Upload
duongkhanh
View
221
Download
2
Embed Size (px)
Citation preview
4-5 Isosceles and Equilateral Triangles
Do Now
Lesson Presentation
Exit Ticket
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.
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.
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.
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.
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.
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.
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.
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,
4-5 Isosceles and Equilateral Triangles
legs of an isosceles triangle
vertex angle
base
base angles
Vocabulary
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.
4-5 Isosceles and Equilateral Triangles
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
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
4-5 Isosceles and Equilateral Triangles
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.
4-5 Isosceles and Equilateral Triangles
The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”
Reading Math
4-5 Isosceles and Equilateral Triangles
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.
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
4-5 Isosceles and Equilateral Triangles
The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.
4-5 Isosceles and Equilateral Triangles
Equilateral Triangle
Equiangular Triangle
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.
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.
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.
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.
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