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CCSS.Math.Content.HSG-CO.A.1, HSG-CO.D.121•

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1•5 Angle RelationshipsPairs of Angles

Some angle pairs are defined by their position in relationship to each other.• Adjacent angles are two angles that lie in the same plane and

have a common vertex and a common side, but no common interior points.

• A pair of adjacent angles with noncommon sides that are opposite rays is called a linear pair.

• Vertical angles are two nonadjacent angles formed by two intersecting lines.

Some angle pairs are defined by the relationship between their measures.• Complementary angles are two angles with measures that have

a sum of 90.• Supplementary angles are two angles with measures that have

a sum of 180.

Angles and Angle Pairs

Name an angle pair that satisfies each condition.

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a. two vertical angles∠ EFI and ∠GFH are nonadjacent angles formed by two intersecting lines. They are vertical angles.

b. two adjacent angles∠ABD and ∠DBE have a common vertex and a common side but no common interior points. They are adjacent angles.

c. two supplementary angles∠EFG and ∠GFH form a linear pair. The angles are supplementary.

d. two complementary anglesm∠CBD + m∠DBE = 90. These angles are complementary.

EXAMPLE

Program: FL MATH REPRINT Component: HANDBOOK1st Pass

Vendor: LASERWORDS Grade: Geometry

52 HotTopic 1

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Perpendicular Lines

Lines, rays, and segments that form right angles are perpendicular. The right angle symbol indicates that the lines are perpendicular.

In the figure, � �� AC is perpendicular to � �� BD , or � �� AC ⊥ � �� BD .

B

CD

A

Perpendicular Lines

Find x so that ⎯⎯ � ZD and

⎯⎯ � ZP are perpendicular.

Z

D

P

Q(9x + 5)°

(3x + 1)°

If �� ZD ⊥

�� ZP , then m∠DZP = 90.

m∠DZQ + m∠QZP = m∠DZP Definition of adjacent angles.

(9x + 5) + (3x + 1) = 90 Substitution

12x + 6 = 90 Combine like terms.

12x + 6 - 6 = 90 - 6 Subtract 6 from each side.

12x = 84 Simplify.

12x _ 12 = 84 _ 12 Divide each side by 12.

x = 7 Simplify.

EXAMPLE

Angle Relationships 53

Program: FL MATH REPRINT Component: HANDBOOK2nd Pass


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Construct Perpendiculars

You can construct the perpendicular bisector of a line segment using a compass and a straightedge.

Construct Perpendiculars

Use a compass and straightedge to construct the perpendicular bisector of −−

XY .

Step 1: Given line segment XY, open the compass wider than half of XY. Put the tip of the compass on X and draw an arc that goes above and below the line segment.

Step 2: Without changing your compass setting, put the tip on Y and draw an arc that intersects the first arc at W and Z.

Step 3: Draw line WZ. � �� WZ is the perpendicular bisector of −−

XY .

X Y

X Y

W

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X Y

W

Z

EXAMPLE

Program: FL MATH REPRINT Component: HANDBOOK1st Pass


54 HotTopic 1


1•5 ExercisesFor Exercises 1–3, use the figure. 1. Identify two obtuse vertical angles.

2. Identify two acute adjacent angles.

3. Identify an angle supplementary to ∠TNU.

4. Find the measures of two complementary angles if the difference in their measures is 18.

5. Find the value of x and y so that � �� NR ⊥ � �� MQ .

6. Find m∠MSN.

7. m∠EBF = 3x + 10, m∠DBE = x, and

�� BD ⊥

�� BF . Find the value

of x.

8. If m∠EBF = 7y - 3 and m∠FBC = 3y + 3, find the value of y so that

�� BE ⊥

�� BC .

9. Find the value of y, m∠RPT, and m∠TPW.

10. Copy −−− MN . Then construct the

perpendicular bisector of −−− MN .

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NU

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(9y + 18)°

B CA

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T(4y - 5)°

(2y + 5)°

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Program: FL MATH Component: HANDBOOKPDF Pass


Angle Relationships 55


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