8. Z Y XY and XZ are ____________. X Angles Opposite rays ___________ are two rays that are part of a the same line and have only theirendpoints in common. opposite rays straight angle The figure formed by opposite rays is also referred to as a ____________.
9. S vertex T Angles There is another case where two rays can have a common endpoint. angle This figure is called an _____. Some parts of angles have special names. The common endpoint is called the ______, vertex and the two rays that make up the sides ofthe angle are called the sides of the angle. side R side
10. S vertex SRT TRS R 1 T Angles There are several ways to name this angle. 1) Use the vertex and a point from each side. or The vertex letter is always in the middle. side 2) Use the vertex only. 1 If there is only one angle at a vertex, then theangle can be named with that vertex. R side 3) Use a number.
12. C A 1 B ABC 1 B CBA BA and BC Angles 1) Name the angle in four ways. 2) Identify the vertex and sides of this angle. vertex: Point B sides:
13. 2) What are other names for ? 3) Is there an angle that can be named ? 1 2 1 XWY or YWX W XWZ Angles 1) Name all angles having W as their vertex. X W 1 2 Y Z No!
14. exterior W Y Z interior A B Angles An angle separates a plane into three parts: interior 1) the ______ exterior 2) the ______ angle itself 3) the _________ In the figure shown, point B and all other points in the blue region are in the interiorof the angle. Point A and all other points in the greenregion are in the exterior of the angle. Points Y, W, and Z are on the angle.
15. P G Angles Is point B in the interior of the angle, exterior of the angle, or on the angle? Exterior B Is point G in the interior of the angle, exterior of the angle, or on the angle? On the angle Is point P in the interior of the angle, exterior of the angle, or on the angle? Interior
16. Vocabulary §1.6 Angle Measure What You'll Learn You will learn to measure, draw, and classify angles. 1) Degrees 2) Protractor 3)Right Angle 4) Acute Angle 5) Obtuse Angle
17. P 75° Q R m PQR = 75 §1.6 Angle Measure degrees In geometry, angles are measured in units called _______. The symbol for degree is °. In the figure to the right, the angle is 75 degrees. In notation, there is no degree symbol with 75 because the measure of an angle is a real number with no unit of measure.
18. A n° C m ABC = n and 0 < n < 180 B §1.6 Angle Measure 0 180
19. Use a protractor to measure SRQ. 1) Place the center point of the protractor on vertex R. Align the straightedge with side RS. 2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale. Q R S §1.6 Angle Measure protractor You can use a _________ to measure angles and sketch angles of givenmeasure.
20. m SRQ = m SRJ = m SRG = m QRG = m GRJ = m SRH H J G Q S R §1.6 Angle Measure 70 Find the measurement of: 180 – 150 = 30 180 45 150 – 45 = 105 150
21. 1) Draw AB 3) Locate and draw point C at the mark labeled 135. Draw AC. C A B §1.6 Angle Measure Use a protractor to draw an angle having a measure of 135. 2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.
22. A A A obtuse angle 90 < m A < 180 acute angle 0 < m A < 90 right angle m A = 90 §1.6 Angle Measure Once the measure of an angle is known, the angle can be classified as oneof three types of angles. These types are defined in relation to a right angle.
23. 40° 110° 90° 50° 75° 130° §1.6 Angle Measure Classify each angle as acute, obtuse, or right. Acute Obtuse Right Obtuse Acute Acute
24. The measure of H is 67.Solve for y. The measure of B is 138.Solve for x. H 9y + 4 5x - 7 B B = 5x – 7 and B = 138 H = 9y + 4 and H = 67 §1.6 Angle Measure Given: (What do you know?) Given: (What do you know?) 9y + 4 = 67 5x – 7 = 138 Check! Check! 9y = 63 5x = 145 9(7) + 4 = ? 5(29) -7 = ? y = 7 x = 29 63 + 4 = ? 145 -7 = ? 67 = 67 138 = 138
27. Vocabulary §1.6 The Angle Addition Postulate What You'll Learn You will learn to find the measure of an angle and the bisectorof an angle. NOTHING NEW!
28. R X 2) Draw and label a point X in the interior of the angle. Then draw SX. S T §1.6 The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 45° 75° 30° 3) For each angle, find mRSX, mXST, and RST.
