11X1 T08 01 angle theorems (2011)

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  • 1. Euclidean Geometry

2. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to 3. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to is perpendicular to 4. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to is perpendicular to is congruent to 5. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to is perpendicular to is congruent to||| is similar to 6. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to is perpendicular to is congruent to||| is similar to therefore 7. Euclidean GeometryGeometry DefinitionsNotation|| is parallel to is perpendicular to is congruent to||| is similar to therefore because 8. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular to is congruent to||| is similar to therefore because 9. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular to X XY is produced to A YA is congruent to||| is similar to therefore because 10. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular to BXXY is produced to A YA is congruent toYX is produced to B||| is similar to therefore because 11. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular toBX XY is produced to AY A is congruent toYX is produced to B||| is similar tofoot the base or the bottom therefore because 12. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular to BXXY is produced to AY A is congruent toYX is produced to B||| is similar tofoot the base or the bottom therefore A B B is the foot of the perpendicular becauseD C 13. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular toBX XY is produced to A YA is congruent toYX is produced to B||| is similar tofoot the base or the bottom therefore A B B is the foot of the perpendicular because D Ccollinear the points lie on the same line 14. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular toBX XY is produced to AYA is congruent toYX is produced to B||| is similar tofoot the base or the bottom therefore A B B is the foot of the perpendicular because D Ccollinear the points lie on the same lineconcurrent the lines intersect at the same point 15. Euclidean GeometryGeometry DefinitionsTerminologyNotationproduced the line is extended|| is parallel to is perpendicular toBX XY is produced to AYA is congruent toYX is produced to B||| is similar tofoot the base or the bottom therefore A B B is the foot of the perpendicular because D Ccollinear the points lie on the same lineconcurrent the lines intersect at the same pointtransversal a line that cuts two(or more) other lines 16. Naming ConventionsAngles and Polygons are named in cyclic order 17. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC A If no ambiguity BC B or B 18. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC PA If no ambiguity V VPQT or QPVTBC B or B Q or QTVP etc T 19. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC PA If no ambiguity V VPQT or QPVTBC B or B Q or QTVP etc TParallel Lines are named in corresponding order 20. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC PA If no ambiguity V VPQT or QPVTBC B or B Q or QTVP etc TParallel Lines are named in corresponding order BDAB||CD A C 21. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC PA If no ambiguity V VPQT or QPVTBC B or B Q or QTVP etc TParallel Lines are named in corresponding order BDAB||CD ACEqual Lines are marked with the same symbol 22. Naming ConventionsAngles and Polygons are named in cyclic order ABC or ABC PA If no ambiguity V VPQT or QPVTBC B or B Q or QTVP etc TParallel Lines are named in corresponding order BDAB||CD ACEqual Lines are marked with the same symbolQRPQ = SR PPS = QRS 23. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 24. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 25. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 2. Construction of new lines, state clearly your construction so thatanyone reading your proof could recreate the construction. 26. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 2. Construction of new lines, state clearly your construction so thatanyone reading your proof could recreate the construction. 3. A recognised geometrical theorem (or assumption), any theoremyou are using must be explicitly stated, whether it is in the algebraicstatement or the reason itself. 27. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 2. Construction of new lines, state clearly your construction so thatanyone reading your proof could recreate the construction. 3. A recognised geometrical theorem (or assumption), any theoremyou are using must be explicitly stated, whether it is in the algebraicstatement or the reason itself.e.g. A 25 120 180sum 180 28. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 2. Construction of new lines, state clearly your construction so thatanyone reading your proof could recreate the construction. 3. A recognised geometrical theorem (or assumption), any theoremyou are using must be explicitly stated, whether it is in the algebraicstatement or the reason itself.e.g. A 25 120 180sum 180Your reasoning should be clear and concise 29. Constructing ProofsWhen constructing a proof, any line that you state must be one of thefollowing; 1. Given information, do not assume information is given, e.g. if youare told two sides are of a triangle are equal dont assume it isisosceles, state that it is isosceles because two sides are equal. 