Math 9 Honours 7a Angles with Parallel Lines

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

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

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

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

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

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

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:

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

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

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

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

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

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

Practice: Prove Given: Statement Reason

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