1. Math 9 Honours 7a Angles with Parallel Lines

2. I) Properties of a Circle
Chord: A line with two endpoints on the circle
Major Arc
Sector
Secant: A line that intersects a circle at two different points
Diameter: A chord with the midpoint on the center of the circle
Radius: A line that runs from the center to the edge of the circle
Chord
Secant
Segment: Area in a circle separated by the chord (Watermelon)
Segment
Sector: Area in a circle separated by two radii’s (Pizza)
Minor Arc
Arc A fraction of the circumference
Major Arc: Arc over 50% of the circumference
Minor Arc Arc less than 50% of the circumference
Click Here for Circle Applet

3. Area of Circle Area of One Sector
Concentric Circles: Circles having the same center
Ex: A circle with a diameter of 20cm is cut into 20 equal sectors. What is the area of one sector?
Area of Circle
Area of One Sector

4. Perpendicular Bisector Theorem
Equilateral Triangle
All sides are equal and all angles equal 60
Isosceles Triangle
Opposite angles and sides are equal
Radii’s in a Circle
All radii’s in a circle are equal in length
Two radii in a circle will create an isosceles triangle
Perpendicular Bisector Theorem
All points on a perpendicular bisector are equal in distance from the line segment

5. Angles in a Triangle add to 180
Complimentary Angles add to 90
Supplementary Angles add to All angles on a line add to 180
Vertically Opposite angles are Equal
Alternate Interior Angles are Equal (Z shape)

6. ii) Angle Property In Triangle
An exterior angle of a triangle is equal to the sum of the other two interior angles

7. Practice: What is the value of angles X and “Y”?

9. Ex: Given that the Diameter of the circle is 18 cm, find the area of the shaded figure:
Radius is 9 cm
The area can be split into 3 equilateral triangle
The height can be obtained using special triangles
The area of one triangle is:
The area of 3 triangles:

10. III) Angle SUM in a Polygon
Count the number of non-overlapping triangles formed with the vertices
Ex: Find the sum of all the interior angles for each shape
12 sides10 triangles
6 sides4 triangles
10 sides8 triangles
Another alternative is count the number of sides, minus 2, then multiply by 180

11. IV) Proving Congruent Triangles
There are three ways to prove that two triangles are congruent
1) (SSS) All three corresponding sides are equal in length
II) (SAS) Two corresponding sides and the angle in between are equal
III) (ASA) Two angles and the side in between are equal

12. Ex: Prove Statement Reason Given: AD and BE are both diameters
O is the center of the circle
Goal: Prove that the two triangles are congruent by SSS, SAS, or ASA
Statement
Reason

13. Practice: Prove
Given:
CA and MA are equal
Statement
Reason

14. V) CPCTC Corresponding Parts of Congruent Triangles are Congruent
If two triangles are congruent  Then all corresponding parts must be equal
All the other corresponding parts must also be equal as well
You could use CPCTC “ONLY” after you proved that two triangles are congruent

15. Ex: Prove
Given:
AE and BD are equal
Statement
Reason

16. Practice: Prove
Given:
Statement
Reason

17. Challenge: Prove Statement Reason Given:
There are two overlapping triangles
Statement
Reason