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GEOMETRY

GEOMETRY. Lines and Angles. Right Angles. - Are 90 ° or a quarter turn through a circle. e.g. - Straight lines are 180 °. Angle Types. - Angles can be named according to their sizes. 1) Acute:. Are angles less than 90 °. 2) Obtuse:. Are angles between 90 ° and 180 °. 3) Reflex:.

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GEOMETRY

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  1. GEOMETRY Lines and Angles

  2. Right Angles - Are 90° or a quarter turn through a circle e.g. - Straight lines are 180° Angle Types - Angles can be named according to their sizes 1) Acute: Are angles less than 90° 2) Obtuse: Are angles between 90° and 180° 3) Reflex: Are angles between 180° and 360°

  3. Naming Angles - Where the rays meet (vertex) gives the middle letter of the angle name e.g. Name the following angles C X a) b)   B  ABC or CBA  XYZ or ZYX Y A Z Measuring Angles Make sure you read from the scale starting from 0 on the line! e.g. Measure the following angles a) b) X C B Y Z ABC = 43° XYZ = 141° A

  4. - When measuring reflex angles, measure smaller angle and subtract from 360° e.g. Measure the following angle C B ABC = 360 - 65 A = 295°

  5. Drawing Angles Don’t forget to add an arc to show the correct angle! e.g. Draw the following angles a) ABC = 62° b) XYZ = 156° C Z A B Y X - When drawing reflex angles, subtract angle from 360° and draw this new angle e.g. Draw the following angle C a) ABC = 274° 360 - 274 = 86 Add the arc to the outside to indicate a reflex angle A B

  6. Estimating Angles - Involves guessing how big an angle is - Firstly decide whether the angle is acute, obtuse or reflex e.g. Estimate the size of the following angles a) b) C X B ABC = 45° XYZ = 220° Y A Z

  7. Special Triangles Equilateral Triangle Isosceles Triangle • All sides are equal • Two sides are equal • All angles are equal • (60°) • Base angles are equal Right Angle Triangle Scalene Triangle • Contains a 90° angle • No sides are equal • No angles are equal Describing Triangles By Angles 1) Acute Triangle: All angles are less than 90° 2) Right Angle Triangle: Contains a 90° angle 3) Obtuse Triangle: Contains an angle greater than 90°

  8. Quadrilaterals Diagonals go between opposite corners - Are four sided figures - Have two diagonals Special Spaces and their Properties Square Rectangle/Oblong Parallelogram Kite/Diamond Arrowhead Trapezium Dashes on lines indicate equal length and arrows indicate parallel lines Isosceles Trapezium Rhombus

  9. ANGLE STATEMENTS Remember: You must supply a geometrical reason when calculating angles! Adjacent Angles On A Straight Line Add To 180° 1. Find x x + 37 = 180 (adj. ’s on a str. line = 180°) - 37 - 37 x 37° x = 143° 2. Find x x + 119 = 180 (adj. ’s on a str. line = 180°) - 119 - 119 119° x x = 61°

  10. Complementary Angles Add To 90° When two angles make up a right angle (i.e. 48° and 42° are complementary angles) e.g. Find x x + 50 = 90 (complementary angles) - 50 - 50 x 50° x = 40° (therefore 40° is the complement of 50°) Supplementary Angles Add To 180° When two angles make up a straight angle (i.e. 125° and 55° are supplementary angles) e.g. What angle is the supplement of 10°? x + 10 = 180 (supplementary angles) - 10 - 10 x = 170° (therefore 170° is the supplement of 10°)

  11. Vertically Opposite Angles Are Equal Vertically opposite angles are formed by two straight lines 1. Find x 2. Find x 12° 38° x 58° x x = 58° (vert. opp. ’s are =) x = 38 + 12 (vert. opp. ’s are =) x = 50° Angles At A Point Add To 360° 1. Find x 2. Find x x + 90 + 82 + 71 + 59 = 360 82° (’s at a point = 360°) x 34° x + 302 = 360 71° x 59° - 302 - 302 x + 34 = 360 (’s at a point = 360°) x = 58° - 34 - 34 x = 326°

  12. Interior Angles In A Triangle Add To 180° 1. Find x 2. Find x x 85° 46° x 52° x + 90 + 46 = 180 x + 85 + 52 = 180 (’s in a triangle add to 180°) (’s in a triangle add to 180°) x + 137 = 180 x + 136 = 180 - 137 - 137 - 136 - 136 x = 43° x = 44° Base Angles In An Isosceles Triangle Are Equal 1. Find x x + x + 40 = 180 (base ’s of an isosceles triangle) 2x + 40 = 180 (’s in a triangle add to 180°) - 40 - 40 40° 2x = 140 ÷ 2 ÷ 2 x x = 70°

  13. Exterior Angles Of A Polygon Add To 360° 1. Find x 2. Find x in this regular pentagon 68° Regular means equal sides and angles 55° x 56° 55° 68° x x + 68 + 55 + 56 + 68 + 55 = 360 5x = 360 (ext. ’s of a polygon add to 360°) (ext. ’s of a polygon add to 360°) x + 298 = 360 ÷ 5 ÷ 5 - 298 - 298 x = 72° x = 62° 3. A regular polygon has exterior angles of 36°. How many sides does it have? 36x = 360 The number of sides = the number of angles ÷ 36 ÷ 36 x = 10

