Angle Geometry

Measuring Angles  AOB = 27o AOC = 122° C A B O A

Naming Angles Angles are named using the three letters that form them. The middle of the three letters is the vertex of the angle (pointy bit) E D F DEF or DÊF or FED or FÊ D

Types of angle Obtuse angle (between 90° and 180°) Acute angle (less than 90°) Right Angle (90°) Reflex angle (more than 180°) Straight angle (180°)

O 1 0 S C Angle Reasons Complementary angles add up to 90° So the complement of 40° is 50° 1 0 S Supplementary angles add up to 180° The supplement of 40° is 140° Angle Reasons x = 50° 130° x Adjacent angles on a straight line sum to 180°

Vertically opposite angles are equal 70° x x = 70° 80° 110° 120° x Angles at a point sum to 360° x = 50°

Angles on Parallel Lines transversal 110° x   x = 110°  Corresponding angles on parallel lines are equal    50° x x = 50°  Alternate angles on parallel lines are equal  110° x x = 70°  Co-interior angles on parallel lines sum to 180° 

Angles in a triangle sum to 180° demo 70° 50° x x = 60° Angles in a triangle sum to 180° demo 50° x x = 50° Base angles of an isosceles triangle are equal x x = 60° Each angle in an equilateral triangle is 60°

Angles in a quadrilateral sum to 360° 50° 70° x x = 120° Exterior angle of a triangle equals the sum of the 2 interior opposite angles 120° 100° 80° x x = 60° Angles in a quadrilateral sum to 360°

Types of Quadrilaterals □ I □ » I I II II ^ ^ » I I Parallelogram □ □ I Rectangle Rhombus □ I □ » I I I I II □ □ II I » Square Trapezium Kite

Angles on Parallel Lines transversal 110° x   x = 110°  Corresponding angles on parallel lines are equal    50° x x = 50°  Alternate angles on parallel lines are equal  110° x x = 70°  Co-interior angles on parallel lines sum to 180° 

Angles on parallel lines Types of Triangles (according to sides) Scalene - no equal sides Isosceles - 2 equal sides Equilateral - 3 equal sides Types of Triangles (according to angles) Acute angled triangle - all angles less than 90° Right angled triangle - one 90° angle Obtuse angled triangle - one angle  90°

Polygons Can you remember the special names for polygons that have five, six, eight and ten sides A polygon is a closed figure made up of straight sides.

Names of polygons: Number of sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 8 Octagon 10 Decagon A regular polygon has all it’s sides equal and all it’s angles equal. E.g. a square is a regular quadrilateral.

Interior and Exterior angles The blue angles are interior angles of the triangle. The grey angles are exterior angles of the triangle.

Exterior Angles of Polygons The sum of the exterior angles of any polygon is always 360° A regular polygon is one with all sides equal in length and all angles equal in size. ° So in a regular polygon with n sides, each exterior angle is The sum of the exterior angles = 360° Each exterior angle of this regular hexagon is 360 ÷ 6, so b = 60°

Interior Angles of Polygons The sum of the interior angles for a polygon with n sides is (n – 2) x 180o example: Find x Number of sides: n = 5 Sum of interior angles = (5 – 2) x 180 = 3 x 180 = 540° x + 120 + 160 + 45 + 170 = 540 x + 495 = 540 x = 45°

eg: Find the size of each interior angle of a regular octagon To find the size of each interior angle of a regular polygon, divide the sum by the number of sides. eg: Find the size of each interior angle of a regular octagon soln: n = 8 Cool, eh? Sum of interior angles = (8 - 2) x 180° = 6 x 180° = 1080° Each interior angle = 1080 ÷ 8 = 135°

. . . . . Constructions 1) To Bisect an Angle Method: P . . M With centre B, scribe an arc to cut the two arms. Label these points P & Q B Q C From P and Q draw arcs to intersect at a point which we will label M Join M to B 2) To Bisect a Line Segment P Method: . . From A and B scribe arcs to intersect at 2 points, P and Q A B Join P to Q Q

• 3) To construct an angle of 90° at a given point P on a line segment AB Method: Q With centre P, scribe an arc to cut the line segment at M and N From M and N scribe 2 arcs to meet at Q A M • N B P Join Q to P 4) To construct an angle of 90° from a given point P off a line segment AB Method: • P With centre P, scribe an arc to cut the line segment at M and N From M and N scribe arcs to meet at Q A M N B Join P to Q Q

5) To Construct a Triangle Given the lengths of the 3 sides, say AB = 5 cm, AC = 4 cm and BC = 3 cm Method: Draw one of the sides, measuring it with a ruler, say AB With centre A scribe an arc of length 4 cm C With centre B scribe an arc of length 3 cm Where the 2 arcs meet is point C Join C to A Join C to B A 5 cm B

. 6) To construct a Hexagon Method: Draw a circle With the same radius, step the radius off, around the circumference . Join the points with straight line segments to form a regular hexagon 7) To Construct an Angle of 60° on a given line segment XY Method: W With centre X and any radius, scribe an arc which cuts XY at Z Z X Y With centre Z and using the same radius, scribe an arc to cut the first one at W Join W to X. Angle WXY is 60°

Parts of the Circle tangent Segment Sector

Compass Directions N to E is 90° N to NE is 45° N to NNE is 22.5° N NW WNW ENE W E N to NE is 45° WSW ESE N to NNE is 22.5° SW SSW SSE SE S

Bearings A bearing is a direction. It is always measured from North in a clockwise direction. It always consists of 3 digits before the decimal point. eg 135°, 067°, 036.5° The compass directions as bearings are: 000° N S W E NE NW SW NNE ENE ESE SSE SSW WSW WNW NNW 315° 045° 337.5° 022.5° 067.5° 292.5° 090° 270° 247.5° 112.5° 225° 157.5° 202.5° 135° SE 180°

Bearings Example N 252° Find the bearing of the boat from the lighthouse. So the bearing is 252°