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Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now...

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4-5 Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket
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Page 1: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Do Now

Lesson Presentation

Exit Ticket

Page 2: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Warm Up # 3

1. Find each angle measure.

True or False. If false explain.

2. Every equilateral triangle is isosceles.

3. Every isosceles triangle is equilateral.

60°; 60°; 60°

True

False; an isosceles triangle can have only two congruent sides.

Page 3: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Knowledge: Justify Mathematical Argument (1)(G)

A builder using the truss shown at the right claims that ACB will have the same measure as ADB. 𝑨𝑪 and 𝑨𝑫represent identical beams, and 𝑨𝑩 bisects CAD. Is the builder correct? Justify your answer.Yes. The builder is correct.

It is given that 𝐴𝐶 ≌ 𝐴𝐷 and by definition of angle bisectors, CAB ≌ DAB.

By the Reflexive Prop. of ≌, 𝐴𝐵 ≌ 𝐴𝐵.

Thus, ∆ACB ≌ ∆ADB by SAS Postulate.

ACB ≌ ADB because of CPCTC.

Page 4: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Knowledge: Making a Conjecture

A.Construct congruent segments to make a conjecture about the angles opposite the congruent sides in an isosceles triangle.

Step 1: Construct an isosceles ∆ABC on your paper, with 𝐴𝐶 ≅ 𝐵𝐶.

Page 5: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Know: Making a Conjecture

Construct congruent segments to make a conjecture about the angles opposite the congruent sides in an isosceles triangle.

Step 2: Fold the paper so that the two congruent sides fit exactly one on top of the other. Create the paper. Notice that A and B appear to be congruent.

Page 6: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Think: How can folding a piece of paper help you tell if two angles are congruent?

Communicate: Connect Mathematical Ideas (1)(F)

When folding the paper, congruent angles will fit exactly one on top of the other.

Page 7: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Knowledge: Making a Conjecture

Angles opposite the congruent sides in an isosceles triangle are congruent.

Write a conjecture that you observed for the angles opposite the congruent sides in an isosceles triangle.

Page 8: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Knowledge: Making a Conjecture

Sides opposite the congruent angles in an isosceles triangle are congruent.

Write a conjecture that you observed for the sides opposite the congruent angles in an isosceles triangle.

Page 9: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Connect to Math

SWBAT 1. Prove theorems about isosceles and equilateral triangles.

2. Apply properties of isosceles and equilateral triangles.

By the end of today’s lesson,

Page 10: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

legs of an isosceles triangle

vertex angle

base

base angles

Vocabulary

Page 11: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and 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.

Page 12: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Page 13: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 1: Proving the Isosceles Triangle Theorem

Begin with isosceles ∆XYZ with 𝑿𝒀 ≅ 𝑿𝒁. Draw 𝑿𝑩, the bisector of vertex angle YXZ.

Given: 𝑋𝑌 ≅ 𝑋𝑍, 𝑋𝐵 bisects YXZProve: Y ≌ Z

Statements Reasons

4. SAS Postulate Steps 1, 2, 34. ∆XYB ∆XZB

3. Reflex. Prop. of

2. Definition of angle bisector2. 1 2

1. Given1. 𝑋𝑌 𝑋𝑍; 𝑋𝐵 bisects YXZ

3. 𝑋𝐵 𝑋𝐵

5. CPCTC5. Y ≌ Z

Page 14: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

A builder using the truss shown at the right claims that ACB will have the same measure as ADB. 𝑨𝑪 and 𝑨𝑫represent identical beams, and 𝑨𝑩 bisects CAD. Is the builder correct? Justify your answer.

Yes. The builder is correct.

It is given that 𝐴𝐶 ≌ 𝐴𝐷 by the Isosceles Triangle Theorem.

Example 2: Proving the Isosceles Triangle Theorem

Page 15: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Page 16: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 3:

Using the Isosceles Triangle Theorem and its Converse

A. Is 𝑨𝑩 congruent to 𝑪𝑩 ? Explain.

B. Is A congruent to DEA ? Explain.

Yes. Since C ≌ A, 𝐴𝐵 ≅ 𝐶𝐵 by the Converse of the Isosceles Triangle Theorem.

Yes. Since 𝐴𝐷 ≅ 𝐸𝐷, A ≌ DEA by the Isosceles Triangle Theorem.

Page 17: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”

Reading Math

Page 18: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Page 19: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 4: Using Algebra

What is the value of x ?

Since 𝐴𝐵 ≅ 𝐶𝐵, ∆ABD is isosceles ∆. By the Isosceles ∆ Theorem A ≌ C.

mC = 54o

Since 𝐵𝐷 bisects ABC, you know by Theorem 4-5 that 𝐵𝐷 ⊥ 𝐴𝐶. So, BDC = 90o.

mC + mBDC + mDBC = 180o

54 + 90 + x = 180o

x = 36o

∆ Sum Theorem.

Substitute.

Subtract 144 from each side.

Page 20: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 5: Complete each Statement.

Explain why it is true.

a. 𝑽𝑻 ≅ ________

b. 𝑼𝑻 ≅ ________ ≌ 𝒀𝑿

c. 𝑽𝑼 ≅ ________

d. 𝑽𝒀𝑼 ≅ ________

𝑉𝑋 Converse of Isosceles ∆ Theorem

𝑈𝑊 Converse of Isosceles ∆ Thrm.

𝑉𝑌 Converse of Isosceles ∆ Thrm.

and Segment Addition Post.

𝑉𝑈𝑌 Isosceles ∆ Theorem

Page 21: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.

Page 22: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Equilateral Triangle

Equiangular Triangle

Page 23: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

B

F

D

A C E G

40o

yo

xo

A. What is the value of x ?

Example 6: Using Algebra

Because x is the measure of an angle

in an equilateral triangle, x = 60o.

Page 24: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

B

F

D

A C E G

40o

yo

xo

B. What is the value of y ?

Example 6: Using Algebra

mDCE + mDEC + mEDC = 180.

60 + 70 + y = 180

y = 50

∆ Sum Theorem.

Substitute.

Subtract 130 from each side.

Page 25: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 7: Using Algebra

A. What is the value of x ?

B. What is the value of y ?

It is given that the triangle is an isosceles ∆. Thus, the base angles are congruent. Since 110o and the base angle to yare linear pair.

x + 2y = 180o

x = 40o

∆ Sum Theorem.

Substitute.

Subtract 140 from each side.

x + 2(70) = 180

Hence, y = 70o by Linear Pair Postulate.

Page 26: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Example 8: Using Algebra

The vertex angle of an isosceles triangle measures (a + 15)°, and one of the base angles measures 7a°. Find a and each angle measure.

Therefore, each angle measure is 26°; 77°; 77°

(a + 15)°

7a° 7a°

a + 15 + 7a + 7a = 180o

15a + 15 = 180

15a = 165

∆ Sum Theorem.

Combined Like Terms

Subtract 15 from each side.

a = 11 Subtract 15 from each side.

Page 27: Do Now Lesson Presentation Exit Ticket - Uplift … Isosceles and Equilateral Triangles Do Now Lesson Presentation Exit Ticket 4-5 Isosceles and Equilateral Triangles Warm Up # 3 1.

4-5 Isosceles and Equilateral Triangles

Exit Ticket:

Find each angle measure.

1. mR

2. mP

Find each value.

3. x 4. y

5. x


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