1.7 angles and perpendicular lines

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1.7 angles and perpendicular lines

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

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