Download Angle Relationships in Polygons

Warm up
The average of 6 numbers is 31. When a new number is introduced, the average is now 33. What is the new
number?
Problem Solving Techniques
Summing Numbers
1. What is the sum of the numbers 1 to 100?
2. What is the sum of the numbers 250 to 500?
Divisibility Rules
For a number to be divisible by...
2-
3-
4-
9-
8-
6-
5-
Question
The digits 1, 2, 3, 4, and 5 are each used once to create a five digit number abcde which satisfies the
following conditions:
i) the three digit number abc is odd;
ii) the three digit number bcd is divisible by 5; and
iii) the three digit number cde is divisible by 3.
Determine every five digit number formed using each of the digits 1 to 5 exactly once that satisfies all
three conditions.
Counting
1. How many numbers between 1 and 100 (inclusive) are divisible by 5?
2. How many two digit numbers contain exactly one 7?
Probabilities
1. a) How many different possible outcomes are there when rolling two dice?
b) What is the probability of rolling 2 or 3 when you roll two dice?
2. A number between 1 and 1000 is drawn from a hat. What is the probability that the number ends in 3?
3. For any three-digit integer greater than or equal to 100, the sum of the digits is calculated.
Determine the probability that the sum of the digits is 10.
(The number 253 has a digit sum of 10 and would be one of the numbers we are interested in.)
Warm up
Match each term with its definition
1. Scalene triangle
2. Isosceles triangle
3. Equilateral triangle
4. Acute angle
5. Obtuse angle
6. Reflex angle
7. Acute triangle
8. Obtuse triangle
9. Right triangle
Angle Relationships
Supplementary Angles
Complementary Angles
Parallel Line Angle Properties
(when a transversal crosses parallel lines)
Alternate Angles 
Corresponding Angles
Examples: Determine the values of x, y and z
Opposite Angles
Co-interior Angles
Sum of Interior Angles
Example: Find the value of x
Exterior Angle Theorem
Example: Find the measure of angle A
Sum of Exterior Angles
Example: Find the measure of b
Practice
p. 362 #1, 2, 5, 6
p. 372 #4 - 11, 14
Warm up
Find the measure of x, y and z and justify your answer.
Angle Relationships in Polygons
Concave Polygon
Convex Polygon
Regular Polygon
Quadrilaterals
Sum of Interior Angles
Sum of Exterior Angles
Angles of Parallelograms
Example 1: Determine the measure of each angle
Example 2: Omar calculates that x represents an angle measure of 50°. Is he correct? How do
you know?
Polygon
Number of
Sides
Sum of Interior
Angles
Sum of Exterior
Angles
Example 1:
a) What is the sum of the interior angles of a 12 sided polygon?
b) What is the measure of each angle if it is a regular polygon?
Example 2: The sum of the interior angles of a polygon is 2700o. How many sides does it have?
Example 3: Each interior angle of a regular polygon measures 140o. How many sides will it have?
p. 381 #1 - 3, 5ac, 6 – 8, 10, 13
p. 391 #1 - 6, 9 – 12
Warm up
1) What is the sum of the interior angles of a icosikaihenagon (21 sided polygon)?
2) If it is a regular icosikaihenagon, what is the measure of each interior angle?
Medians, Midpoints and Diagonals
Median of a Triangle
Example 1: The area of ΔABC is 20 m2.
a) What is the area of ΔABD?
b) What is the area of ΔADC?
Example 2: In this right triangle, the median bisects the right angle.
Does the median bisect the right angle in all right triangles?
Midsegment of a Triangle
Draw a line segment connecting the midpoint of AC and the midpoint
of AB. Call it DE.
a) Measure the length of BC and length of DE. Conclusions?
b) Measure the height of ABC and the height of ADE. Conclusions?
c) How do the areas of ABC and ADE compare?
Example 3: Determine the values of the indicated variables, justify your answers.
Midpoints of a Quadrilateral
Construct any quadrilateral. Then construct a quadrilateral inside it by joining the midpoints of
each side. What are the properties of the inside quadrilateral?
Diagonals of a Parallelogram
p. 398 # 1 – 4, 6, 7, 8
p. 405 #1 - 4, 7, 12
Warm up
1) Name each quadrilateral shown below.
2) Do the diagonals bisect each other?
3) Do the diagonals cross at a 90o angle?
Review for Test
Find the value of x
a)
b)
1) Work on toolkit
2) Practice Questions
p. 408 #1 - 8, 11, 12
c)