Transcript Lesson 1 Contents

Lesson 4-6
Isosceles Triangles
5-Minute Check on Lesson 4-5
Transparency 4-6
Refer to the figure. Complete each congruence statement and the
postulate or theorem that applies.
1. WXY  _____ by _____.
2. WYZ  _____ by _____.
3. VWZ  _____ by _____.
4.
What additional congruence statement is
necessary to prove RST  UVW by the ASA Postulate?
Standardized Test Practice:
A
T  W
B
R  U
C
ST  UW
D
RT  VW
5-Minute Check on Lesson 4-5
Transparency 4-6
Refer to the figure. Complete each congruence statement and the
postulate or theorem that applies.
1. WXY   VZY by ASA .
or AAS
2. WYZ   VYX by AAS .
3. VWZ   WVX by ASA .
or SAS
4.
AAS, SSS
What additional congruence statement is
necessary to prove RST  UVW by the ASA Postulate?
Standardized Test Practice:
A
T  W
B
R  U
C
ST  UW
D
RT  VW
Objectives
• Use properties of isosceles triangles
• Use properties of equilateral triangles
Vocabulary
• Vertex angle – the angle formed by the two
congruent sides
• Base angle – the angle formed by the base
and one of the congruent sides
Theorems
•
Isosceles Triangle Theorem: If two sides of
a triangle are congruent, then the angles
opposite those sides are congruent.
•
Converse of Isosceles Triangle Theorem: If
two angles of a triangle are congruent, then
the sides opposite those angles are
congruent.
Corollaries
•
A triangle is equilateral if, and only if, it is
equiangular
•
Each angle of an equilateral triangle
measures 60°
Isosceles Triangle
B
Vertex angle
leg
A
leg
base
C
Base Angles
A  C
A + B + C = 180°
Write a two-column proof.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Def. of
3. ABC and BCD are
isosceles
4.
3. Def. of isosceles 
5.
6.
5. Given
6. Substitution
segments
4. Isosceles  Theorem
Write a two-column proof.
Given:
.
Prove:
Proof:
Statements
1.
Reasons
1. Given
2. ADB is isosceles.
2. Def. of isosceles triangles
3.
4.
3. Isosceles  Theorem
4. Given
5.
6. ABC ADC
7.
5. Def. of midpoint
6. SAS
7. CPCTC
Multiple-Choice Test Item
If
and
measure of
A. 45.5
B. 57.5
C. 68.5
D. 75
Read the Test Item
CDE is isosceles with base
isosceles with
what is the
Likewise, CBA is
Solve the Test Item
Step 1 The base angles of CDE are congruent. Let
Angle Sum Theorem
Substitution
Add.
Subtract 120 from
each side.
Divide each side by 2.
Step 2
are vertical angles so they have
equal measures.
Def. of vertical angles
Substitution
Step 3 The base angles of CBA are congruent.
Angle Sum Theorem
Substitution
Add.
Subtract 30 from each
side.
Divide each side by 2.
Answer: D
Multiple-Choice Test Item
If
and
measure of
A. 25
Answer: A
B. 35
what is the
C. 50
D. 130
Name two congruent angles.
Answer:
Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the
sides opposite congruent angles are congruent. So,
Answer:
a. Name two congruent angles.
Answer:
b. Name two congruent
segments.
Answer:
ABC is an equilateral triangle.
a. Find x.
Answer: 30
b.
Answer: 90
bisects
Summary & Homework
• Summary:
– Two sides of a triangle are congruent if, and only
if, the angles opposite those sides are congruent.
– A triangle is equilateral if, and only if, it is
equiangular.
• Homework:
– pg 219 - 20: 9, 10, 13-18, 27