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Chapter 1 lines and angles ii [compatibility mode]

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K
Khusaini Majid

1. The document defines key properties of angles associated with parallel lines and transversals, including corresponding angles, alternate angles, and interior angles. 2. It provides examples of using these properties to determine if lines are parallel and calculate angle measures. 3. Exercises at the end provide additional practice problems involving parallel lines and their associated angles.

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1 CHAPTER 1 Mohd Khusaini Majid Mrsm Kota Kinabalu
2 1111....1111 PPPPRRRROOOOPPPPEEEERRRRTTTTIIIIEEEESSSS OOOOFFFF AAAANNNNGGGGLLLLEEEESSSS AAAASSSSSSSSOOOOCCCCIIIIAAAATTTTEEEEDDDD WWWWIIIITTTTHHHH TTTTRRRRAAAANNNNSSSSVVVVEEEERRRRSSSSAAAALLLLSSSS AAAANNNNDDDD PPPPAAAARRRRAAAALLLLLLLLEEEELLLL LLLLIIIINNNNEEEESSSS
3 Identify transversals, corresponding IIIdddeeennntttiiifffyyy tttrrraaannnsssvvveeerrrsssaaalllsss,,, cccooorrrrrreeessspppooonnndddiiinnnggg aaaannnngggglllleeeessss,,,, aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaannnndddd iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss Transversal – a line that intersects two or more parallel lines P P R Q Q Q S P PQ and RS are transversals.
4 Corresponding angles b a e f d c TIPS Corresponding angles can be identified by the F shape. The F can be flipped over or turned upside down. Shaded angles are corresponding angles. Angles a and d are also corresponding angle.
5 Alternate angles TIPS Alternate angles can be identified by the Z shape or Z shape. Shaded angles are alternate angles. Angles a and b also alternate angles.
6 Interior angles TIPS Interior angles can be identified by the Z shape or Z shape. Shaded angles are interior angles. Angles c and d also interior angles.
7 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: aaaa)))) ccccoooorrrrrrrreeeessssppppoooonnnnddddiiiinnnngggg aaaannnngggglllleeeessss aaaarrrreeee eeeeqqqquuuuaaaallll Corresponding angles are equal
8 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: bbbb)))) aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaarrrreeee eeeeqqqquuuuaaaallll Alternate angles are equal
9 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: cccc)))) tttthhhheeee ssssuuuummmm ooooffff iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss iiiissss 111188880000°° The sum of interior angles is 180° or interior angles are supplementary
10 EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111AAAA 1. For each of the following diagrams, list i. four pairs of corresponding angles. ii. two pairs of alternate angles. iii. two pairs of interior angles. a) b)
11 Find the values of corresponding FFFiiinnnddd ttthhheee vvvaaallluuueeesss ooofff cccooorrrrrreeessspppooonnndddiiinnnggg aaaannnngggglllleeeessss,,,, aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaannnndddd iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss,,,, aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ppppaaaarrrraaaalllllllleeeellll lllliiiinnnneeeessss EEEExxxxaaaammmmpppplllleeee 1111 Find the value of x in each of the following. a) b)
12 a) The angles shown in the diagram are corresponding angles. Corresponding angles are equal. Thus, x = 125°. b) The angles shown in the diagram are interior angles. The sum of interior angles is 180°. Thus, x + 74° = 180° x = 180° - 74° = 106° Solution:
13 EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111BBBB Find the value of the unknown in each of the following. 1. 2. 3. 4.
14 Determine if two given DDDeeettteeerrrmmmiiinnneee iiifff tttwwwooo gggiiivvveeennn lllliiiinnnneeeessss aaaarrrreeee ppppaaaarrrraaaalllllllleeeellll bbbbaaaasssseeeedddd oooonnnn tttthhhheeee pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff aaaannnngggglllleeeessss aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ttttrrrraaaannnnssssvvvveeeerrrrssssaaaallllssss When two straight lines are intersected by a transversal, and • if the corresponding angles are equal, then the two lines are parallel. • if the alternate angles are equal, then the two lines are parallel. • if the interior angles are supplementary, then the two lines are parallel.
15 EEEExxxxaaaammmmpppplllleeee 2222 In the diagrams below, KLM, STU and LTV are straight lines. Determine if the lines KLM and STU are parallel. a) b)
Corresponding angle 16 Solution: a) The corresponding angles are equal. Thus, the lines KLM and STU are parallel. b) If the lines are parallel, then, LTU = 153°. Given STU is a straight line. Hence, STU = 180°. But LTU + LTS = 153° + 26° = 179° 180° Thus, the lines KLM and STU are not parallel.
EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111CCCC 1. Determine if the bold lines shown below are 17 parallel. a) b)
18 2. Determine which two lines are parallel in each of the following. a) b)
The diagram shows a parallelogram ABCD. Given EDC is a straight line. Calculate all the angles in the parallelogram. 19 Solve problems involving SSSooolllvvveee ppprrrooobbbllleeemmmsss iiinnnvvvooolllvvviiinnnggg tttthhhheeee pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff aaaannnngggglllleeeessss aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ttttrrrraaaannnnssssvvvveeeerrrrssssaaaallllssss
Corresponding angles Interior angles 20 DCB = EDA = 105° ADC = 180° - DCB = 180° - 105° = 75° DAB = EDA = 105° ABC = 180° - DAB = 180° - 105° = 75° Solution: Alternate angles Interior angle
EEEExxxxaaaammmmpppplllleeee 4444 The diagram shows two parallel lines intersected by two transversals. Find the values of x and y. 21
22 BSV = ABS x = 34° Angle y is the sum of ASW and WSC. ASW = 180° - UAS = 180° - 93° = 87° WSC = UBS = 34° Thus, y = 87° + 34° = 121° Solution:
EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111DDDD 1. In the diagram, ABCD is a trapezium and the line EF is an extension of AB. Given EAD = 77° and FBC = 25°. Calculate all the angles in the trapezium. 23
24 2. The following diagrams consist of straight lines. Find the values of x and y. a) b) c)
Alternate angles Interior angles 25 SSSSuuuummmmmmmmaaaarrrryyyy LLLLIIIINNNNEEEESSSS AAAANNNNDDDD AAAANNNNGGGGLLLLEEEESSSS IIIIIIII Parallel lines * The lines that will never meet * The lines that are always the same distance apart. Transversal * A line that intersects two or more parallel lines. Transversal and parallel lines Corresponding angles Corresponding angles * are equal * can be identified by the F shape. Alternate angles * are equal * can be identified by the Z shape. Interior angles * have a sum of 180° * can be identified by the U shape.
26 Reference Yoong Kwee Soon, Sim Kwang Yaw, Ding Hong Eng, Noorliah Binti Ahmat, Lai Jun Siew, Mathematics FORM 3, Ebiza Sdn. Bhd., 2004. Prepared by: mohd khusaini majid Mrsm Kota Kinabalu

