CH 4-2
Isosceles & Equilateral Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Warm Up
1. Find each angle measure.
60°; 60°; 60°
True or False. If false explain.
2. Every equilateral triangle is isosceles.
True
3. Every isosceles triangle is equilateral.
False; an isosceles triangle can have
only two congruent sides.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Objectives
Prove theorems about isosceles and
equilateral triangles.
Apply properties of isosceles and
equilateral triangles.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
California
Standards
12.0 Students find and use measures of
sides and of interior and exterior angles of triangles
and polygons to classify figures and solve problems.
15.0 Students use the Pythagorean theorem
to determine distance and find missing lengths of
sides of right triangles.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Vocabulary
legs of an isosceles triangle
vertex angle
base
base angles
Geometry
Isosceles & Equilateral Triangles
Triangle
CH 4-2
Geometry
Isosceles & Equilateral Triangles
Triangle External Angle
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Corollary: The measure of an exterior angle of a
triangle is greater than the measure of either of its
remote angles.
m4  m1 and m4  m2
Geometry
Isosceles & Equilateral Triangles
Triangle Angle Sum Thm.
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Geometry
Isosceles & Equilateral Triangles
Triangle Classification
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
A corollary is a theorem whose proof follows
directly from another theorem. Here are two
corollaries to the Triangle Sum Theorem.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Recall that 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, and the base angles are the two angles that
have the base as a side.
3 is the vertex angle.
1 and 2 are the base angles.
Geometry
Isosceles & Equilateral Triangles
Isosceles Triangle
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
If AB  AC , then B  C.
If E  F, then DE  DF .
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Bisector of the Vertex
The bisector of the vertex angle of an
isosceles triangle is the perpendicular
bisector of the base.
If AB  BC and BD bisects ABC,
then BD  AC and BD bisects AC.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Example 1: Finding the Measure of an Angle
Find mF.
mF = mD = x°
Isosc. ∆ Thm.
mF + mD + mA = 180 ∆ Sum Thm.
Substitute the
x + x + 22 = 180 given values.
Simplify and subtract
2x = 158 22 from both sides.
x = 79 Divide both
sides by 2.
Thus mF = 79°
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Example 2: Finding the Measure of an Angle
Find mG.
mJ = mG Isosc. ∆ Thm.
(x + 44) = 3x
44 = 2x
Substitute the
given values.
Simplify x from
both sides.
Divide both
sides by 2.
Thus mG = 22° + 44° = 66°.
x = 22
Geometry
CH 4-2
Isosceles & Equilateral Triangles
TEACH! Example 1
Find mH.
mH = mG = x°
Isosc. ∆ Thm.
mH + mG + mF = 180 ∆ Sum Thm.
Substitute the
x + x + 48 = 180 given values.
Simplify and subtract
2x = 132 48 from both sides.
x = 66 Divide both
sides by 2.
Thus mH = 66°
Geometry
CH 4-2
Isosceles & Equilateral Triangles
TEACH! Example 2
Find mN.
mP = mN Isosc. ∆ Thm.
(8y – 16) = 6y
2y = 16
y = 8
Substitute the
given values.
Subtract 6y and
add 16 to both
sides.
Divide both
sides by 2.
Thus mN = 6(8) = 48°.
Geometry
Isosceles & Equilateral Triangles
Equilateral Triangle
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
The following corollary and its converse show the
connection between equilateral triangles and
equiangular triangles.
Geometry
Isosceles & Equilateral Triangles
Equiangular Triangle
CH 4-2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Ex. 3: Using Properties of Equilateral Triangles
Find the value of x.
∆LKM is equilateral.
Equilateral ∆  equiangular ∆
(2x + 32) = 60
2x = 28
x = 14
The measure of each  of an
equiangular ∆ is 60°.
Subtract 32 both sides.
Divide both sides by 2.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Ex. 4: Using Properties of Equilateral Triangles
Find the value of y.
∆NPO is equiangular.
Equiangular ∆  equilateral ∆
5y – 6 = 4y + 12
y = 18
Definition of
equilateral ∆.
Subtract 4y and add 6 to
both sides.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Geometry
CH 4-2
Isosceles & Equilateral Triangles
TEACH! Example 3
Find the value of JL.
∆JKL is equiangular.
Equiangular ∆  equilateral ∆
4t – 8 = 2t + 1
2t = 9
t = 4.5
Definition of
equilateral ∆.
Subtract 4y and add 6 to
both sides.
Divide both sides by 2.
Thus JL = 2(4.5) + 1 = 10.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Remember!
A coordinate proof may be easier if you
place one side of the triangle along the
x-axis and locate a vertex at the origin or
on the y-axis.
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Example 5: Using Coordinate Proof
Prove that the segment joining the midpoints of
two sides of an isosceles triangle is half the base.
Given: In isosceles ∆ABC, X is the mdpt. of AB, and
Y is the mdpt. of BC.
Prove: XY =
1
AC.
2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Example 5 Continued
Proof:
Draw a diagram and place the coordinates as shown.
By the Midpoint Formula,
the coordinates of X are
(a, b), and Y are (3a, b).
By the Distance Formula,
XY = √4a2 = 2a, and AC
= 4a.
1
Therefore XY =
AC.
2
Geometry
CH 4-2
Isosceles & Equilateral Triangles
TEACH! Example 5
The coordinates of isosceles ∆ABC are
A(0, 2b), B(-2a, 0), and C(2a, 0). M is the midpoint
of AB, N is the midpoint of AC, and Z(0, 0), .
Prove ∆MNZ is isosceles.
Proof:
Draw a diagram and place the
coordinates as shown.
y
A(0, 2b)
M
N
x
B (–2a, 0)
Z
C (2a, 0)
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Check It Out! Example 6 Continued
By the Midpoint Formula, the coordinates. of M are
(–a, b), the coordinates. of N are (a, b), and the
coordinates of Z are (0, 0) . By the Distance
y
Formula, MZ = NZ = √a2+b2 .
A(0, 2b)
So MZ  NZ and ∆MNZ is isosceles.
M
N
Z
B(–2a, 0)
x
C(2a, 0)
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Lesson Quiz: Part I
Find each angle measure.
1. mR
28°
2. mP
124°
Find each value.
3. x
5. x
20
4. y
6
26°
Geometry
CH 4-2
Isosceles & Equilateral Triangles
Lesson Quiz: Part II
6. The vertex angle of an isosceles triangle
measures (a + 15)°, and one of the base
angles measures 7a°. Find a and each angle
measure.
a = 11; 26°; 77°; 77°
Geometry