29. R X S T §1.6 The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ? Their sum is equal to the measure of RST . mXST = 30 + mRSX = 45 = mRST = 75 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. 45° The sum of the measures of the twosmaller angles is equal to the measureof the larger angle. 75° 30°
30. P 1 Q A 2 R §1.6 The Angle Addition Postulate m1 + m2 = mPQR. There are two equations that can be derived using Postulate 3 – 3. m1 = mPQR –m2 These equations are true no matter where A is locatedin the interior of PQR. m2 = mPQR –m1
31. X 1 Y W 2 Z §1.6 The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. Postulate 3 – 3. m2 + m1 = mXYZ m2 = mXYZ –m1 m2 = 86 – 22 m2 = 64
32. C D (5x – 6)° 2x° B A §1.6 The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD Angle Addition Postulate Substitution 2x + (5x – 6) = 120 7x – 6 = 120 Combine like terms 7x = 126 Add 6 to both sides x = 18 Divide each side by 7 36 + 84 = 120 mCBD = 5x – 6 mABC = 2x mCBD = 5(18) – 6 mABC = 2(18) mCBD = 90 – 6 mABC = 36 mCBD = 84
33. §1.6 The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle has a ___ that bisects the angle. ray angle bisector This ray is called an ____________ .
34. is the bisector of PQR. P 1 Q A 2 R §1.6 The Angle Addition Postulate m1 = m2
35. Since bisects CAN, 1 = 2. N T 2 1 A C §1.6 The Angle Addition Postulate If bisects CAN and mCAN = 130, find 1 and 2. 1 + 2 = CAN Angle Addition Postulate Replace CAN with 130 1 + 2 = 130 1 + 1 = 130 Replace 2 with 1 2(1) = 130 Combine like terms (1) = 65 Divide each side by 2 Since 1 = 2, 2 = 65
37. A B D C Adjacent Angles and Linear Pairs of Angles What You'll Learn You will learn to identify and use adjacent angles and linear pairs of angles. When you “split” an angle, you create two angles. The two angles are called _____________ adjacent angles 2 1 adjacent = next to, joining. 1 and 2 are examples of adjacent angles. They share a common ray. Name the ray that 1 and 2 have in common. ____
38. J 2 common side R M 1 1 and 2 are adjacent with the same vertex R and N Adjacent Angles and Linear Pairs of Angles Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common
39. B 2 1 1 2 G N L 1 J 2 Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No. They have a common vertex B, but _____________ no common side Yes. They have the same vertex G and a common side with no interior points in common. No. They do not have a common vertex or ____________ a common side The side of 1 is ____ The side of 2 is ____
40. 1 2 1 2 Z D X Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No. Yes. In this example, the noncommon sides of the adjacent angles form a ___________. straight line linear pair These angles are called a _________
41. D A B 2 1 C Note: Adjacent Angles and Linear Pairs of Angles Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays 1 and 2 are a linear pair.
42. In the figure, and are opposite rays. H T E 3 A 4 2 1 C ACE and 1 have a common side , the same vertex C, and opposite rays and M Adjacent Angles and Linear Pairs of Angles 1) Name the angle that forms a linear pair with 1. ACE 2) Do 3 and TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.
44. §1.6 Complementary and Supplementary Angles What You'll Learn You will learn to identify and use Complementary and Supplementary angles
45. E D A 60° 30° F B C §1.6 Complementary and Supplementary Angles Two angles are complementary if and only if (iff) the sum of their degree measure is 90. mABC + mDEF = 30 + 60 = 90
46. E D A 60° 30° F B C §1.6 Complementary and Supplementary Angles If two angles are complementary, each angle is a complement of the other. ABC is the complement of DEF and DEF is the complement of ABC. Complementary angles DO NOT need to have a common side or even the same vertex.
47. I 75° 15° H P Q 40° 50° H S U V 60° T 30° Z W §1.6 Complementary and Supplementary Angles Some examples of complementary angles are shown below. mH + mI = 90 mPHQ + mQHS = 90 mTZU + mVZW = 90
48. D C 130° 50° E B F A §1.6 Complementary and Supplementary Angles If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. mABC + mDEF = 50 + 130 = 180
49. I 75° 105° H Q 130° 50° H S P U V 60° 120° 60° Z W T §1.6 Complementary and Supplementary Angles Some examples of supplementary angles are shown below. mH + mI = 180 mPHQ + mQHS = 180 mTZU + mUZV = 180 and mTZU + mVZW = 180
51. §1.6 Congruent Angles What You'll Learn You will learn to identify and use congruent and vertical angles. Recall that congruent segments have the same ________. measure Congruent angles _______________ also have the same measure.
