5-2 Isosceles & Equilateral Triangles

5.2 Isosceles and Equilateral Triangles

I can apply theorems about isosceles and equilateral triangles to solve problems.

What is an Isosceles Triangle?

An isosceles triangle is a triangle with at least two sides of the same length.

In an isosceles triangle, the angle between the two sides of equal length is called the vertex angle, and the other two angles are called base angles. The side between the two base angles is called the base of the isosceles triangle.

Labelling

Label the Isosceles Triangle with these words:

Drag labels to their correct position on the image

(Left) Base Angle

Vertex Angle

(Rt) Base Angle

(Right) Leg

Base

(Left) Leg

Draw

How are the base angles of an isosceles triangle related?

Let’s use Reflectional Symmetry. 

We know that the legs are equal. 

*Draw in the line of Symmetry with this in mind.*

Multiple Choice

An architect is designing a community park between N. First St. and S. First St. The pathways on either side of the pool will be equal in length and will provide effective access and circulation around the pool. To protect the landscaping and to minimize erosion, the architect will place a triangular section of triangular cobblestones at the corners along Park Plaza.

What angle measure should the architect specify for the corners in her design?

• 50

• 65

• 130

• 30

Multiple Choice

What is the value of x?

• 13.4

• 28.6

• 23

• 59

Example 2: What is the value of x?

Multiple Choice

What is the value of x? 

• 3

• 43.5

• 1.5

• 1

Example 3: What is the value of x?

Multiple Choice

Think about which two sides must be equal. Set up an equation. Solve for x.

What are the lengths of all three sides of the triangle?

• PQ = 60,

  PR = 32,

  RQ = 32

• PQ = 26.4,

  PR = 26.4,

  QR = 17.1

• PQ = 9.6,

  PR = 23.6,

  QR = 9.6

Example 4:
What are the lengths of all three sides of the triangle?

Multiple Choice

Is $$\overline{AB}\$$ congruent to $$\overline{CB}$$ ?

• YES

• NO

Example 5:

Yes, so by

the Converse of the

Isos. ∆ Thm

Multiple Choice

Is $$\angle A$$ congruent to $$\angle DEA$$ ?

• YES

• NO

Example 6:

Yes, so by
the Isos. ∆ Thm

Let’s go back to Symmetry. Where is the line of
symmetry in an Isosceles Triangle?

What is created when you fold on this line?

What kind of angles are created?

Can this line be called something else?

Answers are on the next slides...

Vertex Angle Bisector Theorem

Multiple Choice

$$m\angle A=54\degree,\ m\angle DBC=x$$

Use the given picture to solve for the value of x.

• 54

• 90

• 36

• 72

Example 7:

There are a few ways to find the value of x. I will only show one method. If you found x another way please ask.

By the Vertex Angle Bisector Theorem
we know that BD is a perpendicular
bisector creating a 90 degree angle.

What is an Equilateral Triangle?

An equilateral triangle is a triangle with three sides of the same length.

Corollary to The Isosceles Triangle Theorem

An equilateral triangle is equiangular.

An equiangular triangle is equilateral.

Multiple Choice

If $$m\angle CBD=130,$$ what is $$m\angle BAD?$$

• 60

• 70

• 130

• 55

Example 8:

Open Ended

Explain why or why not an equilateral triangle is an isosceles triangle.