2. Construction of new lines, state clearly your construction so thatanyone reading your proof could recreate the construction. 3. A recognised geometrical theorem (or assumption), any theoremyou are using must be explicitly stated, whether it is in the algebraicstatement or the reason itself.e.g. A 25 120 180sum 180Your reasoning should be clear and concise 4. Any working out that follows from lines already stated. 30. Angle Theorems 31. Angle Theorems Angles in a straight line add up to 180 32. Angle Theorems DAngles in a straight line add up to 180x 30ABC 33. Angle Theorems DAngles in a straight line add up to 180Ax 30 x 30 180straight ABC 180 BC 34. Angle Theorems DAngles in a straight line add up to 180Ax 30 x 30 180straight ABC 180 BCx 150 35. Angle Theorems DAngles in a straight line add up to 180Ax 30 x 30 180straight ABC 180 BCx 150Vertically opposite angles are equal 36. Angle TheoremsDAngles in a straight line add up to 180A x 30 x 30 180straight ABC 180 BCx 150 Vertically opposite angles are equal3x 39 37. Angle TheoremsDAngles in a straight line add up to 180A x 30 x 30 180straight ABC 180 BCx 150 Vertically opposite angles are equal3x 393x 39vertically opposite s are 38. Angle TheoremsDAngles in a straight line add up to 180A x 30 x 30 180straight ABC 180 BCx 150 Vertically opposite angles are equal3x 393x 39vertically opposite s are x 13 39. Angle TheoremsDAngles in a straight line add up to 180A x 30 x 30 180straight ABC 180 BCx 150 Vertically opposite angles are equal3x 393x 39vertically opposite s are x 13 Angles about a point equal 360 40. Angle TheoremsD Angles in a straight line add up to 180Ax 30 x 30 180straight ABC 180 BCx 150Vertically opposite angles are equal 3x 39 3x 39vertically opposite s are x 13 150 Angles about a point equal 360 15 y120 41. Angle TheoremsDAngles in a straight line add up to 180Ax 30 x 30 180 straight ABC 180 BCx 150Vertically opposite angles are equal 3x 39 3x 39 vertically opposite s are x 13 150 Angles about a point equal 360 15 y120 y 15 150 120 360revolution 360 42. Angle TheoremsDAngles in a straight line add up to 180Ax 30 x 30 180 straight ABC 180 BCx 150Vertically opposite angles are equal 3x 39 3x 39 vertically opposite s are x 13 150 Angles about a point equal 360 15 y120 y 15 150 120 360revolution 360 y 75 43. Parallel Line Theoremsa bdc fe h g 44. Parallel Line TheoremsAlternate angles (Z) are equal a bd c f e hg 45. Parallel Line TheoremsAlternate angles (Z) are equala bdc fc f alternate s , || lines e h g 46. Parallel Line TheoremsAlternate angles (Z) are equala bdc fc f alternate s , || lines e hd e g 47. Parallel Line TheoremsAlternate angles (Z) are equala bdc fc f alternate s , || lines e hd e gCorresponding angles (F) are equal 48. Parallel Line TheoremsAlternate angles (Z) are equal a bd c fc f alternate s , || lines e hd egCorresponding angles (F) are equalae corresponding s , || lines 49. Parallel Line TheoremsAlternate angles (Z) are equal a bd c fc f alternate s , || lines e hd egCorresponding angles (F) are equal aecorresponding s , || lines b f cg d h 50. Parallel Line TheoremsAlternate angles (Z) are equal a bd c fc f alternate s , || lines e hd egCorresponding angles (F) are equal aecorresponding s , || lines b f cg d hCointerior angles (C) are supplementary 51. Parallel Line TheoremsAlternate angles (Z) are equal a bd c fc falternate s , || lines e hd egCorresponding angles (F) are equal ae corresponding s , || lines b f cg d hCointerior angles (C) are supplementary c e 180cointerior s 180, || lines 52. Parallel Line TheoremsAlternate angles (Z) are equal a bd c fc falternate s , || lines e hd egCorresponding angles (F) are equal ae corresponding s , || lines b f cg d hCointerior angles (C) are supplementary c e 180cointerior s 180, || lines d f 180 53. ABe.g. 120 P yx 65 CQ D 54. ABe.g.120 P yx 65 CQDx 65 180 straight CQD 180 55. ABe.g.120 P yx 65 CQDx 65 180 straight CQD 180 x 115 56. A Be.g. 120X P Yy x 65CQDx 65 180straight CQD 180 x 115 Construct XY||CD passing through P 57. ABe.g. 120 X P Yy x 65 C Q Dx 65 180straight CQD 180 x 115 Construct XY||CD passing through PXPQ 65alternate s , XY || CQ 58. ABe.g. 120 X PYy x 65 C QDx 65 180straight CQD 180 x 115 Construct XY||CD passing through PXPQ 65alternate s , XY || CQ XPB 120 180 cointerior s 180 , AB || XY 59. ABe.g. 120 X PYy x 65 C QDx 65 180straight CQD 180 x 115 Construct XY||CD passing through PXPQ 65alternate s , XY || CQ XPB 120 180 cointerior s 180 , AB || XY XPB 60 60. ABe.g. 120 X PYy x 65 C QDx 65 180straight CQD 180 x 115 Construct XY||CD passing through PXPQ 65alternate s , XY || CQ XPB 120 180 cointerior s 180 , AB || XY XPB 60BPQ XPQ XPB common 61. ABe.g. 120 X PYy x 65 C QDx 65 180straight CQD 180 x 115 Construct XY||CD passing through PXPQ 65alternate s , XY || CQ XPB 120 180 cointerior s 180 , AB || XY XPB 60BPQ XPQ XPB common y 65 60 y 125 62. ABe.g. 120 X P Yy x 65 C QDx 65 180straight CQD 180 x 115 Construct XY||CD passing through P Book 2XPQ 65alternate s , XY || CQ Exercise 8A;XPB 120 180 cointerior s 180 , AB || XY 1cfh, 2bdeh, 3bd, 5bcf,XPB 60 6bef, 10bd,BPQ XPQ XPB common 11b, 12bc, y 65 6013ad, 14, 15 y 125