1.5 describe angle pair relationships
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1.5 Describe Angle Pair Relationships. Objectives. Identify and use special pairs of angles Identify perpendicular lines. Pairs of Angles. Adjacent Angles – two angles that lie in the same plane, have a common vertex and a common side, but no common interior points - PowerPoint PPT PresentationTRANSCRIPT
1.5 Describe Angle Pair Relationships
Objectives Identify and use special pairs of
angles Identify perpendicular lines
Pairs of Angles Adjacent Angles – two angles that lie in the
same plane, have a common vertex and a common side, but no common interior points
Vertical Angles – two nonadjacent angles formed by two intersecting lines
Linear Pair – a pair of adjacent angles whose non-common sides are opposite rays
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 1a:
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.
Example 1b:
Name an angle pair that satisfies each condition.
a. two acute vertical angles
b. two adjacent angles whose sum is less than 90
Answer: BAC and CAD or EAF and FAN
Answer: BAC and FAE, CAD and NAF, or BAD and NAE
Your Turn:
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.
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
Example 2:
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
Your Turn:
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 LinesThe 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 ┴.
ALGEBRA Find x so that .Example 3:
If , then mKJH 90. To find x, use KJI and IJH.
Substitution
Add.
Subtract 6 from each side.
Divide each side by 12.
Answer:
Sum of parts whole
Example 3:
ALGEBRA Find x and y so that and are perpendicular.
Answer:
Your Turn:
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.
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.
Determine whether the following statement can be assumed from the figure below. Explain.mVYT 90
Example 4a:
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 4b:
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.
Example 4c:
Determine whether each statement can be assumed from the figure below. Explain.a.
b. TAU and UAY are complementary.
c. UAX and UXA are adjacent.
Answer: Yes; lines TY and SX are perpendicular.
Answer: No; they do not share a common side.
Answer: No; the sum of the two angles is 180, not 90.
Your Turn: