GEOMETRY Lines and Angles

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

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

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

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

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 45 °  ABC = a)b) Z Y X  XYZ = 220 °

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

Quadrilaterals - Are four sided figures - Have two diagonals Diagonals go between opposite corners Special Spaces and their Properties SquareRectangle/OblongParallelogram Kite/DiamondArrowheadTrapezium Isosceles TrapeziumRhombus Dashes on lines indicate equal length and arrows indicate parallel lines

ANGLE STATEMENTS Remember: You must supply a geometrical reason when calculating angles! Adjacent Angles On A Straight Line Add To 180° x37° x + 37 = 180 (adj.  ’s on a str. line = 180°) - 37 x = 143° x119° x = 180 (adj.  ’s on a str. line = 180°) x = 61° 1. Find x 2. Find x

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° x + 50 = 90(complementary angles) - 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 x = 170°(therefore 170° is the supplement of 10°)

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

Interior Angles In A Triangle Add To 180° x52° 85° 1. Find x x = 180 (  ’s in a triangle add to 180°) x = x = 43° 2. Find x x 46° x = 180 (  ’s in a triangle add to 180°) x = x = 44° Base Angles In An Isosceles Triangle Are Equal 1. Find x x 40° x + x + 40 = 180 2x + 40 = x = 140 ÷ 2 x = 70° (base  ’s of an isosceles triangle) (  ’s in a triangle add to 180°)

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

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 divide into triangles from one corner Interior angle sum = (6 – 2) x 180 n = 6 (or 4 triangles) (interior angle sum of a polygon) Interior angle sum = 720° 2. Find x x Interior angle sum = (5 – 2) x 180 n = 5 or 3 triangles (interior angle sum of a polygon) 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 72° Exterior angle = 72° x + 72 = 180 x = 108° (adjacent. angles on a straight line = 180°)

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

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. e.g. Find x 122° x x = 122° (Corresponding angles on parallel lines are equal)

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 77° x x + 77 = 180(Co-interior angles in parallel lines add to 180°) - 77 x = 103° Remember to always add ‘on parallel lines’ with your angle statements

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 xy First calculate ratio between corresponding sides A B C F E G AC = 15 EG 6 = 2.5 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 = 10 y = 20 ÷ 2.5 = 8(Similar Triangles)

Parts Of A Circle RadiusDiameterChordArc SectorSegmentCircumferenceTangent

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

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

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

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. 79° x x + 79 = x = 101° (opp.  ’s, cyc. quad) Proof: B 2B A2A 2A + 2B = 360 A + B = 180 Exterior Angle Of A Cyclic Quadrilateral Equals Opposite Interior Angle 110° x x = 110° (ext. , cyc. quad) e.g.Proof: A B C A + B = 180 B + C = 180 B = 180 – C A – C = 180 A = C