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 Wednesday, 19 April 2017 Created by Mr Lafferty

Revision Angle Properties www.mathsrevision.com Learning Intention Success Criteria We are revising all the basic properties in Level 3 and 4. To know the basic properties for angles. Solve problems using properties. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty

Revision Angle Properties www.mathsrevision.com 65o DEMO 145o 90o 146o Two angles making a straight line add to 180o DEMO 145o Angles round a point Add up to 360o 50o 40o www.mathsrevision.com 90o 146o 34o 146o 3 angles in a triangle ALWAYS add up to 180o. angles opposite each other at a cross are equal. 19-Apr-17 Created by Mr. Lafferty

Angles and Triangles Think ! www.mathsrevision.com 360o xo 90 100 + Example Find angle x. 360o www.mathsrevision.com 90 100 105 + 295 Angle xo = 360o - (90o + 100o + 105o) = 360o – 295o = 65o Wednesday, 19 April 2017 Created by Mr Lafferty

Right-angled triangle 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 Wednesday, 19 April 2017 Created by Mr Lafferty

Calculate angles a, b and c Angles and Triangles Example 1 a 65o Calculate angle a. Angle a = 180 – (90 + 65) = 180 – 155 = 25o Example 2 Calculate angles a, b and c a b c www.mathsrevision.com Since the triangle is equilateral, angles a, b and c are all 60o (180/3) Wednesday, 19 April 2017 Created by Mr Lafferty

Angles and Triangles www.mathsrevision.com Example 3 b Calculate angle a. Angle a = 65o (base angles of an isosceles triangle are equal). b Angle b = 180 –(65 + 65) = 180 – 130 = 50o 65o a Example 4 Calculate angles x and y y 130o x www.mathsrevision.com Wednesday, 19 April 2017 Created by Mr Lafferty

Calculate angles a and b. Angles and Triangles Example 5 Calculate angles a and b. a b Isosceles triangle www.mathsrevision.com Wednesday, 19 April 2017 Created by Mr Lafferty

Sum of Angles in a Triangle Copy out the following triangles and find the missing angles. 50o xo 38o 87o www.mathsrevision.com xo 32o xo xo Remember all the angles add up to 180o 19-Apr-17 Created by Mr.Lafferty Math Dept

Revision Angle Properties DEMO ALL angles in an equilateral triangle are 60o Two angles in a isosceles are equal d = 115o ao co bo go fo ho eo 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 19-Apr-17 Created by Mr. Lafferty

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 ao do A D But for any triangle the sum of the angles is 1800 Hence for the quadrilateral we have 2 x 180o=360o Wednesday, 19 April 2017

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 Wednesday, 19 April 2017 Created by Mr.Lafferty

Circle Angle Properties www.mathsrevision.com Learning Intention Success Criteria We are learning about isosceles triangles within circles. Understand why isosceles triangles can be formed within circles. Solve problems using properties. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty

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 Wednesday, 19 April 2017 Created by Mr Lafferty

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 Wednesday, 19 April 2017 Created by Mr Lafferty

Isosceles triangles in Circles Q. Find the angle xo. Solution Angle at C is equal to: B www.mathsrevision.com xo C Since the triangle is isosceles we have A 280o Wednesday, 19 April 2017 Created by Mr Lafferty

Circle Angle Properties www.mathsrevision.com Learning Intention Success Criteria We are learning about angle in a semi-circle property. Understand how a right angle is formed using semi-circle knowledge. Solve problems using angle properties. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty

Angles in a Semi-Circle KeyPoint for Angles in a Semi-circle P A B 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 Wednesday, 19 April 2017 Created by Mr Lafferty

Angles in a Semi-Circle National 4 Example 1 : Sketch diagram and find all the missing angles. 20o Hints 43o Look for right angle triangles www.mathsrevision.com Remember ! Angles in any triangle sum to 180o 47o 70o Wednesday, 19 April 2017 Created by Mr Lafferty

Angles in a Semi-Circle National 4 Example 2 : Sketch the diagram. (a) Right down two right angle triangles (a) Calculate all missing angles. D C www.mathsrevision.com 60o E 25o A B Wednesday, 19 April 2017 Created by Mr Lafferty

Circle Angle Properties www.mathsrevision.com Learning Intention Success Criteria We are learning the tangent property to a circle. Understand the tangent property to a circle. Solve problems using angle properties. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty

Angles in a Semi-Circle Tangent Line A tangent line is a line that touches a circle at only one point. Which of the lines are tangent to the circle? www.mathsrevision.com Wednesday, 19 April 2017 Created by Mr Lafferty

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 Wednesday, 19 April 2017 Created by Mr Lafferty

Polygons www.mathsrevision.com Interior and Exterior Angles Learning Intention Success Criteria We are learning about interior and exterior angles for polygons. Understand the terms interior and exterior angles. Be able to calculate interior and exterior angles for a polygon. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty

Polygons www.mathsrevision.com 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 19-Apr-17 Created by Mr. Lafferty

Polygons www.mathsrevision.com 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 19-Apr-17 Created by Mr. Lafferty

Polygons www.mathsrevision.com 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 Pentagon 72o Hexagon 60o Octagon 45o Interior Angles Interior Angle 19-Apr-17 Created by Mr. Lafferty

Level 3/4 - Polygons Worksheet Pentagon (5 sided) Hexagon (6 sided) Heptagon (7 sided) Nat 5 www.mathsrevision.com Octagon (8 sided) Nonagon (9 sided) Decagon (10 sided) 19-Apr-17 Created by Mr. Lafferty

Polygons www.mathsrevision.com 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 19-Apr-17 Created by Mr. Lafferty

Polygons www.mathsrevision.com 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 19-Apr-17 Created by Mr. Lafferty

Level 3/4 - Polygons Worksheet Pentagon (5 sided) Hexagon (6 sided) Heptagon (7 sided) Nat 5 www.mathsrevision.com Octagon (8 sided) Nonagon (9 sided) Decagon (10 sided) 19-Apr-17 Created by Mr. Lafferty

3 marks