and Let AB TEST 534

An angle is the union of two rays that share a common endpoint. DEFINITION In Figure 1.46, the angle is symbolized by ⬔ABC or ⬔CBA. The rays BA and BC are known as the sides of the angle. B, the common endpoint of these rays, is known as the vertex of the angle. When three letters are used to name an angle, the vertex is al- ways named in the middle. Recall that a single letter or numeral may be used to name the angle. The angle in Figure 1.46 may be described as ⬔B the vertex of the angle or as ⬔1. In set notation, ⬔B = . BA ´ BC 1 B C A Figure 1.46 B C E A F D Figure 1.47 C B D A Figure 1.48 The measure of an angle is a unique positive number. POSTULATE 8 왘 Protractor Postulate NOTE: In Chapters 1 to 10, the measures of most angles will be between 0° and 180°, including 180°. Angles with measures between 180° and 360° are introduced in this section; these angles are not used often in our study of geometry. TYPES OF ANGLES An angle whose measure is less than 90° is an acute angle. If the angle’s measure is ex- actly 90°, the angle is a right angle. If the angle’s measure is between 90° and 180°, the angle is obtuse. An angle whose measure is exactly 180° is a straight angle; alterna- tively, a straight angle is one whose sides form opposite rays a straight line. A reflex angle is one whose measure is between 180° and 360°. See Table 1.4 on page 32. In Figure 1.47, ⬔ABC contains the noncollinear points A, B, and C. These three points, in turn, determine a plane. The plane containing ⬔ABC is separated into three subsets by the angle: Points like D are said to be in the interior of ⬔ABC. Points like E are said to be on ⬔ABC. Points like F are said to be in the exterior of ⬔ABC. With this description, it is possible to state the counterpart of the Segment-Addition Postulate Consider Figure 1.48 as you read Postulate 9. If a point D lies in the interior of an angle ABC, then m ⬔ABD + m ⬔DBC = m ⬔ABC. POSTULATE 9 왘 Angle-Addition Postulate TABLE 1.4 Angles Angle Example Acute 1 m ⬔ 1 = 23° Right 2 m ⬔ 2 = 90° Obtuse 3 m ⬔ 3 = 112° Straight 4 m ⬔ 4 = 180° Reflex 5 m ⬔ 5 = 337° NOTE: An arc is necessary in indicating a reflex angle and can be used to indicate a straight angle as well. 2 4 5 3 EXAMPLE 1 Use Figure 1.48 on page 31 to find m ⬔ABC if: a m ⬔ABD = 27° and m⬔DBC = 42° b m ⬔ABD = x° and m⬔DBC = 2x - 3° Solution a Using the Angle-Addition Postulate, m ⬔ABC = m⬔ABD + m⬔DBC. That is, m⬔ABC = 27° + 42° = 69°. b m ⬔ABC = m⬔ABD + m⬔DBC = x° + 2x - 3° = 3x - 3° 쮿 Discover When greater accuracy is needed in angle measurement, a degree can be divided into 60 minutes. In symbols, 1° = 60⬘. Convert 22.5° to degrees and minutes. ANSWER Discover An index card can be used to categorize the types of angles displayed. In each sketch, an index card is placed over an angle. A dashed ray indicates that a side is hidden. What type of angle is shown in each figure? Note the placement of the card in each figure. Technology Exploration Use software if available. 1. Draw ⬔RST. 2. Through point V in the interior of ⬔RST, draw . 