Corollary: a statement that follows immediately from a theorem a statement that follows immediately from a theorem Corollary to Corollary to Thm Thm 4 4- -3: 3: If a triangle is equilateral, then it is equiangular. If a triangle is equilateral, then it is equiangular. Corollary to Corollary to Thm Thm 4 4- -4: 4: If a triangle is equiangular, then it is equilateral. If a triangle is equiangular, then it is equilateral.
Ex. 3) Ex. 3) BAC m = 38 C If find A A m B B C C M M
Ex. 4) Ex. 4) = A A 23 m BAM m CAM If find m BMA B B C C M M
Ex. 5) Ex. 5) = 2 x A A and x 3 = 20 m BAC If find x. B m B B C C M M
Ex. 6) Find the missing angles and sides. Ex. 6) Find the missing angles and sides.