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Chapter 1 lines and angles ii [compatibility mode]

  • 1. 1 CHAPTER 1 Mohd Khusaini Majid Mrsm Kota Kinabalu
  • 2. 2 1111....1111 PPPPRRRROOOOPPPPEEEERRRRTTTTIIIIEEEESSSS OOOOFFFF AAAANNNNGGGGLLLLEEEESSSS AAAASSSSSSSSOOOOCCCCIIIIAAAATTTTEEEEDDDD WWWWIIIITTTTHHHH TTTTRRRRAAAANNNNSSSSVVVVEEEERRRRSSSSAAAALLLLSSSS AAAANNNNDDDD PPPPAAAARRRRAAAALLLLLLLLEEEELLLL LLLLIIIINNNNEEEESSSS
  • 3. 3 Identify transversals, corresponding IIIdddeeennntttiiifffyyy tttrrraaannnsssvvveeerrrsssaaalllsss,,, cccooorrrrrreeessspppooonnndddiiinnnggg aaaannnngggglllleeeessss,,,, aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaannnndddd iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss Transversal – a line that intersects two or more parallel lines P P R Q Q Q S P PQ and RS are transversals.
  • 4. 4 Corresponding angles b a e f d c TIPS Corresponding angles can be identified by the F shape. The F can be flipped over or turned upside down. Shaded angles are corresponding angles. Angles a and d are also corresponding angle.
  • 5. 5 Alternate angles TIPS Alternate angles can be identified by the Z shape or Z shape. Shaded angles are alternate angles. Angles a and b also alternate angles.
  • 6. 6 Interior angles TIPS Interior angles can be identified by the Z shape or Z shape. Shaded angles are interior angles. Angles c and d also interior angles.
  • 7. 7 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: aaaa)))) ccccoooorrrrrrrreeeessssppppoooonnnnddddiiiinnnngggg aaaannnngggglllleeeessss aaaarrrreeee eeeeqqqquuuuaaaallll Corresponding angles are equal
  • 8. 8 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: bbbb)))) aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaarrrreeee eeeeqqqquuuuaaaallll Alternate angles are equal
  • 9. 9 Determine that for parallel linesDDDeeettteeerrrmmmiiinnneee ttthhhaaattt fffooorrr pppaaarrraaalllllleeelll llliiinnneeesss:::: cccc)))) tttthhhheeee ssssuuuummmm ooooffff iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss iiiissss 111188880000°° The sum of interior angles is 180° or interior angles are supplementary
  • 10. 10 EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111AAAA 1. For each of the following diagrams, list i. four pairs of corresponding angles. ii. two pairs of alternate angles. iii. two pairs of interior angles. a) b)
  • 11. 11 Find the values of corresponding FFFiiinnnddd ttthhheee vvvaaallluuueeesss ooofff cccooorrrrrreeessspppooonnndddiiinnnggg aaaannnngggglllleeeessss,,,, aaaalllltttteeeerrrrnnnnaaaatttteeee aaaannnngggglllleeeessss aaaannnndddd iiiinnnntttteeeerrrriiiioooorrrr aaaannnngggglllleeeessss,,,, aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ppppaaaarrrraaaalllllllleeeellll lllliiiinnnneeeessss EEEExxxxaaaammmmpppplllleeee 1111 Find the value of x in each of the following. a) b)
  • 12. 12 a) The angles shown in the diagram are corresponding angles. Corresponding angles are equal. Thus, x = 125°. b) The angles shown in the diagram are interior angles. The sum of interior angles is 180°. Thus, x + 74° = 180° x = 180° - 74° = 106° Solution:
  • 13. 13 EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111BBBB Find the value of the unknown in each of the following. 1. 2. 3. 4.
