Aim: Isosceles Triangle Course: Applied Geometry

Aim: What is an Isosceles Triangle?

Do Now:

What type of triangle has sides of 3, 6, 8?

Aim: Isosceles Triangle Course: Applied Geometry

TrianglesA triangle is a three sided polygon enclosing three

angles.The sum of the measure of the angles of a triangle

is 180 degrees (1800)

3 equal 2 equal No equal sides sides sides

Aim: Isosceles Triangle Course: Applied Geometry

Isosceles Triangle

A triangle with two sides that are equal in length.

AB BC

A C

Bleg

Base anglesBase

Base angles of an isosceles triangle are congruent

Isosceles Triangle

leg

Aim: Isosceles Triangle Course: Applied Geometry

The Special Lines of a Triangle

Altitude

BH is an altitude from B to AC

Altitude of a Triangle - A line segment from a vertex and perpendicular to the opposite side.

Angle Bisector

BQ is the bisector of B: mABQ = mCBQ Angle bisector of a triangle - A line segment that divides an angle of a triangle into two halves.

Aim: Isosceles Triangle Course: Applied Geometry

Median

BM is the median from B to the midpoint of AC: AM = MC

Median of a triangle - A line segment from a vertex of a triangle to the midpoint of the

opposite side.

Special lines of various triangles

Aim: Isosceles Triangle Course: Applied Geometry

Special Lines of an Isosceles Triangle

E G

F

H

Altitude - line segment from a vertex and perpendicular to the opposite side.

Median - A line segment from a vertex to the midpoint of the opposite side.

Angle bisector - A line segment that divides an angle of a triangle into two halves.

In an isosceles triangle, all of three of these lines, drawn from the vertex angle, are the same line.

Aim: Isosceles Triangle Course: Applied Geometry

Complete each statement. Explain.

Model Problem

K

J

I

M

L

Na. _____ b. ____

. _____

KI KN

c ML

Aim: Isosceles Triangle Course: Applied Geometry

Find the measure of the vertex angle of an isosceles triangle if a base angle measures:

A. 44o

180o - (44o + 44o)180o - (88o) = 92o

92o

44o 44o

Find the measure of the base angles of an isosceles triangle if the vertex angle measures:

B. 44o

180o - 44o = 136o

2x = 136o

x = 68o

xo xo68o 68o

44o

Model Problem

Aim: Isosceles Triangle Course: Applied Geometry

Triangle ABC is isosceles with AB BC, AB = 3x - 2 and BC = 5x – 14. Find the value of x:

3x - 2 = 5x - 14-3x -3x - 2 = 2x - 14 +14 +14 +12 = 2x 6 = x

3(6) - 2= 163x - 2

5x - 145(6) - 14= 16

Model Problem

A

B

C

3x -

2 5x - 14

16 16

Aim: Isosceles Triangle Course: Applied Geometry

Model Problem

The measure of the vertex angle of an isosceles triangle is 100o. Find the number of degrees in one of the base angles of the triangle.

If the degree measure of each angle of a triangle is 60, which of the following statements is false?

a) The triangle is equiangularb) The triangle is equilateralc) The triangle is scalene.d) The sum of the measure of the interior

angles of the triangle is 1800.

Find the degree measure of each of the acute angles of an isosceles right triangle.

Aim: Isosceles Triangle Course: Applied Geometry

Model Problem

Find the values of x and y.

y

x

63 D

B

C

AABC is isoscelesBC AB

CBD ABD

C A = 63

What the diagram tells me:

Base angles of an isosceles triangle are congruent

mCBD = 54 = mx

mCDB = 90 = m yIn an isosceles triangle, the angle bisector and altitude drawn from the vertex angle, are the same line.

angle bisector

63 + 90 + CBD = 180 Sum of angles of a triangle equal 180.

( x)90

Aim: Isosceles Triangle Course: Applied Geometry

Equilateral Triangle

P O

R

If a triangle is equilateral, then it is equiangular with each angle of the triangle measuring 60o.

An equilateral triangle has three equal sides.

All three special lines drawn from the each angle of an equilateral triangle are the same line.

Aim: Isosceles Triangle Course: Applied Geometry

Model Problem

x

Find x.

Find x.

65

x

Aim: Isosceles Triangle Course: Applied Geometry

Model Problem

Find x.

VQ and YZ are angle bisectors

x

Z

V Y

W

m n

50

Find m & n.

Aim: Isosceles Triangle Course: Applied Geometry

In triangle ABC, mA = x – 2, mB = 3x + 20 and mC = 5x. Find the value of x and the measure of each angle

x – 2 + 3x + 20 + 5x = 1809x + 18 = 180 - 18 - 189x = 1629 9 x = 18

mA = x - 2

mB = 3x + 20

mC = 5x

18 - 2 = 16

3(18) + 20 = 74

5(18)= 90

What type of triangle is this?

Right Triangle

Model Problem

mA + mB + mC = 180.