Download 1 - Mira Costa High School

1.5 Describe Angle Pair Relationships
Goal  Use special angle relationships to find angle measures.
Your Notes
VOCABULARY
Complementary angles
Two angles whose sum is 90°
Supplementary angles
Two angles whose sum is 180°
Adjacent angles
Two angles that share a common vertex or side but have no common interior points
Linear pair
Two adjacent angles are a linear pair if their noncommon sides are opposite rays.
Vertical angles
Two angles are vertical angles if their sides form two pairs of opposite rays.
Example 1
Identify complements and supplements
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a
pair of adjacent angles.
In Example 1, BDE
and CDE share a
common vertex. But
they share common
__interior__ points, so
they are not adjacent
angles.
Solution
Because _52°_ + _38°_ = 90°, _ABD_ and _CDB_ are _complementary_ angles
Because _52°_ + _128°_ = 180°, _ABD_ and _EDB_ are _supplementary_ angles.
Because _CDB_ and _BDE_ share a common vertex and side, they are _adjacent_
angles.
Your Notes
Example 2
Find measures of complements and supplements
a. Given that 1 is a complement of 2 and m2 = 57°, find m1.
b. Given that 3 is a complement of 4 and m4 = 41°, find m3.
Angles are
sometimes
named with
numbers. An
angle measure
in a diagram
has a degree
symbol. An
angle name
does not.
Solution
a. You can draw a diagram with complementary adjacent angles to illustrate the
relationship.
m1 = _90°_ – m2__ = _90°_ – _57°_ = _33°_
b. You can draw a diagram with supplementary adjacent angles to illustrate the
relationship.
m 3 = _180°_ – _m4_= _180°_ – _41°_ = _139°_
Checkpoint Complete the following exercises.
1.
In the figure, name a pair of complementary angles, a pair of supplementary
angles, and a pair of adjacent angles.
Complementary: DEF and ABC, supplementary; FEG and ABC, adjacent: DEF
and FEG
2. Given that 1 is a complement of 2 and m1 = 73°, find m2.
17°
3. Given that 3 is a supplement of 4 and m4 = 37°, find m3.
143°
Your Notes
Example 3
Find angle measures
Basketball
The basketball pole forms a pair of supplementary angles with the ground. Find mBCA
and mDCA.
In a diagram, you can
assume that a line that
looks straight is
straight. In Example 3,
B, C, and D lie on
BD . So,  BCD is a
__straight__ angle.
Solution
Step 1
Use the fact that __180°__ is the sum of the measures of supplementary angles.
mBCA + mDCA = _180°_
(_3x + 8_)° + (_4x – 3_)° = _180°_
_7x + 5_ = _180°_
_7x_ = _175_
_x_ = 25
Write equation.
Substitute.
Combine like terms.
Subtract.
Divide.
Step 2
Evaluate the original expressions when x = _25_.
mBCA = (_3x + 8_)° = (_3 • 25 + 8_)° = _83°_.
mDCA = (_4x – 3_)° = (_4 • 25 – 3_)° = _97°_.
The angle measures are _83°_ and _97°_.
Checkpoint Complete the following exercise.
4. In Example 3, suppose the angle measures are (5x + 1)° and (6x + 3)°. Find
mBCA and mDCA.
81° and 99°
Your Notes
Example 4
Identify angle pairs
Identify all of the linear pairs and all of the vertical angles in the figure at the right.
Solution
In the diagram,
one side of 1
and one side of
4 are opposite
rays. But the
angles are not a
linear pair
because they are
not _adjacent_.
To find vertical angles, look for angles formed by _intersecting lines_.
_1_ and _3_ are vertical angles.
To find linear pairs, look for adjacent angles whose noncommon sides are _opposite
rays_.
_1_ and _2_ are a linear pair. _2_ and _3_ are a linear pair.
Checkpoint Complete the following exercise.
5. Identify all of the linear pairs and all of the vertical angles in the figure.
linear pairs none; vertical angles: 1 and 4, 2 and 5, 3 and 6
Example 5
Find angle measures in a linear pair
Two angles form a linear pair. The measure of one angle is 4 times the measure of the
other. Find the measure of each angle.
You may find it
useful to draw a
diagram to
represent a word
problem like the
one in Example
5.
Solution
Let x° be the measure of one angle. The measure of the other angle is _4x°_. Then use the
fact that the angles of a linear pair are _supplementary_ to write an equation.
_x°_ + _4x°_ = 180°
_5x_ = _180_
_x_ = _36_
Write an equation.
Combine like terms.
Divide each side by _5_.
The measures of the angles are _36°_ and _4(36°)_ = _144°_.
Your Notes
Checkpoint Complete the following exercise.
6. Two angles form a linear pair. The measure of the other. Find the measure of each
angle.
45° and 135°
CONCEPT SUMMARY: INTERPRETING A DIAGRAM
There are some things you can conclude from a diagram, and some you cannot. For
example, here are some things that you can conclude from the diagram at the right.





All points shown are _coplanar_.
Points, A, B, and C are _collinear_, and B is between A and C.
AC , BD , and BE_intersect_ at point B.
DBE and EBC are _adjacent_ angles, and  ABC is a _straight angle_.
Point E lies in the _interior_ of DBC.
In the diagram above, you cannot conclude that AB  BC ,, that DBE  EBC, or that
ABD is a right angle. This information must be indicated, as shown at the right.
Homework
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