3. Measure ⬔RST, ⬔RSV, and ⬔VST. 4. Show that m⬔RSV + m⬔VST = m⬔RST. SV 1 22⬚ 30⬘ ANSWERS Obtuse angle Acute angle Right angle Straight angle One edge of the index card coincides with both of the angle’s sides Sides of the angle coincide with two edges of the card Card hides the second side of the angle Card exposes the second side of the angle Exs. 1–6 CLASSIFYING PAIRS OF ANGLES Many angle relationships involve exactly two angles a pair—never more than two an- gles and never less than two angles In Figure 1.48, ⬔ABD and ⬔DBC are said to be adjacent angles. In this descrip- tion, the term adjacent means that angles lie “next to” each other; in everyday life, one might say that the Subway sandwich shop is adjacent to the Baskin-Robbins ice cream shop. When two angles are adjacent, they have a common vertex and a common side between them. In Figure 1.48, ⬔ABC and ⬔ABD are not adjacent because they have in- terior points in common. Two angles are adjacent adj. ⬔s if they have a common vertex and a common side between them. DEFINITION We now recall the meaning of congruent angles. Congruent angles ⬵ ⬔s are two angles with the same measure. DEFINITION Congruent angles must coincide when one is placed over the other. Do not con- sider that the sides appear to have different lengths; remember that rays are infinite in length In symbols, ⬔1 ⬵ ⬔2 if m⬔1 = m⬔2. In Figure 1.49, similar markings arcs indicate that ⬔1 ⬵ ⬔2. EXAMPLE 2 GIVEN: ⬔1 ⬵ ⬔2 m ⬔1 = 2x + 15 m ⬔2 = 3x - 2 FIND: x Solution ⬔1 ⬵ ⬔2 means m⬔1 = m⬔2. Therefore, or NOTE: m ⬔1 = 217 + 15 = 49° and m⬔2 = 317 - 2 = 49°. 쮿 x = 17 17 = x 2x + 15 = 3x - 2 The bisector of an angle is the ray that separates the given angle into two congruent angles. DEFINITION 1 2 Figure 1.49 Q P N M Figure 1.50 With P in the interior of ⬔MNQ so that ⬔MNP ⬵ ⬔PNQ, is said to bisect ⬔MNQ. Equivalently, is the bisector or angle-bisector of ⬔MNQ. On the basis of Figure 1.50, possible consequences of the definition of bisector of an angle are m ⬔MNP = m⬔PNQ m ⬔MNQ = 2m⬔PNQ m ⬔MNQ = 2m⬔MNP m ⬔PNQ = m⬔MNQ m ⬔MNP = m⬔MNQ 1 2 1 2 NP NP C B D A Figure 1.48 Two angles are complementary if the sum of their measures is 90°. Each angle in the pair is known as the complement of the other angle. DEFINITION Two angles are supplementary if the sum of their measures is 180°. Each angle in the pair is known as the supplement of the other angle. DEFINITION EXAMPLE 3 Given that m ⬔1 = 29°, find: a the complement x b the supplement y Solution a x + 29 = 90, so x = 61°; complement = 61° b y + 29 = 180, so y = 151°; supplement = 151° 쮿 Angles with measures of 37° and 53° are complementary. The 37° angle is the com- plement of the 53° angle, and vice versa. If the measures of two angles are x and y and it is known that x + y = 90°, then these two angles are complementary. EXAMPLE 4 GIVEN: ⬔P and ⬔Q are complementary, where m ⬔P = and m ⬔Q = FIND: x , m ⬔P, and m⬔Q Solution m ⬔P + m⬔Q = 90 Multiplying by 6 the least common denominator, or LCD, of 2 and 3, we have m ⬔P = = 54° m ⬔Q = NOTE: m ⬔P = 54° and m⬔Q = 36°, so their sum is exactly 90°. 쮿 x 3 = 108 3 = 36° x 2 = 108 2 x = 108 5x = 540 3x + 2x = 540 6 x 2 + 6 x 3 = 6 90 x 2 + x 3 = 90 x 3 x 2 When two straight lines intersect, the pairs of nonadjacent angles in opposite posi- tions are known as vertical angles. In Figure 1.51, ⬔5 and ⬔6 are vertical angles as are ⬔7 and ⬔8. In addition, ⬔5 and ⬔7 can be described as adjacent and supplemen- tary angles, as can ⬔5 and ⬔8. If m⬔7 = 30°, what is m⬔5 and what is m⬔8? It is true in general that vertical angles are congruent, and we will prove this in Example 3 of Section 1.6. We apply this property in Example 5 of this section. Recall the Addition and Subtraction Properties of Equality: If a = b and c = d, then a ⫾ c = b ⫾ d . These principles can be used in solving a system of equations such as the following: left and right sides are added We can substitute 4 for x in either equation to solve for y: by substitution If x = 4 and y = 1, then x + y = 5 and 2x - y = 7. When each term in an equation is multiplied by the same nonzero number, the so- lutions of the equation are not changed. For instance, the equations and each term multiplied by 3 both have the solution . Likewise, the values of x and y that make the equation true also make the equation each term multiplied by 4 true. We use this method in Example 5. 