Transcript Interactive Chalkboard

1.5 Angle Relationships
Objectives
Identify and use special pairs of angles
 Identify perpendicular lines

Pairs of Angles

Adjacent Angles – two angles that lie in the same

Vertical Angles – two nonadjacent angles formed by

Linear Pair – a pair of adjacent angles whose
plane, have a common vertex and a common side, but
no common interior points
two intersecting lines
noncommon sides are opposite rays
Example 1a:
Name two angles that form a linear pair.
A linear pair is a pair of
adjacent angles whose
noncommon sides are
opposite rays.
Answer: The angle pairs that satisfy this definition are
Example 1b:
Name two acute vertical angles.
There are four acute
angles shown. There is one
pair of vertical angles.
Answer: The acute vertical angles are VZY
and XZW.
Your Turn:
Name an angle pair that satisfies each condition.
a. two acute vertical angles
Answer: BAC and FAE,
CAD and NAF, or
BAD and NAE
b. two adjacent angles whose
sum is less than 90
Answer: BAC and CAD or
EAF and FAN
Angle Relationships

Complementary Angles – two angles whose
measures have a sum of 90º

Supplementary Angles – two angles whose
measures have a sum of 180º

Remember, angle measures are real numbers, so the operations
for real numbers and algebra can apply to angles.
Example 2:
ALGEBRA Find the measures of two supplementary
angles if the measure of one angle is 6 less than five
times the other angle.
Explore
The problem relates the measures of two
supplementary angles. You know that the sum
of the measures of supplementary angles is 180.
Plan
Draw two figures to represent the angles.
Example 2:
Let the measure of one angle be x.
Solve
Given
Simplify.
Add 6 to each side.
Divide each side by 6.
Example 2:
Use the value of x to find each angle measure.
Examine Add the angle measures to verify that the
angles are supplementary.
Answer: 31, 149
Your Turn:
ALGEBRA Find the measures of two complementary
angles if one angle measures six degrees less than
five times the measure of the other.
Answer: 16, 74
Perpendicular Lines

Lines that form right angles are
perpendicular.

We use the symbol “┴” to illustrate two
lines are perpendicular. ┴ is read “ is
perpendicular to.”
AB ┴ CD
Perpendicular Lines
The following is true for all ┴ lines:
1. ┴ lines intersect to form 4 right angles.
2. ┴ lines intersect to form congruent
adjacent angles.
3. Segments and rays can be ┴ to lines or to
other segments and rays.
4. The right angle symbol (┐) indicates that
lines are ┴.
Example 3:
ALGEBRA Find x so that
.
Example 3:
If
IJH.
, then mKJH
90. To find x, use KJI and
Sum of parts
whole
Substitution
Add.
Subtract 6 from each side.
Divide each side by 12.
Answer:
Your Turn:
ALGEBRA Find x and y so that
are perpendicular.
Answer:
and
Assumptions in Geometry

As we have discussed previously, we
cannot assume relationships among
figures in geometry. Figures are not drawn
to reflect total accuracy of the situation,
merely to provide or depict it. We must be
provided with given information or be able
to prove a situation from the given
information before we can state truths
about it.
Example 4a:
Determine whether the following statement can be
assumed from the figure below. Explain.
mVYT
90
The diagram is marked to
show that
From the definition of
perpendicular, perpendicular
lines intersect to form
congruent adjacent angles.
Answer: Yes;
and
are perpendicular.
Example 4b:
Determine whether the following statement can be
assumed from the figure below. Explain.
TYW and TYU are supplementary.
Answer: Yes; they form a
linear pair of angles.
Example 4c:
Determine whether the following statement can be
assumed from the figure below. Explain.
VYW and TYS are adjacent angles.
Answer: No; they do not
share a common side.
Your Turn:
Determine whether each statement can be assumed
from the figure below. Explain.
a.
Answer: Yes; lines TY and SX
are perpendicular.
b. TAU and UAY are
complementary.
Answer: No; the sum of the
two angles is 180, not 90.
c. UAX and UXA are adjacent.
Answer: No; they do not share
a common side.
Assignment

Geometry
Pg. 42 #11 – 35, omit #29 and #30

Pre-AP Geometry
Pg. 42 #11 - 35