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4-9 Isosceles and Equilateral Triangles
Objectives
Use CPCTC to prove parts of triangles
are congruent.
Prove theorems about isosceles and
equilateral triangles.
Apply properties of isosceles and
equilateral triangles.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Vocabulary
CPCTC
legs of an isosceles triangle
vertex angle
base
base angles
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
CPCTC
 an abbreviation for the phrase
“Corresponding Parts of Congruent
Triangles are Congruent.”
 It can be used as a justification in a
proof after you have proven two
triangles congruent.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Remember!
SSS, SAS, ASA, AAS, and HL
 use corresponding parts
 to prove triangles congruent
CPCTC
 uses congruent triangles
 to prove corresponding parts
congruent
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Example: Engineering Application
A and B are on the edges of a ravine.
What is AB?
One angle pair is congruent,
because they are vertical angles.
Two pairs of sides are congruent,
because their lengths are equal.
Therefore the two triangles are congruent by SAS.
By CPCTC, the third side pair is congruent.
AB = 18 mi.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Example: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ.
Prove: XYW  ZYW
Z
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
ZW
WY
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
YOU TRY
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
SOULTION
PR bisects QPS
and QRS
Given
RP  PR
Reflex. Prop. of 
∆PQR  ∆PSR
ASA
PQ  PS
CPCTC
Holt McDougal Geometry
QRP  SRP
QPR  SPR
Def. of  bisector
4-9 Isosceles and Equilateral Triangles
Example: Using CPCTC In the Coordinate Plane
Given: D(–5, –5), E(–3, –1), F(–2, –3),
G(–2, 1), H(0, 5), and I(1, 3)
Prove: DEF  GHI
Step 1
Plot the points on a
coordinate plane.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Step 2
Use the Distance Formula to find the lengths
of the sides of each triangle.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Step 3
Write statements justifying the triangles are
congruent and proving angles congruent.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
Recall:
 an isosceles triangle has at least two congruent
sides.
 The congruent sides are called the legs.
 The vertex angle is the angle formed by the
legs.
 The side opposite the vertex angle is called the
base.
 The base angles are the two angles that have
the base as a side.
Holt McDougal Geometry
4-9 Isosceles and Equilateral Triangles
B
1 and 2 are
the base angles.
3 is the
vertex angle.
𝑨𝑩 is the base.
A
𝑩𝑪 and 𝑨𝑪 are legs.
Holt McDougal Geometry
C