Isosceles and Equilateral Triangles

Academic Geometry

Isosceles and Equilateral Triangles

Draw a large isosceles triangle ABC, with exactly two congruent sides, AB and AC.

What is symmetry?

How many lines of symmetry does it have?

Label the point of intersection D.

Isosceles and Equilateral Triangles

What is the relationship between AD and BC?

Isosceles and Equilateral Triangles

Draw a Triangle XYX with exactly two congruent angles, <Y and <Z. Find the line of symmetry.

What can you conclude about the sides?

Isosceles Triangle Theorems

The congruent sides of an isosceles trianlge are its legs.

The third side is the base.

The two congruent sides form the vertex angle.

The other two angles are base angles.

Theorem 4-3

Isosceles Triangle Theorem

The base angles of an isosceles triangle are congruent.

If the two sides of a triangle are congruent, then the angles opposite those sides are congruent.

a b

c

Theorem 4-4

Converse of Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

a b

c

Theorem 4-5

The line of symmetry for an isosceles triangle bisects the vertex angle and is the perpendicular bisector of the base.

CD AB

CD bisects ABa b

c

d

Example

EC is a line of symmetry for isosceles triangle MCJ.

Draw and label the triangle.

M<MCJ = 72. Find m<MEC, m<CEM and EJ.

ME = 3

Proof of the Isosceles Triangle Theorem

Begin with isosceles triangle XYZ. XY is congruent XZ. Draw XB, the bisector of the vertex angle YXZ

Prove <Y congruent <Z

Statements Reasons

Using the Isosceles Triangle Theorems

Why is each statement true?

<WVS congruent <S

TR congruent TS

Can you deduce that Triangle RUV is isosceles? Explain

t

u

r

w

sv

Using Algebra

Find the value of y m

lo n

y

63

Equilateral Triangles

Draw a large equilateral triangle, EFG.

Find all the lines of symmetry. How many are there?

What do we know about the sides?

The angles?

Isosceles and Equilateral Triangles

We learned in the last chapter that equilateral triangles are also isosceles.

A corollary is a statement that immediate follows from a theorem.

Corollary to Theorem 4-3

If a triangle is equilateral, then the triangle is equiangular.

<X is congruent to <Y is congruent to <Z

x

y

z

Corollary to Theorem 4-4

If the triangle is equiangular, then the triangle is equilateral.

XY is congruent to YZ is congruent to ZX

x

y

z