Download 4-5 PPT Isosceles Triangles

Section 4-5 Isosceles and Equilateral Triangles
SPI 32C: determine congruence or similarity between triangles
SPI 32M: justify triangle congruence given a diagram
Objectives:
• Use and Apply Properties of isosceles triangles
Isosceles Triangle
Isosceles Triangle Theorem
Triangle Proofs
Examine the diagram below. Suppose that you draw XB YZ.
Can you use SAS to prove XYB
XZB? Explain.
It is given that XY XZ.
By the definition of perpendicular, XBY = XBZ.
By the Reflexive Property of Congruence, XB XB.
However, because the congruent angles are not included between the
congruent corresponding sides, the SAS Postulate does not apply.
You cannot prove the triangles congruent using SAS.
Triangle Proofs
Suppose that mL = y. Find the values of x and y.
MO
LN
The bisector of the vertex angle of an
isosceles triangle is the perpendicular
bisector of the base.
x = 90
Definition of perpendicular
mN = mL Isosceles Triangle Theorem
mL = y
Given
mN = y
Transitive Property of Equality
mN + mNMO + mMON = 180 Triangle Angle-Sum Theorem
y + y + 90 = 180 Substitute.
2y + 90 = 180 Simplify.
2y = 90
Subtract 90 from each side.
y = 45
Divide each side by 2.
Therefore, x = 90 and y = 45.
Triangle Proofs
Suppose the raised garden bed is a regular hexagon. Suppose that a segment is
drawn between the endpoints of the angle marked x. Find the angle measures of
the triangle that is formed.
Because the garden is a regular hexagon, the sides
have equal length, so the triangle is isosceles.
By the Isosceles Triangle Theorem, the unknown
angles are congruent.
The measure of the angle marked x is 120. The sum
of the angle measures of a triangle is 180.
If you label each unknown angle y, 120 + y + y = 180.
120 + 2y = 180
2y = 60
y = 30
So the angle measures in the triangle are 120, 30 and 30.