Download ISOSCELES AND EQUILATERAL TRIANGLES Isosceles Triangle

LESSON 4-5 NOTES: ISOSCELES AND EQUILATERAL TRIANGLES
Isosceles Triangle: A triangle with two congruent sides.
X
Vertex Angle
1 2
Leg
Leg
Base
Y
Base Angles
Z
B
Diagram for proof of Theorem 4-3
Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Proof of Theorem 4-3: Refer to the diagram of isosceles triangle XYZ above.
Given:
,
bisects ∠YXZ
Prove:
STATEMENTS
∠Y
∠Z
REASONS
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
EXAMPLES/PRACTICE: Find the value of a, b and c in each isosceles triangle.
1)
2)
3)
a
a
c
a
c
a
50
a
a
b 110
a
b
ac
a
a
a
b
a
75
EXAMPLES/PRACTICE: Find the value of x and the measure of each unknown angle in
each isosceles triangle.
40
3)
4)
5)
6x − 4
65
9x + 7
8x − 5
5(x + 5)
Remember that a converse of a statement "switches" the hypothesis ("if" part) and the conclusion
("then" part) of the statement.
Theorem 4-4 Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Theorem 4-5
If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular
bisector of the base.
bisects the vertex angle ∠YXZ. We proved earlier by SAS that
∆XYB ∆XZB. So by CPCTC, ∠XBY ∠XBZ and
.
Proof of Theorem 4-5:
X
∠XBY and ∠XBZ form a linear pair. By the Linear Pair Postulate,
m∠XBY + m∠XBZ = 180. Because ∠XBY ∠XBZ, both angles
are right angles.
1 2
Because
Y
B
Z
, B is the midpoint of
.
So
, which bisects the vertex angle, ∠YXZ, is the perpendicular
bisector of the base, , of isosceles triangle YXZ,
A corollary is a theorem that can be proved easily using another theorem. Since a corollary is a
theorem, you can use it as a reason in a proof.
Corollary to Theorem 4-3
If a triangle is equilateral, then the triangle is equiangular.
If:
then:
Corollary to Theorem 4-4
If a triangle is equiangular, then the triangle is equilateral.
If:
then:
Because the sum of the interior angles of a triangle is 180 degrees, each angle of an
equiangular/equilateral triangle is 60 degrees.
EXAMPLES/PRACTICE: Find the values of x and y in each.
1)
2)
x
60
y
8x - 12
y