1. Angle Properties Revision of Basic Angle Properties
Isosceles Triangles in Circles
Angles in a semi-circle
Tangent line on a circle
Interior / Exterior Angles in Polygon
Exam Questions
Wednesday, 19 April 2017
Created by Mr Lafferty

2. 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.
19-Apr-17
Created by Mr. Lafferty

3. 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
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

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

5. Right-angled triangle
Angles and Triangles
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

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

7. 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 –( )
= 180 – 130 = 50o
65o a
Example 4
Calculate angles x and y y
130o x
Wednesday, 19 April 2017
Created by Mr Lafferty

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

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

10. 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 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

12. 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 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

13. Angles in a Quadrilateral
Question : Find the missing angle below.
The four angles of a quadrilateral add to = 360o w x
34o
100o yo z y
Wednesday, 19 April 2017
Created by Mr.Lafferty

19. 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.
19-Apr-17
Created by Mr. Lafferty

20. 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 C
Wednesday, 19 April 2017
Created by Mr Lafferty

21. Isosceles triangles in Circles
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180o
Wednesday, 19 April 2017
Created by Mr Lafferty

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

25. 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.
19-Apr-17
Created by Mr. Lafferty

26. 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.
Remember - Angles in any triangle sum to 180o
Wednesday, 19 April 2017
Created by Mr Lafferty

27. 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
Remember !
Angles in any triangle
sum to 180o
47o
70o
Wednesday, 19 April 2017
Created by Mr Lafferty

28. 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
60o E
25o A B
Wednesday, 19 April 2017
Created by Mr Lafferty

30. 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.
19-Apr-17
Created by Mr. Lafferty

31. 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?
Wednesday, 19 April 2017
Created by Mr Lafferty

32. 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.
Wednesday, 19 April 2017
Created by Mr Lafferty

36. 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.
19-Apr-17
Created by Mr. Lafferty

37. 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
Irregular Pentagon
19-Apr-17
Created by Mr. Lafferty

38. 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.
Pentagon
Hexagon
Octagon
19-Apr-17
Created by Mr. Lafferty

39. 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
Pentagon
72o
Hexagon
60o
Octagon
45o
Interior Angles
Interior Angle
19-Apr-17
Created by Mr. Lafferty

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

41. Polygons www.mathsrevision.com Interior and Exterior Angles
What you should have found :
Interior angle = 180 – (360÷n)
n = Number of sides
eg . A hexagonal has interior angle is:
Interior angle = 180 – (360÷6) = 120o
19-Apr-17
Created by Mr. Lafferty

42. Polygons www.mathsrevision.com Interior and Exterior Angles A E B
This is called the
“Exterior angle” O
Pentagon Q D C
Exterior angle = 180 – interior angle
eg . For the pentagon above :
Exterior angle = 180 – 108 = 72o
19-Apr-17
Created by Mr. Lafferty

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

46. 3 marks