16x ⫹ 4y ⫽ 720 4x ⫹ y ⫽ 180 x ⫽ 5 6x ⫺ 9 ⫽ 21 2x ⫺ 3 ⫽ 7 y = 1 4 + y = 5 x + y = 5 x = 4 3x = 12 2x - y = 7 x + y = 5 m 8 7 5 6 Figure 1.51 EXAMPLE 5 GIVEN: In Figure 1.51, and m intersect so that FIND: x and y Solution ⬔5 and ⬔8 are supplementary adjacent and exterior sides form a straight angle. Therefore, m ⬔5 + m⬔8 = 180. ⬔5 and ⬔6 are congruent vertical. Therefore, m ⬔5 = m⬔6. Consequently, we have Using the Multiplication Property of Equality, we multiply the first equation by 4. Then the equivalent system allows us to eliminate variable y by addition. x = 40 18x = 720 2x - 4y = 0 16x + 4y = 720 2x - 4y = 0 4x + y = 180 2x + 2y = 4x - 2y 2x + 2y + 2x - y = 180 m ⬔6 = 4x - 2y m ⬔8 = 2x - y m ⬔5 = 2x + 2y supplementary ⬔s 5 and 8 ⬵⬔s 5 and 6 Simplifying, adding left, right sides Exs. 7–12 CONSTRUCTIONS WITH ANGLES In Section 1.2, we considered Constructions 1 and 2 with line segments. Now consider two constructions that involve angle concepts. In Section 3.4, it will become clear why these methods are valid. However, intuition suggests that the techniques are appropriate. a b H G c K d K R S T R S T P Q N P Q Figure 1.52 Using the equation , it follows that Summarizing, and . NOTE: m ⬔ 5 = 120°, m ⬔ 8 = 60°, and m ⬔ 6 = 120°. 쮿 y ⫽ 20 x ⫽ 40 y = 20 160 + y = 180 440 + y = 180 4x ⫹ y ⫽ 180 Construction 4 To construct the angle bisector of a given angle. GIVEN: ⬔PRT in Figure 1.53a CONSTRUCT: so that ⬔PRS ⬵ ⬔SRT RS Just as a line segment can be bisected, so can an angle. This takes us to a fourth construction method. Construction 3 To construct an angle congruent to a given angle. GIVEN: ⬔RST in Figure 1.52a CONSTRUCT: With as one side, ⬔NPQ ⬵ ⬔RST CONSTRUCTION: Figure 1.52b: With a compass, mark an arc to intersect both sides of ⬔RST at points G and H, respectively. Figure 1.52c: Without changing the radius, mark an arc to intersect at K and the “would-be” second side of ⬔NPQ. Figure 1.52b: Now mark an arc to measure the distance from G to H. Figure 1.52d: Using the same radius, mark an arc with K as center to intersect the would-be second side of the desired angle. Now draw the ray from P through the point of intersection of the two arcs. The resulting angle is the one desired, as we will prove in Section 3.4, Example 1. PQ PQ a P R T b M N T P R c M N S T P R Figure 1.53 Exs. 13–20 CONSTRUCTION: Figure 1.53b: Using a compass, mark an arc to intersect the sides of ⬔PRT at points M and N. Figure 1.53c: Now, with M and N as centers, mark off two arcs with equal radii to intersect at point S in the interior of ⬔PRT, as shown. Now draw ray RS, the desired angle bisector. Reasoning from the definition of an angle bisector, the Angle-Addition Postulate, and the Protractor Postulate, we can justify the following theorem. There is one and only one angle bisector for a given angle. THEOREM 1.4.1 This theorem is often stated, “The angle bisector of an angle is unique.” This statement is proved in Example 5 of Section 2.2. Exercises 1.4 1. What type of angle is each of the following? a 47° b 90° c 137.3° 2. What type of angle is each of the following? a 115° b 180° c 36°