  • 14. 14 Determine if two given DDDeeettteeerrrmmmiiinnneee iiifff tttwwwooo gggiiivvveeennn lllliiiinnnneeeessss aaaarrrreeee ppppaaaarrrraaaalllllllleeeellll bbbbaaaasssseeeedddd oooonnnn tttthhhheeee pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff aaaannnngggglllleeeessss aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ttttrrrraaaannnnssssvvvveeeerrrrssssaaaallllssss When two straight lines are intersected by a transversal, and • if the corresponding angles are equal, then the two lines are parallel. • if the alternate angles are equal, then the two lines are parallel. • if the interior angles are supplementary, then the two lines are parallel.
  • 15. 15 EEEExxxxaaaammmmpppplllleeee 2222 In the diagrams below, KLM, STU and LTV are straight lines. Determine if the lines KLM and STU are parallel. a) b)
  • 16. Corresponding angle 16 Solution: a) The corresponding angles are equal. Thus, the lines KLM and STU are parallel. b) If the lines are parallel, then, LTU = 153°. Given STU is a straight line. Hence, STU = 180°. But LTU + LTS = 153° + 26° = 179° 180° Thus, the lines KLM and STU are not parallel.
  • 17. EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111CCCC 1. Determine if the bold lines shown below are 17 parallel. a) b)
  • 18. 18 2. Determine which two lines are parallel in each of the following. a) b)
  • 19. The diagram shows a parallelogram ABCD. Given EDC is a straight line. Calculate all the angles in the parallelogram. 19 Solve problems involving SSSooolllvvveee ppprrrooobbbllleeemmmsss iiinnnvvvooolllvvviiinnnggg tttthhhheeee pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff aaaannnngggglllleeeessss aaaassssssssoooocccciiiiaaaatttteeeedddd wwwwiiiitttthhhh ttttrrrraaaannnnssssvvvveeeerrrrssssaaaallllssss
  • 20. Corresponding angles Interior angles 20 DCB = EDA = 105° ADC = 180° - DCB = 180° - 105° = 75° DAB = EDA = 105° ABC = 180° - DAB = 180° - 105° = 75° Solution: Alternate angles Interior angle
  • 21. EEEExxxxaaaammmmpppplllleeee 4444 The diagram shows two parallel lines intersected by two transversals. Find the values of x and y. 21
  • 22. 22 BSV = ABS x = 34° Angle y is the sum of ASW and WSC. ASW = 180° - UAS = 180° - 93° = 87° WSC = UBS = 34° Thus, y = 87° + 34° = 121° Solution:
  • 23. EEEExxxxeeeerrrrcccciiiisssseeee 1111....1111DDDD 1. In the diagram, ABCD is a trapezium and the line EF is an extension of AB. Given EAD = 77° and FBC = 25°. Calculate all the angles in the trapezium. 23
  • 24. 24 2. The following diagrams consist of straight lines. Find the values of x and y. a) b) c)
  • 25. Alternate angles Interior angles 25 SSSSuuuummmmmmmmaaaarrrryyyy LLLLIIIINNNNEEEESSSS AAAANNNNDDDD AAAANNNNGGGGLLLLEEEESSSS IIIIIIII Parallel lines * The lines that will never meet * The lines that are always the same distance apart. Transversal * A line that intersects two or more parallel lines. Transversal and parallel lines Corresponding angles Corresponding angles * are equal * can be identified by the F shape. Alternate angles * are equal * can be identified by the Z shape. Interior angles * have a sum of 180° * can be identified by the U shape.
  • 26. 26 Reference Yoong Kwee Soon, Sim Kwang Yaw, Ding Hong Eng, Noorliah Binti Ahmat, Lai Jun Siew, Mathematics FORM 3, Ebiza Sdn. Bhd., 2004. Prepared by: mohd khusaini majid Mrsm Kota Kinabalu