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- 1. Angles<br /><ul><li>§1.6Angles
- 2. §1.6Angle Measure
- 3. §1.6The Angle Addition Postulate
- 4. §1.6Adjacent Angles and Linear Pairs of Angles
- 5. §1.6Complementary and Supplementary Angles
- 6. §1.6Congruent Angles
- 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. 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. 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. 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. D<br />2<br />F<br />DEF<br />2<br />E<br />FED<br />E<br />Angles<br />Symbols:<br />
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Is m a larger than m b ?<br />? ? ?<br />60°<br />60°<br />
- 26. End of Lesson<br />
- 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. 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 mRSX, mXST, and RST.<br />
- 29. R<br />X<br />S<br />T<br />§1.6 The Angle Addition Postulate<br />1) How does the sum of mRSX and mXST compare to mRST ?<br />Their sum is equal to the measure of RST .<br />mXST = 30<br />+ mRSX = 45<br />= mRST = 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. P<br />1<br />Q<br />A<br />2<br />R<br />§1.6 The Angle Addition Postulate<br />m1 + m2 = mPQR.<br />There are two equations that can be derived using Postulate 3 – 3.<br />m1 = mPQR –m2 <br />These equations are true no matter where A is locatedin the interior of PQR. <br />m2 = mPQR –m1 <br />
- 31. X<br />1<br />Y<br />W<br />2<br />Z<br />§1.6 The Angle Addition Postulate<br />Find m2 if mXYZ = 86 and m1 = 22.<br />Postulate 3 – 3.<br />m2 + m1 = mXYZ<br />m2 = mXYZ –m1 <br />m2 = 86 – 22<br />m2 = 64<br />
- 32. C<br />D<br />(5x – 6)°<br />2x°<br />B<br />A<br />§1.6 The Angle Addition Postulate<br />Find mABC and mCBD if mABD = 120.<br />mABC + mCBD = mABD<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 />mCBD = 5x – 6 <br />mABC = 2x<br />mCBD = 5(18) – 6 <br />mABC = 2(18)<br />mCBD = 90 – 6 <br />mABC = 36<br />mCBD = 84 <br />
- 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. 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 />m1 = m2<br />
- 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 mCAN = 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. End of Lesson<br />
- 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. 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. 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. 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. 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. 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. End of Lesson<br />
- 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. 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 />mABC + mDEF = 30 + 60 = 90<br />
- 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. 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 />mH + mI = 90<br />mPHQ + mQHS = 90<br />mTZU + mVZW = 90<br />
- 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 />mABC + mDEF = 50 + 130 = 180<br />
- 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 />mH + mI = 180<br />mPHQ + mQHS = 180<br />mTZU + mUZV = 180<br />and<br />mTZU + mVZW = 180<br />
- 50. End of Lesson<br />
- 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. 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 />mB = mV<br />
- 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 />mX = mZ<br />
- 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. §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. 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 />m1 + m4 = 180<br />Hands-On<br />B. 3 is supplementary to 4.<br />m3 + m4 = 180<br />Therefore, it can be shown that<br />1 3<br />Likewise, it can be shown that<br />24<br />
- 57. 1<br />4<br />2<br />3<br />§1.6 Congruent Angles<br />1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3<br />x = 4; 3 = 19°<br />2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4<br />x = 56; 4 = 65°<br />3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3<br />x = 9; 3 = 127°<br />4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4<br />x = 10; 4 = 97°<br />
- 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. 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. §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. §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. 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. 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. 52°<br />52°<br />A<br />B<br />§1.6 Congruent Angles<br />Suppose A B and mA = 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. §1.6 Congruent Angles<br />If 1 is complementary to 3,<br /> 2 is complementary to 3,<br /> and m3 = 25,<br /> What are m1 and m2 ?<br />m1 + m3 = 90 Definition of complementary angles.<br />m1 = 90 - m3 Subtract m3 from both sides.<br />m1 = 90 - 25Substitute 25 in for m3.<br />m1 = 65Simplify the right side.<br />You solve for m2<br />m2 + m3 = 90 Definition of complementary angles.<br />m2 = 90 - m3 Subtract m3 from both sides.<br />m2 = 90 - 25Substitute 25 in for m3.<br />m2 = 65Simplify the right side.<br />
- 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 m1 = 2x + 3 and the m3 = 3x - 14, then find the m3<br />x = 17; 3 = 37°<br />2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC<br />x = 29; EBC = 121°<br />3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4<br />x = 16; 4 = 39°<br />4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1<br />x = 18; 1 = 43°<br />
- 67. Suppose you draw two angles that are congruent and supplementary.<br />What is true about the angles?<br />
- 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. 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 = m3<br />m 3 = 64<br />3 is supplementary to 4<br />Given<br />Definition of supplementary.<br />m3 + m4 = 180<br />64 + m4 = 180<br />m4 = 180 – 64<br />m4 = 116<br />
- 70. End of Lesson<br />
- 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. 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. m<br />n<br />§1.6 Perpendicular Lines<br />Perpendicular lines are lines that intersect to form a<br />right angle.<br />
- 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 />m4 + 90 = 180, m4 = 90<br />Vertical angles are congruent<br />6) 2 is a right angle.<br />
- 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. §1.6 Perpendicular Lines<br />1) PRN is an acute angle.<br />False.<br />2) 4 8<br />True<br />
- 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. Homework: p. 40-42 (2-46even, 50-56 all)<br />

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