Isosceles and Equilateral Triangle Theorems

Use Isosceles and Equilateral Triangles

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Multiple Choice

Classify the triangle by its sides: 2cm, 2cm, 2cm

• equilateral

• isosceles

• scalene

Multiple Choice

Classify the triangle by its sides: 7ft, 11ft, 7ft

• equilateral

• isosceles

• scalene

Multiple Choice

Classify the triangle by its sides: 9m, 8m, 10m

• equilateral

• isosceles

• scalene

Math spoken here.

Vocabulary:

Legs: when an isosceles triangle has exactly two congruent sides, these sides are called legs.

Vertex angle: the angle formed by the legs.

Base: The third side of the isosceles triangle.

Base Angle: The two angles adjacent to the base.

Multiple Choice

Which segment is the base?

• Segment AB

• Segment BC

• Segment AC

Multiple Choice

Which angle is the vertex angle?

• <C

• <A

• <B

Theorems:

*Base Angles Theorem

*Converse of the Base Angles Theorem

Base Angles Theorem:

If two sides are congruent, then the angles opposite them are congruent.

Converse of the Base Angles Theorem:

If two angles are congruent, then the sides opposite them are congruent.

Multiple Choice

For isosceles triangles, when we are given that two sides are congruent we can prove that ______________________.

• two angles are congruent

• the third side is congruent

• three angles are congruent

• no other relationships exist

Multiple Select

$$In\ \Delta DEF,\ \overline{DE\ }\approx\overline{DF}.\$$  Use the base angles theorem to choose the two congruent angles.

• $$\angle D$$  

• $$\angle E$$  

• $$\angle F$$  

Multiple Choice

Complete the statement.

$$If\ \overline{HG}\approx\overline{HK},\ then\ \angle HGK\ \approx\angle\ldots$$  

• GHK

• KGH

• HKG

• GKH

Multiple Choice

Complete the statement.  $$If\ \angle KHJ\approx\angle KJH,\ then\ \overline{KH}\approx\ldots$$  

• $$\overline{KH}$$  

• $$\overline{KJ}$$  

• $$\overline{JH}$$  

Finding Measures: Angles

Isosceles Triangles

Multiple Choice

Use the triangle sum theorem and the base angles theorem to find a.

• 45

• 50

• 80

• 25

Finding Measures: Legs

Isosceles Triangles

By the converse of the base angles theorem c = 14.

Multiple Choice

Find the measure of the missing side.

• 7

• 13

• 20

• 14.8

Corollaries

Recall:

Equilateral triangles are special cases of isosceles triangles that have three congruent sides.

Multiple Choice

$$Find\ m\angle P.$$  

• 60

• 80

• 45

• 90

Challenge!

Find the value of x and y.

Multiple Choice

Find the value of x and y.

• x = 50, y = 1/2

• x = 1/2, y = 50

• x = 50, y = 2

• x = 2, y = 50

Takeaways:

1. If two sides of a triangle are congruent, then the angles opposite are also congruent. The converse is also true.
2. If a triangle is equilateral, then it is also equiangular.

Multiple Choice

What is the value of the missing angle?

• 61
• 50
• 60
• 69

Multiple Choice

What is the measure of BC?

• 12

• 6

• 9

• 10

Multiple Choice

Find x

• 90
• 45
• 22
• 35

Multiple Choice

Find x

• 10
• 32
• 116
• 16

Multiple Choice

Solve for x.

• 72^o

• 144^o

• 46^o

• 36^o

Fill in the Blanks

What is the

$m\angle U$  ?

Multiple Choice

Find the value of x. (Hint: What type of triangle is this?)

• 4

• 1

• -4

• 10

Multiple Choice

Find the value of y.

• 60

• 30

• 20

• 12

Multiple Choice

Find the value of x.

• $$30\degree$$

• $$45\degree$$

• $$60\degree$$

• $$12\degree$$

Multiple Choice

Find the value of x. (Hint: what kind of triangle is this?)

• $$43\degree$$

• $$47\degree$$

• $$94\degree$$

• $$137\degree$$

Multiple Choice

Find the value of y.

• 5

• 10

• 15

• 20

Multiple Choice

Triangle KLM is equilateral with KM = x + 2, LM = 12 – x, and KL = 4x – 13.

Find the value of x.

(Hint: draw a picture)

• 2

• 3

• 4

• 5