MFM 1P. Foundations of Mathematics Grade 9 Applied Mitchell District High School. Unit 4 Geometry There are 5 formal lessons in this unit.
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1 MFM 1P Foundations of Mathematics Grade 9 Applied Mitchell District High School Unit 4 Geometry There are 5 formal lessons in this unit. Lesson # Lesson Title Practice Questions Date Completed 1 Interior Angles of Triangles Page 77 # Exterior Angles of Triangles 3 Parallel Lines Theorem 4 Interior Angles of Polygons 5 Exterior Angles of Polygons Page 83 #1-8 Page 89 #1-8 Page 97 #1,2,4,6 Page 101 #2, 4, 6 Page 83 #2, 3, 7 Page 98 #7 Page 101 #3, 5, 8 Test Written on :
2 MFM1P U5L1 Angles in a Triangle Topic : Angles in a Triangle Goal : I know that all triangles have interior angles that add to 180 degrees and I can use this knowledge to solve problems. Interior Angles in a Triangle The interior angles in a triangle will ALWAYS add up to 180º. This is called the Angle Sum of a Triangle Theorem. Example 1. Determine the unknown angle in the following triangles. a) b) Example 2. Can a triangle be formed that has the following angles a) 40º, 60º, 75º? b) 14º, 52º, 118º
3 MFM1P U5L1 Angles in a Triangle Example 3. An isosceles triangle has two sides and two angles equal. a) if the two equal angles are 40º, what is the remaining angle? b) if the third angle is 92º, what are the equal angles? Practice Questions - Page 77 #1-10
4 MFM1P U5L3 Exterior Angles of a Triangle Topic : Goal : Exterior Angles in a Triangle I know how the exterior angle in a triangle is related to the interior angles and I can use the Exterior Angle Theorem to solve for missing angles. Exterior Angles in a Triangle Think about what you know from interior angles and straight angles to find the EXTERIOR angle in these triangles. We can find the missing interior angle, by subtracting the two we know from the 180 o of the whole triangle. Then we can subtract the interior angle we found from the 180 o that make up a straight line. This gives us the exterior angle. What do you notice?
5 MFM1P U5L3 Exterior Angles of a Triangle This works everytime! The two interior angles will always add up to the opposite exterior angle. We call this the Exterior Angle Theorem of Triangles or EAT for short. Example 1. Find the missing angles
6 MFM1P U5L3 Exterior Angles of a Triangle Now what about ALL the outside angles Example 2. Find the missing angle Practice Questions - Page 83 #1-8
7 MFM1P U5L3 Parallel Lines Theorem Topic : Goal : parallel lines theorem I can find alternate, corresponding and co-interior angles where parallel lines are involved. Parallel Lines Theorem Parallel lines are lines that will never cross. A transversal is a line that crosses other lines. The diagrams below show special angles in pairs. Alternate angles Corresponding angles Interior angles Alternate angles make what we call a Z-pattern. Alternate angles are equal. Corresponding angles make what we call an F-pattern. Corresponding angles are equal. Interior angles make what we call a C-pattern. Interior angles add to 180 o
8 MFM1P U5L3 Parallel Lines Theorem Opposite Angles Opposite angles appear directly across from each other. Opposite angles are equal. Example 1. What kind of angles (opposite, alternate, cooresponding or interior) are marked in the diagrams? Example 2. Find the value of the indicated angle.
9 MFM1P U5L3 Parallel Lines Theorem Practice Questions - Page 89 #1-8
10 MFM1P U5L4 Interior Angles in a Polygon Topic : Goal : Angles in polygons I know the relationship between the number of degrees in a polygon and its number of sides. Interior Angles in a Polygon In class we determined that the interior angles of ANY triangle all add up to 180 o. How about other kinds of shapes. Use the applet at the following link to fill in the chart below. # of sides Total degrees o Can you come up with a rule to use for any number of sides? Note : a regular polygon has all sides and all angles the exact same. A square and an equilateral triangle are examples of regular polygons. Example 1 : Calculate the missing angle. Since this is a, it has o Example 2 : A loonie is a regular, 11-sided polygon. What is the measure of each angle inside a loonie.
11 MFM1P U5L4 Interior Angles in a Polygon Example 3 : For the following parallelogram, determine the missing angles. Practice Questions - Page 97 #1, 2, 4, 6 Page 101 #2, 4, 6,
12 MFM1P U5L5 Exterior Angles in a Polygon Topic : Goal : Exterior Angles in a Polygon I understand what is meant by the exterior angle of a polygon, and I can find the sum of all the exterior angles. Exterior Angles of Polygons Triangles An exterior angle of a polygon is created when a side of a polygon is extended. Notice that an interior and exterior angle always form a straight line and add up to. There are 3 pairs of interior/exterior angles in a triangle - so if we added up all the interior and exterior pairs, they will total... Since we know the interior angles of the triangle add up to 180 o, what do the exterior angles add up to? INTERIOR + EXTERIOR = TOTAL EXTERIOR = TOTAL - INTERIOR
13 MFM1P U5L5 Exterior Angles in a Polygon Quadrilaterals Let's try that with a quadrilateral (4 sides) How many interior/exterior pairs? Total of all interior/exterior pairs? Interior angles in a quadrilateral Exterior angles in a quadrilateral What do you notice about triangles and quadrilaterals? What do you think the total of all exterior angles in a pentagon will be? See if this applet confirms your suspicions The exterior angles in ANY polygon will all add up to.
14 MFM1P U5L5 Exterior Angles in a Polygon Example 1. Find "d" Example 2. Find x, y and z. Example 3. Find x if the pictures shown below are regular polygons. x x Practice Questions - Page 83 #2, 3, 7 Page 98 #7 Page 101 #3, 5, 8
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