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Angle Properties

Angle Properties. Revision of Basic Angle Properties. Isosceles Triangles in Circles. Angles in a semi-circle. Tangent line on a circle. www.mathsrevision.com. Interior / Exterior Angles in Polygon. Exam Questions. Revision Angle Properties. Learning Intention. Success Criteria.

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Angle Properties

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  1. Angle Properties Revision of Basic Angle Properties Isosceles Triangles in Circles Angles in a semi-circle Tangent line on a circle www.mathsrevision.com Interior / Exterior Angles in Polygon Exam Questions Created by Mr Lafferty

  2. Revision Angle Properties Learning Intention Success Criteria • To know the basic properties for angles. • We are revising all the basic properties in Level 3 and 4. • Solve problems using properties. www.mathsrevision.com Created by Mr. Lafferty

  3. 115o 95o 120o 50o 40o 34o 34o Revision Angle Properties 65o Two angles making a straight line add to 180o DEMO 145o Angles round a point Add up to 360o www.mathsrevision.com 90o 146o 146o 3 angles in a triangle ALWAYS add up to 180o. angles opposite each other at a cross are equal. Created by Mr. Lafferty

  4. 105o Think ! xo 100o 90 100 + 105 295 Angles and Triangles Example Find angle x. 360o www.mathsrevision.com Angle xo = 360o - (90o + 100o + 105o) = 360o – 295o = 65o Created by Mr Lafferty

  5. Angles and Triangles www.mathsrevision.com Equilateral Triangle Isosceles triangle Right-angled triangle 3 equal sides 3 equal angles. 2 equal sides 2 equal angles (base) One angle is 90o Created by Mr Lafferty

  6. Example 1 65o Calculate angle a. a b Example 2 Calculate angles a, b and c a c Angles and Triangles Angle a = 180 – (90 + 65) = 180 – 155 = 25o www.mathsrevision.com Since the triangle is equilateral, angles a, b and c are all 60o(180/3) Created by Mr Lafferty

  7. Example 3 Calculate angle a. Example 4 130o y Calculate angles x and y x Angles and Triangles Angle a = 65o (base angles of an isosceles triangle are equal). b Angle b = 180 –(65 + 65) = 180 – 130 = 50o a 65o www.mathsrevision.com Created by Mr Lafferty

  8. Example 5 Calculate angles a and b. Angles and Triangles a b Isosceles triangle www.mathsrevision.com Created by Mr Lafferty

  9. Sum of Angles in a Triangle 38o 50o Copy out the following triangles and find the missing angles. xo 87o www.mathsrevision.com xo 32o xo xo Remember all the angles add up to 180o Created by Mr.Lafferty Math Dept

  10. d = 115o ao co bo ho eo go fo RevisionAngle Properties DEMO ALL angles in an equilateral triangle are 60o Two angles in a isosceles are equal www.mathsrevision.com h is corresponding to d and must be 115o b is opposite to d and must be 115o c is must be 65o (straight line) e is alternate to c and must also be 65o DEMO Created by Mr. Lafferty

  11. Angles in a Quadrilateral IMPORTANT : The angles in a quadrilateral ALWAYS add up to 360o B C bo co We have split the quadrilateral into two triangles www.mathsrevision.com do ao But for any triangle the sum of the angles is 1800 A D Hence for the quadrilateral we have 2 x 180o=360o

  12. Angles in a Quadrilateral Question : Find the missing angle below. The four angles of a quadrilateral add to = 360o w x 34o www.mathsrevision.com 100o yo z y Created by Mr.Lafferty

  13. Circle Angle Properties Learning Intention Success Criteria • Understand why isosceles triangles can be formed within circles. • We are learning about isosceles triangles within circles. • Solve problems using properties. www.mathsrevision.com Created by Mr. Lafferty

  14. Isosceles triangles in Circles When two radii are drawn to the ends of a chord, An isosceles triangle is formed. DEMO A B xo xo www.mathsrevision.com C Created by Mr Lafferty

  15. Isosceles triangles in Circles Special Properties of Isosceles Triangles Two equal lengths www.mathsrevision.com Two equal angles Angles in any triangle sum to 180o Created by Mr Lafferty

