Transcript Adjacent angles

Angle Relationships
Warm Up
Identify the type of angle.
1. 70°
acute
2. 90°
right
3. 140° obtuse
4. 180° straight
Today we will learn to understand new
relationships of angles.
Vocabulary
congruent
vertical angles
adjacent angles
complementary angles
supplementary angles
Angle Relationships
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
Course 1
P
You can use what you know about right angles
to identify an unknown angle measure.
Find each unknown angle measure.
The angles are complementary.
We know one is 71.
71° + a =
–71°
a=
We know the sum must be 90.
90°
–71°
19°
The sum of
the measures
is 90°.
a
71°
Find the unknown angle measure.
The angles are complementary.
65° + d = 90°
–65°
–65°
d = 25°
The sum of
the measures
is 90°.
d
65°
Supplementary angles are two angles whose
measures have a sum of 180°.
65° + 115° = 180°
GHK and
KHJ are supplementary.
K
65°
G
Connecting the two makes a semicircle on a line.
115°
H
J
We can use what we know about straight
angles to find a missing angle measure.
Find each unknown angle measure.
The angles are
supplementary.
The sum of
the measures
is 180°.
125° + b = 180°
–125°
–125°
b=
55°
b
125°
Find the unknown angle measure.
The angles are supplementary.
145° + s = 180°
–145°
–145°
s=
35°
The sum of
the measures
is 180°.
145°
s
When angles have the same measure, they are
said to be congruent.

Use what we know about straight angles to find a missing
angle measure. JKL and MKN are congruent.
We want to find the measures of
two congruent angles.We will let n stand
for the measure of one. Same
measure, same variable….
M
L
n
J
80°
K
n1
N
2n + 80° = 180° The sum of the measures is
–80°
–80° 180°.
2n = 100°
So…both = 50°
n = 50° and n1 = 50°
Each angle measures half of
100°.
Find each unknown angle measure.
C
ABC and DBE are
congruent.
D
50°
n
A
n1
B
E
2n + 50° = 180° The sum of the measures is
–50°
–50° 180°.
2n = 130°
n = 65° and n1 = 65°
Each angle measures half of
130°.
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.
MRP and NRQ are vertical angles.
MRN and
PRQ are vertical angles
Identifying Types of Angle Pairs: Extra Info
Identify the type of each angle pair shown.
5
6
Imagine two triangles pointing
towards each other. These would
be inside the vertical angles.
5 and 6 are opposite each other and
are formed by two intersecting lines.
They are vertical angles.
These two angles are also
vertical angles.
Find each unknown angle measure.
The angles are vertical
angles. (given)
c = 82°
Vertical angles
are congruent.
c
82°
d
WHY?????
If we know that angle C = 82 degrees, what is the measure of
angle d?
180 = 82 + d
180 = 82 + 92
Let’s use the previous example in your notes to discuss
adjacent angles. 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
MRN and NRQ are adjacent angles. They
share vertex R and RN. How else could these
two angles be classified?
They are also supplementary.
NRQ and QRP are adjacent angles. They
share vertex R and RQ. They are also ???????
They are also supplementary.
These two angles are adjacent. They are not
supplementary or complementary.
7 and 8 are side by side and
have a common vertex and
ray.
7
8
They are adjacent angles.
3
4
3 and 4 are side by side and
have a common vertex and
ray.
They are adjacent angles.
They are also
supplementary.
Lesson Quiz, Part I
Give the complement of each angle.
1. 70°
What do I need
2.
n=20°to = 90?
42° n=48°
What do I need
to = 90?
Give the supplement of each angle.
3. 120° n=60
4. 17° 163°
°
5. Identify the type of angle pair shown.
adjacent
Why are they NOT supplementary?
What do I need
to = 180?
Time Check
Oh yes, there is more. Not today.
We have several new angle relationships
and the algebraic application for finding
missing angles.
Angle Relationships II
Vocabulary
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•
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Corresponding Angles
Alternate Exterior Angles
Alternate Interior Angles
Transversal
More Angle Relationships
In this picture, we have two parallel lines intersected by another line.
This intersecting line is called a transversal. When a transversal intersects
two parallel lines, we create several new angle relationships.
Corresponding angles are angles that lie on the same side
of the transversal and the same side of the (respective) parallel line.
I call this the Walgreens’ Property.
We’ll get to your notes in a minute.
This could also be called the Starbucks’ Property…..
or the Dollar General Property…..
Now let’s identify the corresponding
angles….The letters in your notes represent the
different angles created by the parallel lines and the transversal.
a
h
e
c
f
b
g
d
The corresponding angles are:
a and c, e and h, b and d, f and g
Corresponding angles are congruent .
Alternate interior angles lie inside the region of the parallel
lines, on alternate sides of the transversal.
h
c
g
d
Alternate Interior Angles: c and g, d and h
Alternate interior angles are congruent to each other.
Alternate exterior angles are angles that lie on opposite sides
of the transversal, outside of the parallel lines.
a
e
f
Alternate exterior angles are congruent.
A and f, b and e are alternate exterior angles…..
b
If we know the measure of one angle, we can find the rest. We will find them
in this order: a (given), g, f, c, e, d, h, b
a has a measure of 135o
g is also 135o
supplementary
f is 135o
c is 135o
vertical
supplementary
45o h
alt. interior
corresponding
corresponding
vertical
alt. interior
b
45o
d
f 135o
alt exterior
e must be
d is 45o
h is 45o
b is 45o
vertical
supplementary
corresponding
45o
corresponding
alternate ext.
45o
g 135o
vertical
135oc
45o e
supplementary
135o a
We will use the letter on the
diagram to represent the
angles created by the
intersection of these lines.
a has a measure of 80o
80o a
b
100o g
80o
100o
h 80o
c
100o f
o
d 100
e
80o
Geometry + Algebra???
6(8) + 60 =
48 + 60 =
108°
(6x + 60)° 108°
72°(9x)°
6x + 60 + 9x = 180
15x + 60 = 180
-60
-60
15x = 120
x=8
9x = 9(8)
9x = 72°
1 72° Corresponding angles
2
These two lines are parallel.
108° Alternate exterior angles….and?
52°
1
15x – 7 + 5x + 7 = 180
20x = 180
(15x – 7)°
128°
Supplementary
x = 9°
52°
15x - 7 =
15(9) - 7 =
135 - 7 = 128°
Vertical
(5x + 7)°
5x + 7 =
5(9) + 7 =
45 + 7 = 52°
2 128°
Corresponding
Did we learn more relationships of angles?