52. 50° 50° B V §1.6 Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure B V iff mB = mV
53. 1 2 X Z §1.6 Congruent Angles arcs To show that 1 is congruent to 2, we use ____. To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: X Z mX = mZ
54. §1.6 Congruent Angles four When two lines intersect, ____ angles are formed. There are two pair of nonadjacent angles. vertical angles These pairs are called _____________. 1 4 2 3
55. §1.6 Congruent Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 1 and 3 1 4 2 2 and 4 3
56. 1 4 2 3 §1.6 Congruent Angles 1) On a sheet of paper, construct two intersecting lines that are not perpendicular. 2) With a protractor, measure each angle formed. 3) Make a conjecture about vertical angles. Consider: A. 1 is supplementary to 4. m1 + m4 = 180 Hands-On B. 3 is supplementary to 4. m3 + m4 = 180 Therefore, it can be shown that 1 3 Likewise, it can be shown that 24
57. 1 4 2 3 §1.6 Congruent Angles 1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3 x = 4; 3 = 19° 2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4 x = 56; 4 = 65° 3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3 x = 9; 3 = 127° 4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4 x = 10; 4 = 97°
58. §1.6 Congruent Angles Vertical angles are congruent. n m 2 1 3 3 1 2 4 4
59. 130° x° §1.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°.
60. §1.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° x – 10 = 125. 125° x = 135.
61. §1.6 Congruent Angles Suppose two angles are congruent. What do you think is true about their complements? 1 2 2 + y = 90 1 + x = 90 y is the complement of 2 x is the complement of 1 y = 90 - 2 x = 90 - 1 Because 1 2, a “substitution” is made. y = 90 - 1 x = 90 - 1 x = y x y If two angles are congruent, their complements are congruent.
62. 60° 60° B A 1 2 3 4 §1.6 Congruent Angles If two angles are congruent, then their complements are _________. congruent The measure of angles complementary to A and B is 30. A B If two angles are congruent, then their supplements are _________. congruent The measure of angles supplementary to 1 and 4 is 110. 110° 110° 70° 70° 4 1
63. 3 1 2 §1.6 Congruent Angles If two angles are complementary to the same angle, then they are _________. congruent 3 is complementary to 4 5 is complementary to 4 4 3 5 5 3 If two angles are supplementary to the same angle, then they are _________. congruent 1 is supplementary to 2 3 is supplementary to 2 1 3
64. 52° 52° A B §1.6 Congruent Angles Suppose A B and mA = 52. Find the measure of an angle that is supplementary to B. 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128°
65. §1.6 Congruent Angles If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25, What are m1 and m2 ? m1 + m3 = 90 Definition of complementary angles. m1 = 90 - m3 Subtract m3 from both sides. m1 = 90 - 25Substitute 25 in for m3. m1 = 65Simplify the right side. You solve for m2 m2 + m3 = 90 Definition of complementary angles. m2 = 90 - m3 Subtract m3 from both sides. m2 = 90 - 25Substitute 25 in for m3. m2 = 65Simplify the right side.
66. G D 1 2 A C 4 B 3 E H §1.6 Congruent Angles 1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3 x = 17; 3 = 37° 2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC x = 29; EBC = 121° 3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4 x = 16; 4 = 39° 4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1 x = 18; 1 = 43°
67. Suppose you draw two angles that are congruent and supplementary. What is true about the angles?
68. 1 2 C A B §1.6 Congruent Angles If two angles are congruent and supplementary then each is a __________. right angle 1 is supplementary to 2 1 and 2 = 90 All right angles are _________. congruent A B C
69. B A 2 E 3 1 4 C D §1.6 Congruent Angles If 1 is supplementary to 4, 3 is supplementary to 4, and m 1 = 64, what are m 3 and m 4? They are vertical angles. 1 3 m 1 = m3 m 3 = 64 3 is supplementary to 4 Given Definition of supplementary. m3 + m4 = 180 64 + m4 = 180 m4 = 180 – 64 m4 = 116
71. §1.6 Perpendicular Lines What You'll Learn You will learn to identify, use properties of, and construct perpendicular lines and segments.
72. In the figure below, lines are perpendicular. A 1 2 C D 4 3 B §1.6 Perpendicular Lines perpendicular lines Lines that intersect at an angle of 90 degrees are _________________.
73. m n §1.6 Perpendicular Lines Perpendicular lines are lines that intersect to form a right angle.
74. 1 2 4 3 §1.6 Perpendicular Lines In the figure below, l m. The following statements are true. m l Definition of Perpendicular Lines 1) 1 is a right angle. Vertical angles are congruent 2) 1 3. Definition of Linear Pair 3) 1 and 4 form a linear pair. Linear pairs are supplementary 4) 1 and 4 are supplementary. 5) 4 is a right angle. m4 + 90 = 180, m4 = 90 Vertical angles are congruent 6) 2 is a right angle.
75. a 1 2 4 3 b §1.6 Perpendicular Lines If two lines are perpendicular, then they form four rightangles.
76. §1.6 Perpendicular Lines 1) PRN is an acute angle. False. 2) 4 8 True
77. m T §1.6 Perpendicular Lines If a line m is in a plane and T is a point in m, then there exists exactly ___ line in that plane that is perpendicular to m at T. one
![Angles ,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/85/17-angles-and-perpendicular-lines-1-320.jpg)