  14. The Sum Of The Interior Angles Of A Polygon Is (n – 2)  180 n = number of sides of a polygon 1. Find the angle sum of this regular hexagon 2. Find x 72° x divide into triangles from one corner n = 6 (or 4 triangles) n = 5 or 3 triangles Interior angle sum = (6 – 2) x 180 Interior angle sum = (5 – 2) x 180 (interior angle sum of a polygon) (interior angle sum of a polygon) Interior angle sum = 720° Interior angle sum = 540° x = 540 ÷ 5 x = 108° Another method is to calculate an exterior angle first then use adjacent angles on a straight line to calculate interior angle Exterior angle = 72° x + 72 = 180 (adjacent. angles on a straight line = 180°) x = 108°

  15. Perpendicular Lines - Always cross at right angles. e.g. B AB is perpendicular to CD or AB  CD C D A Parallel Lines - Never meet and are always the same distance apart. e.g. E A B AB is parallel to CD or AB ⁄⁄ CD C D EF is known as a transversal F

  16. Angle Statements and Parallel Lines Alternate Angles On Parallel Lines Are Equal - There are two pairs of alternate angles between parallel lines and a transversal. e.g. e.g. Find x 113° x x = 113° (Alternate angles on parallel lines are equal) Corresponding Angles On Parallel Lines Are Equal - There are four pairs of corresponding angles between parallel lines and a transversal. e.g. Find x e.g. x = 122° (Corresponding angles on parallel lines are equal) 122° x

  17. Co-Interior Angles On Parallel Lines Add To 180 - There are two pairs of co-interior angles between parallel lines and a transversal. e.g. e.g. Find x x 77° x + 77 = 180 (Co-interior angles in parallel lines add to 180°) - 77 - 77 x = 103° Remember to always add ‘on parallel lines’ with your angle statements

  18. Bearings - Bearings are used to indicate directions - Are measured clockwise from North - Must be expressed using 3 digits (i.e. 000° to 360°) - Compass directions such as NW give directions but are not bearings 000° e.g. The compass points and their bearings: 315° 045° e.g. Draw a bearing of 051°: NW NE N W E 270° 090° 51° SW SE 225° 135° e.g. What is the bearing of R from N? S N 180° Bearing = 180 + 37 = 217° 37°

  19. Similar Triangles And Other Shapes - One shape is similar to another if they have exactly the same shape. The ratios of the corresponding sides are therefore the same. - Triangles are similar if they have the same angles e.g. The following two triangles are similar. Work out the lengths x and y B F First calculate ratio between corresponding sides x 4 y 20 AC = 15 EG 6 E G A 6 C = 2.5 15 To find x we need to multiply the corresponding side by the ratio: To find y we need to divide the corresponding side by the ratio: x = 4 × 2.5 y = 20 ÷ 2.5 = 10 (Similar Triangles) = 8 (Similar Triangles)

  20. Parts Of A Circle Radius Diameter Chord Arc Sector Segment Circumference Tangent

  21. Angle Properties of Circles Base Angles Of An Isosceles Triangle Are Equal Because two sides of the triangle are radii, an isosceles triangle is formed e.g. 40° x r r x = 40° (base ’s of an isosceles triangle) The Angle At Centre Is Twice The Angle At The Circumference e.g. Find x Proof: x = 180 – 2A x = 2 × 42 x + C = 180 42° A B ( at centre = 2 ×  at circumf.) 180 – 2A + C = 180 x C = 2A A x C D D = 2B x = 84° C + D = 2A + 2B C + D = 2(A + B)

  22. Angle In A Semi Circle Is A Right Angle - This case is a special version of the previous rule e.g. x = 90° ( in a semi-circle) x Angles On The Same Arc Are Equal e.g. Find x: Proof: C = 2A x = 32° C = 2B (’s on the same arc) 2A = 2B C A = B 32° A x B There are 2 arcs joining angles

  23. The Angle Between Tangent And Radius Is A Right Angle e.g. x = 90° (tangent  radius) x If Two Tangents Are Drawn From A Point To A Circle They Are The Same Length e.g. 2x + 54 = 180 ( sum isos. triangle) 54° - 54 - 54 2x = 126 ÷ 2 ÷ 2 y x = 63° x y + 63 = 90 (tangent  radius) - 63 - 63 y = 27°

  24. Cyclic Quadrilaterals - Are four sided figures with all four vertices (corners) lying on the same circle. Opposite Angles Of A Cyclic Quadrilateral Add To 180 e.g. Proof: x + 79 = 180 x (opp. ’s, cyc. quad) 2A + 2B = 360 - 79 - 79 2B A + B = 180 B 2A A x = 101° 79° Exterior Angle Of A Cyclic Quadrilateral Equals Opposite Interior Angle e.g. Proof: A + B = 180 A 110° x = 110° B + C = 180 B = 180 – C (ext. , cyc. quad) A + 180 – C = 180 B A = C x C

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