Download CHAPTER 3: PARALLEL LINES AND PLANES

CHAPTER 3: PARALLEL
LINES AND PLANES
Section 3-4: Angles of a Triangle
TRIANGLES
•
•
•
A triangle is a figure formed by three
segments that join three non-collinear
points.
The three non-collinear points of
triangles are known as vertices (plural
for vertex).
The three segments that form the
triangle are known as sides.
TRIANGLES
Triangle QRS is shown ( QRS):
Q
• Vertices: points Q, R, and S
• Sides: QR, RS, and SQ
• Angles: Q,
R, and
S
S.
R
CLASSIFYING TRIANGLES
Triangles can be classified by the number of
congruent sides it has.
Scalene
Triangle
Isosceles
Triangle
No sides congruent.
At least 2 sides
congruent.
Equilateral
Triangle
All sides
congruent.
CLASSIFYING TRIANGLES
Triangles can also be classified by their
angles.
• Acute (3 acute angles):
• Obtuse
• Right
(1 obtuse angle):
(1 right angle):
• Equiangular
(All angles congruent):
AUXILIARY LINE
Definition:
An auxiliary line is a line, ray, or segment,
that is added to a diagram to help derive
information for proofs.
We will use an auxiliary line in the proof for
Theorem 3-11.
THEOREM 3-11 PROOF
D
Given:
4
ABC
Prove:
m
1+m
B
A
2+m
1
3 = 180
1. Through B draw BD
parallel to AC.
2. m DBC + m 5 = 180;
m DBC = m 4 + m 2
3. m 4 + m 2 + m 5 = 180
4. m 4 = m 1;
m 5=m 3
5. m 1 + m 2 + m 3 = 180
2 5
3
C
1.
Theorem 3-8
2.
Angle Addition Post.
3.
4.
Substitution Property
Theorem 3-2
5.
Substitution Property
THEOREM 3-11 & Corollaries
Theorem 3-11: The sum of the measures of
the angles of a triangle is 180.
Corollary 1: If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are
congruent.
3-11 COROLLARIES
Corollary 2: Each angle of an equiangular
triangle has measure 60.
Corollary 3: In a triangle, there can be at
most one right angle or obtuse angle.
Corollary 4: The acute angles of a right
triangle are complementary.
EXAMPLES
State whether a triangle is acute, right, or obtuse if
it has angles with measures:
1.
2.
3.
4.
5.
55, 43
47, 43
80, 75
33, 33
24, 66
1.
2.
3.
4.
5.
Acute
Right
Acute
Obtuse
Right
PRACTICE
Find the m A:
1.
1. 80
50
80
A
50
2.
2. 60
40
A
60
3.
40
3. 40
80
120
A
THEOREM 3-12
Theorem 3-12: The measure of an exterior
angle of a triangle equals the sum of the
measures of the two remote interior angles.
50
Remote Interior
Angles
Exterior Angle
120
60
70
CLASSWORK/HOMEWORK
• CW: Pg. 96, Classroom Exercises 1-15
• HW: Pg. 97, Written Exercises 2-18 even