3. What relationship, if any, exists between two angles:

a with measures of 37° and 53°? b with measures of 37° and 143°?

4. What relationship, if any, exists between two angles:

a with equal measures? b that have the same vertex and a common side between them? In Exercises 5 to 8, describe in one word the relationship between the angles.

5. ⬔ABD and ⬔DBC

6. ⬔7 and ⬔8

10. Suppose that

, , , , and are coplanar. AF AE AD AC AB C A D B E F Exercises 10–13 C B D A 8 7 5 6 m 7. ⬔1 and ⬔2 8. ⬔3 and ⬔4 1 C B D A 2

3 4

G F H E Use drawings as needed to answer each of the following questions. 9. Must two rays with a common endpoint be coplanar? Must three rays with a common endpoint be coplanar? Classify the following as true or false: a m ⬔BAC + m ⬔CAD = m ⬔BAD b ⬔BAC ⬵ ⬔CAD c m ⬔BAE - m ⬔DAE = m ⬔BAC d ⬔BAC and ⬔DAE are adjacent e m ⬔BAC + m ⬔CAD + m ⬔DAE = m ⬔BAE

11. Without using a protractor, name the type of angle

represented by: a ⬔BAE b ⬔FAD c ⬔BAC d ⬔FAE

12. What, if anything, is wrong with the claim

m ⬔FAB + m ⬔BAE = m ⬔FAE?

13. ⬔FAC and ⬔CAD are adjacent and

and are opposite rays. What can you conclude about ⬔FAC and ⬔CAD? For Exercises 14 and 15, let m ⬔1 = x and m ⬔2 = y.

14. Using variables x and y, write an equation that expresses

the fact that ⬔1 and ⬔2 are: a supplementary b congruent

15. Using variables x and y, write an equation that expresses

the fact that ⬔1 and ⬔2 are: a complementary b vertical For Exercises 16, 17, see figure on page 38.

16. Given: m

⬔RST = 39° m ⬔TSV = 23° Find: m ⬔RSV

17. Given: m

⬔RSV = 59° m ⬔TSV = 17° Find: m ⬔RST AD AF

18. Given:

m ⬔RST = 2x + 9 m ⬔TSV = 3x - 2 m ⬔RSV = 67° Find: x

19. Given:

m ⬔RST = 2x - 10 m ⬔TSV = x + 6 m ⬔RSV = 4x - 6 Find: x and m ⬔RSV

20. Given:

m ⬔RST = 5x + 1 - 3 m ⬔TSV = 4x - 2 + 3 m ⬔RSV = 42x + 3 - 7 Find: x and m ⬔RSV

21. Given:

m ⬔RST = m ⬔TSV = m ⬔RSV = 45° Find: x and m ⬔RST

22. Given: m

⬔RST = m ⬔TSV = m ⬔RSV = 49° Find: x and m ⬔TSV

23. Given:

bisects ⬔RSV m ⬔RST = x + y m ⬔TSV = 2x - 2y m ⬔RSV = 64° Find: x and y

24. Given:

bisects ⬔RSV Find: x and y

25. Given:

and in plane P as shown; intersects P at point A ⬔CAB ⬵ ⬔DAC ⬔DAC ⬵ ⬔DAB What can you conclude? Í AD Í AC Í AB m ⬔RSV = 80° m ⬔TSV = 3x - y + 2 m ⬔RST = 2x + 3y ST ST x 2 2x 3 x 4 x 2 R S T V Exercises 16–24 D C A P B

26. Two angles are complementary. One angle is 12° larger

than the other. Using two variables x and y, find the size of each angle by solving a system of equations.

27. Two angles are supplementary. One angle is 24° more

than twice the other. Using two variables x and y, find the measure of each angle.

28. For two complementary angles, find an expression for the

measure of the second angle if the measure of the first is: a x° b 3x - 12° c 2x + 5y°

29. Suppose that two angles are supplementary. Find

expressions for the supplements, using the expressions provided in Exercise 28, parts a to c.

30. On the protractor shown, bisects

⬔MNQ. Find x. NP x 53 ° 92° M N P Q Exercises 30, 31

31. On the protractor shown for Exercise 30,

⬔MNP and ⬔PNQ are complementary. Find x.

32. Classify as true or false:

a If points P and Q lie in the interior of ⬔ABC, then lies in the interior of ⬔ABC. b If points P and Q lie in the interior of ⬔ABC, then lies in the interior of ⬔ABC. c If points P and Q lie in the interior of ⬔ABC, then lies in the interior of ⬔ABC. In Exercises 33 to 40, use only a compass and a straightedge to perform the indicated constructions. PQ Í PQ PQ M R P Exercises 33–35

33. Given: Obtuse

⬔MRP Construct: With as one side, an angle ⬵ ⬔MRP

34. Given: Obtuse

⬔MRP Construct: , the angle bisector of ⬔MRP

35. Given: Obtuse

⬔MRP Construct: Rays , , and so that ⬔MRP is RU RT RS RS OA D F E divided into four ⬵ angles

36. Given: Straight

⬔DEF Construct: A right angle with vertex at E HINT: Use Construction 4.