Download Vertical Angles

7-5 Triangles
Warm Up
Problem of the Day
Lesson Presentation
Course 1
Warm Up
1. What are two angles whose sum is
90°? complementary angles
2. What are two angles whose sum is
180°? supplementary angles
3. A part of a line between two points is
called a _________.
segment
4. Two lines that intersect at 90° are
______________.
perpendicular
Problem of the Day
Find the total number of shaded triangles
in each figure.
3
6
10
Problem of the Day
Find the total number of Total triangles in
each figure.
5
13
24
Learn to classify triangles and solve
problems involving angle and side
measures of triangles.
Insert Lesson Title Here
Vocabulary
acute triangle
obtuse triangle
right triangle
scalene triangle
isosceles triangle
equilateral triangle
A triangle is a closed figure with three line
segments and three angles. Triangles can be
classified by the measures of their angles. An
acute triangle has only acute angles. An
obtuse triangle has one obtuse angle. A
right triangle has one right angle.
Acute triangle
Obtuse triangle
Right triangle
To decide whether a triangle is acute, obtuse,
or right, you need to know the measures of
its angles.
The sum of the measures of the
angles in any triangle is 180°. You
can see this if you tear the
corners from a triangle and
arrange them around a point on a
line.
By knowing the sum of the measures of
the angles in a triangle, you can find
unknown angle measures.
Additional Example 1: Application
Sara designed this triangular trophy. The
measure of E is 38°, and the measure
of F is 52°. Classify the triangle.
To classify the triangle, find the
measure of D on the trophy.
m
D = 180° – (38° + 52°)
E
D
F
D = 180° – 90° Subtract the sum of the
known angle measures
m D = 90°
from 180°
So the measure of D is 90°. Because DEF has
one right angle, the trophy is a right triangle.
m
Try This: Example 1
Sara designed this triangular trophy. The
measure of E is 22°, and the measure
of F is 22°. Classify the triangle.
To classify the triangle, find the
measure of D on the trophy.
m
E
D
F
D = 180° – (22° + 22°)
m D = 180° – 44° Subtract the sum of the
known angle measures
m D = 136°
from 180°
So the measure of D is 136°. Because DEF has
one obtuse angle, the trophy is an obtuse triangle.
You can use what you know about
vertical, adjacent, complementary,
and supplementary angles to find
the measures of missing angles.
Take two pencils (or pens)
and have them intersect them
in front of you like this…
Finger Dance
(Geometry)
Finger Dance
(Geometry)
Adjacent Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Adjacent Angles
Finger Dance
(Geometry)
Adjacent Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Adjacent Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Vertical Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Supplementary Angles
Finger Dance
(Geometry)
Supplementary Angles
For today’s warm-up, take out your list of vocabulary words, and TURN THEM OVER
SO YOU CAN’T SEE THEM. Then, answer these three questions alone!
1) What are complementary angles?
2) What are supplementary angles?
3) What is the sum of all the angle measurements in a triangle?
Supplementary Angles
?
50
Get a Partner and see if you can
make Complimentary angles with
your hands/arms…
Find
Complimentary
Angles
Find
Complimentary
Angles
Find
Complimentary
Angles
When angles have the same measure, they are
said to be congruent.
M
N
20°
P
160°
R
160°
20°
Q
Vertical angles are formed opposite each other
when two lines intersect. Vertical angles have
the same measure, so they are always
congruent.
Adjacent angles are side by side and have a
common vertex and ray. Adjacent angles may or
may not be congruent.
M
N
20°
P
160°
R
160°
20°
Q
Identify the type of each angle pair shown.
A.
5
6
They are vertical angles.
Identify the type of each angle pair shown.
B.
7
8
7 and 8 are side by side and
have a common vertex and
ray.
They are adjacent angles.
Identify the type of each angle pair shown.
A.
3 and 4 are side by side and
have a common vertex and
ray.
3
4
They are adjacent angles.
Identify the type of each angle pair shown.
B.
7
8
7 and 8 are opposite each other and
are formed by two intersecting lines.
They are vertical angles.
Complementary angles are two angles
whose measures have a sum of 90°.
65° + 25° = 90°
LMN and
NMP are complementary.
L
N
65°
25°
M
P
Supplementary angles are two angles whose
measures have a sum of 180°.
65° + 115° = 180°
GHK and
KHJ are supplementary.
K
65°
G
115°
H
J
Find each unknown angle measure.
A. The angles are complementary.
71° + a =
–71°
a=
90°
–71°
19°
The sum of
the measures
is 90°.
a
71°
Find each unknown angle measure.
B. The angles are
supplementary.
125° + b = 180°
–125°
–125°
b=
55°
The sum of
the measures
is 180°.
125° b
Find each unknown angle measure.
C. The angles are
vertical angles.
c = 82°
Vertical angles
are congruent.
c
82°
Q
Do you see any “straight
angles” in this figure? Where?
