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 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 noncommon 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 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 ┴.

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:

Assignment

Honors GeometryPgs. 37 – 40 #1 – 30, 37, 40

Foundations GeometryPgs. 40 – 41 #1 – 13, 22-28, 35