Download Chapter 1 Points, Lines, Planes, and Angles page 1

Lesson 3-4
Angles of a Triangle
(page 93)
Essential Question
How can you apply parallel
lines (planes) to make
deductions?
Angles of a Triangle
.
A
.
B
.C
TRIANGLE:
The figure formed by 3 segments joining 3
.
A
B
.
.
∆ ACB
noncollinear
points.
∆ BAC
∆ BCA
∆ ABC
Symbol: ____________
∆ CAB
three points.
VERTEX: each of the ______
∆ CBA
C
Vertices:
A _____
B _____
C
_____
Sides:
_____ _____ _____
Angles:
∠A ∠B
_____
_____∠C
_____
AB BC AC
Classifications of Triangles by Sides
Scalene Triangle
No sides are congruent .
Classifications of Triangles by Sides
Isosceles Triangle
At least two sides are congruent .
Classifications of Triangles by Sides
Equilateral Triangle
All sides are congruent .
Classifications of Triangles by Angles
Acute Triangle
Three acute angles.
Classifications of Triangles by Angles
Right Triangle
One right (90º) angle.
Classifications of Triangles by Angles
Obtuse Triangle
One obtuse angle.
Classifications of Triangles by Angles
Equiangular Triangle
All angles are congruent .
Auxiliary Line:
A line (ray or segment) added to a
diagram to help in a
proof .
 Please note that this is to HELP in a proof.
 This does not give you license to add lines to every diagram.
 There are times when this may be done, but please BEWARE!
Theorem 3-11
The sum of the measures of the
angles of a triangle is 180º .
B
Given: ∆ ABC
Prove: m∠1 + m∠2 + m∠3 = 180º
1
A
2
3
C
D
Given: ∆ ABC
B

4
Prove: m∠1 + m∠ 2 + m∠3 = 180º
2
5
See page 94!
Proof:
Statements
A
1
3
Reasons
C
1.
Through B draw line BD parallel to line AC
Through a point outside a line, there is exactly one line
parallel to the given line.
2.
m∠ DBC +m∠5 = 180º
__________________________________________
∠ - Addition Postulate
_____________________________________________
3.
m∠ DBC = m∠2 + m∠4
m∠2 + m∠4 + m∠5 = 180º
__________________________________________
4.
∠1  ∠4 OR m∠1 = m∠4
__________________________________________
_____________________________________________
5.
m∠1 + m∠2 + m∠ 3 = 180º
__________________________________________
_____________________________________________
∠3  ∠5 OR m∠3 = m∠ 5
Substitution Property
_____________________________________________
|| - lines ⇒ AIA 
Substitution Property
Example # 1.
Find the value of “x”.
xº
x + 40 + 90 = 180
x + 130 = 180
50
x = _____
40º
Example # 2.
Find the value of “x”.
x + 100 + 35 = 180
xº
x + 135 = 180
45
x = _____
100º
35º
Example # 3.
Find the value of “x”.
x + x + 70 = 180
xº
2x + 70 = 180
2x = 110
70º
xº
55
x = _____
COROLLARY:
A statement that can be
proved easily
by applying a theorem.
Corollary 1
If two angles of one triangle are congruent
to two angles of another triangle,
congruent .
then the third angles are
A
B
X
Y
Z
C
If ∠A  ∠X and ∠B  ∠Y, then ∠C ∠Z
Corollary 2
Each angle of an equiangular triangle
has a measure 60º .
xº
xº
xº
If 3 x = 180, then x = 60.
Corollary 3
In a triangle, there can be at most one
right angle or obtuse angle.
Sum = 90º
Sum < 90º
m > 90º
Corollary 4
The acute angles of a right triangle
are complementary .
Sum = 90º
If 2 ∠’s sum = 90º, then they are complementary.
Example # 4.
Find the value of “x”.
40º
50º
xº
40º
50
x = _____
Example # 5.
Find the value of “x”.
xº
xº
xº
60
x = _____
EXTERIOR ANGLE: (of a triangle) the angle
formed when one side of a triangle is
extended .
Example:
∠4
4
3
Example:
2
∠1 & ∠2
1
REMOTE INTERIOR ANGLES: the angles of
the triangle not adjacent to the
exterior angle.
EXTERIOR ANGLE:
Example:
∠4
1
4
2
3
REMOTE INTERIOR ANGLES:
Example:
∠1 & ∠2
Theorem 3-12
The measure of an exterior angle of a triangle equals the
sum
of the measures of the two remote interior angles.
B
2
1
A
Given: ∆ ABC
Prove: m∠1 + m∠2 = m∠ 4
3
4
C
Given: ∆ ABC
B
Prove: m∠1 + m∠2 = m∠4
2
See page 96
C.E. #15!
1
Proof:
A
3
4
C
1.
Statements
m∠1 + m∠2+ m∠ 3 = 180º
__________________________________________
Reasons
Sum of m. of∠‘s of ∆ = 180º
_____________________________________________
2.
m∠3 + m∠ 4 = 180º
__________________________________________
_____________________________________________
3.
m∠1+
m∠2 + m∠3 = m∠3 + m∠4
__________________________________________
_____________________________________________
4.
m∠3 = m∠3
__________________________________________
_____________________________________________
5.
m∠1 + m∠ 2 = m∠4
__________________________________________
_____________________________________________
∠- Addition Postulate
Substitution Property
Reflexive Property
Subtraction Property
Example # 6.
Find the value of “x”.
80º
x + 80 = 120
40
x = _____
120º
xº
Example # 7.
Find the value of “x”.
3x = x + 100
2x = 100
xº
50
x = _____
100º
3xº
Example # 8.
Find the value of “x”.
60º
x = 90 + 60
150
x = _____
xº
Assignment
Written Exercises on pages 97 to 99
RECOMMENDED: 1 to 9 odd numbers
REQUIRED: 10, 11, 13, 15, 17, 18, 19, 20, 25, 26
Do the Paper Triangle Proofs
How can you apply parallel
lines (planes) to make
deductions?