P
T
What measure do
supplementary angles have?
180 degrees
68°
55°
R
What are the “straight angles”
called? SUPLEMENTARY
Do you see any
complementary angles?
What measure do
S complementary angles have?
90 degrees
Additional Example 2A: Using Properties of
Angles to Label Triangles
Use the diagram to find the measure of each
indicated angle.
Q
A. QTR
P
QTR and STR are
supplementary angles, so the
sum of m QTR and m STR
is 180°.
m
QTR = 180° – 68°
= 112°
T
68°
R
55°
S
Additional Example 2B: Using Properties of
Angles to Label Triangles
B.
QRT
QRT and SRT are
complementary angles, so the
sum of m QRT and m SRT
is 90°.
m SRT = 180° – (68° + 55°)
= 180° – 123°
= 57°
Q
m
R
QRT = 90° – 57°
= 33°
P
T
68°
55°
S
Try This: Example 2A
Use the diagram to find the measure of each
indicated angle.
A.
M
MNO
MNO and PNO are
supplementary angles, so the
sum of m MNO and m PNO
is 180°.
m
MNO = 180° – 44°
= 136°
L
N
44°
O
60°
P
Try This: Example 2B
B.
MON
MON and PON are
complementary angles, so the
sum of m MON and m PON
is 90°.
m
m
M
N
PON = 180° – (44° + 60°)
= 180° – 104°
= 76°
MON = 90° – 76°
= 14°
L
44°
O
60°
P
CLASSIFYING TRIANGLES
BY ANGLE
MEASURES
1. Acute – has all
acute angles
2. Obtuse – has an
obtuse angle
3. Right – has a right
angles (90 degrees)
CLASSIFYING TRIANGLES
BY LENGTH OF
SIDES
1.Scalene Triangles
– All sides are
different
2. Isosceles
Triangle – two
sides are the same
3. Equilateral
Triangle- All sides
are congruent
BY ANGLE
MEASURES
1. Acute – has all
acute angles
2. Obtuse – has an
obtuse angle
3. Right – has a right
angles (90 degrees)
Triangles can be classified by the lengths
of their sides. A scalene triangle has no
congruent sides. An isosceles triangle
has at least two congruent sides. An
equilateral triangle has three congruent
sides.
Additional Example 3: Classifying Triangles by
Lengths of Sides
Classify the triangle. The sum of the lengths
of the sides is 19.5 in.
M
c + (6.5 + 6.5) = 19.5
c + 13 = 19.5
6.5 in.
6.5 in.
c + 13 – 13 = 19.5 – 13
c = 6.5
L
c
Side c is 6.5 inches long. Because LMN has
three congruent sides, it is equilateral.
N
Try This: Example 3
Classify the triangle. The sum of the lengths
of the sides is 21.6 in.
B
d + (7.2 + 7.2) = 21.6
d + 14.4 = 21.6
7.2 in.
7.2 in.
d + 14.4 – 14.4 = 21.6 – 14.4
d = 7.2
A
d
Side d is 7.2 inches long. Because ABC has
three congruent sides, it is equilateral.
C
Insert Lesson Title Here
WARM-UP
If the angles can form a triangle, classify the
triangle as acute, obtuse, or right.
not a
1. 37°, 53°, 90° right 2. 65°, 110°, 25°
triangle
3. 61°, 78°, 41° acute 4. 115°, 25°, 40° obtuse
The lengths of three sides of a triangle are
given. Classify the triangle.
5. 12, 16, 25 scalene
6. 10, 10, 15
isosceles
Making Sense of It
M
50°
80°
L
100° N100°
80°
30°
60°
O
40°
P
How many squares do you see?
5
How many squares do you see?
14
How many squares do you see?
How many squares do you see?
How many squares do you see?
How many squares do you see?
How many squares do you see?
30
How many squares do you see?
How many squares do you see?
Try this next one as a team!
How many squares do you see?
How many squares do you see?
How many squares do you see?
How many squares do you see?
37
For today’s warm-up, take out your list of vocabulary words, and TURN THEM OVER
SO YOU CAN’T SEE THEM. Then, answer these three questions alone!
1) What are complementary angles?
2) What are supplementary angles?
3) What is the sum of all the angle measurements in a triangle?
M
L
N
44°
60°
O
P
Can you see Complimentary
Angles?
M
L
N
44°
60°
O
P
Can you seeSupplementary
Angles?
M
L
N
44°
60°
O
P
Can you see Triangles?
M
L
N
44°
60°
O
P
Can you see Vertical Angles?
M
L
N
44°
60°
O
P
M
30°
L
44°
136° N 136°
44°
14°
O
76°
60°
P
M
L
N
42°
58°
O
P
M
32°
42°
L
138° N 138°
42°
10°
80°
O
58°
P
M
L
N
40°
60°
O
P
M
30°
40°
L
140° N 140°
40°
10°
80°
O
60°
P