  16. Solution Angle at C is equal to: Isosceles triangles in Circles Q. Find the angle xo. B www.mathsrevision.com C xo Since the triangle is isosceles we have A 280o Created by Mr Lafferty

  17. Circle Angle Properties Learning Intention Success Criteria • Understand how a right angle is formed using semi-circle knowledge. • We are learning about angle in a semi-circle property. • Solve problems using angle properties. www.mathsrevision.com Created by Mr. Lafferty

  18. P A B Angles in a Semi-Circle KeyPoint for Angles in a Semi-circle DEMO A triangle APB inscribed within a semicircle with hypotenuse equal to the diameter will ALWAYS be right angled at P on the circumference. www.mathsrevision.com Remember - Angles in any triangle sum to 180o Created by Mr Lafferty

  19. Hints Angles in a Semi-Circle National 4 Example 1 : Sketch diagram and find all the missing angles. 20o 43o Look for right angle triangles www.mathsrevision.com Remember ! Angles in any triangle sum to 180o 47o 70o Created by Mr Lafferty

  20. Angles in a Semi-Circle National 4 Example 2 : Sketch the diagram. (a) Right down two right angle triangles (a) Calculate all missing angles. C D www.mathsrevision.com 60o E 25o A B Created by Mr Lafferty

  21. Circle Angle Properties Learning Intention Success Criteria • Understand the tangent property to a circle. • We are learning the tangent property to a circle. • Solve problems using angle properties. www.mathsrevision.com Created by Mr. Lafferty

  22. Which of the lines are tangent to the circle? Angles in a Semi-Circle Tangent Line A tangent line is a line that touches a circle at only one point. www.mathsrevision.com Created by Mr Lafferty

  23. Angles in a Semi-Circle Tangent Line The radius of the circle that touches the tangent line is called the point of contact radius. DEMO Special Property The point of contact radius is always perpendicular (right-angled) to the tangent line. www.mathsrevision.com Created by Mr Lafferty

  24. Polygons Interior and Exterior Angles Learning Intention Success Criteria • Understand the terms interior and exterior angles. • We are learning about interior and exterior angles for polygons. • Be able to calculate interior and exterior angles for a polygon. www.mathsrevision.com Created by Mr. Lafferty

  25. Polygons Interior and Exterior Angles A polygon is a “many-sided closed straight-lined figure” This 5-sided (polygon) is called a PENTAGON www.mathsrevision.com Irregular Pentagon Created by Mr. Lafferty

  26. Polygons Interior and Exterior Angles A polygon is a “many-sided closed straight-lined figure” If all the sides and angles are the same it is called REGULAR POLYGON. We will only be dealing with regular polygons in this section. www.mathsrevision.com Pentagon Hexagon Octagon Created by Mr. Lafferty

  27. 45o 72o 60o Pentagon Hexagon Octagon Interior Angle Interior Angles Polygons Interior and Exterior Angles Some useful points about regular polygons : • All the triangles around the centre are isosceles. • Angle at the centre is 360o • To find one angle at the centre, take 360o and divide it by how many triangles you have www.mathsrevision.com Created by Mr. Lafferty

  28. Pentagon (5 sided) Hexagon (6 sided) Heptagon (7 sided) Octagon (8 sided) Nonagon (9 sided) Decagon (10 sided) Level 3/4 - Polygons Worksheet Nat 5 www.mathsrevision.com Created by Mr. Lafferty

  29. Polygons Interior and Exterior Angles What you should have found : Interior angle = 180 – (360÷n) n = Number of sides www.mathsrevision.com eg . A hexagonal has interior angle is: Interior angle = 180 – (360÷6) = 120o Created by Mr. Lafferty

  30. Polygons Interior and Exterior Angles A E B This is called the “Exterior angle” O Pentagon Q D C www.mathsrevision.com Exterior angle = 180 – interior angle eg . For the pentagon above : Exterior angle = 180 – 108 = 72o Created by Mr. Lafferty

  31. Pentagon (5 sided) Hexagon (6 sided) Heptagon (7 sided) Octagon (8 sided) Nonagon (9 sided) Decagon (10 sided) Level 3/4 - Polygons Worksheet Nat 5 www.mathsrevision.com Created by Mr. Lafferty

  32. 3 marks

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