B. Angles are Everywhere

Math 20: FoundationsFM20.4 Demonstrate understanding of properties of angles and triangles including:deriving proofs based on theorems and postulates about congruent trianglessolving problems.

Geometric Art p.68

Let’s Get Started!

What DO YOU Think? P. 69

FM20.4 Demonstrate understanding of properties of

angles and triangles including: deriving proofs based on theorems and

postulates about congruent triangles solving problems.

1. What are Parallel Lines?

Take a look at the two basketball backboards on p. 70.

The Backboard and support posts are parallel because they are both perpendicular to the ground a all times.

The adjusting arms are the transversals because they intersect both the backboard and support posts

1. What are Parallel Lines?

Measure the Indicated Angles

These are called Corresponding Angles and are always equal when dealing with parallel lines.

Reflecting Questions p. 70

When a transversal intersects a pair of non-parallel lines, the corresponding angles are not equal

Note:

There are other relationships that exist when a transversal intersects 2 parallel lines but we will see those in the next sections

Interior Angles – Any angles formed by a transversal and two parallel lines that lie inside the parallel lines.

Exterior Angles – Any angles formed by a transversal and two parallel lines that lie outside the parallel lines.

Corresponding Angles – One interior angle and one exterior angle that are non-adjacent and on the same side of a transversal.

Converse - A statement that is formed by switching the premise and the conclusion of another statement.

Summary p. 71

Ex. 2.1 (p.72) #1-5

Practice

FM20.4 Demonstrate understanding of properties of

angles and triangles including: deriving proofs based on theorems and

postulates about congruent triangles solving problems.

2. What about Angles Formed from Parallel Lines?

How can we draw a pair of parallel lines with only a compass and a straight edge?

2. What about Angles Formed from Parallel Lines?

Prove the following conjectures:

a) When a transversal intersects a pair of parallel lines, the alternate interior angles are equal.b) When a transversal intersects a pair of parallel lines, the interior angles on the same side of the transversal are supplementary.c) Alternate exterior angles are equal.

Example 1

Example 2

Example 3

Summary p.78

Ex. 2.2 (p.78) #1-15#4-21

Practice

FM20.4 Demonstrate understanding of properties of

angles and triangles including: deriving proofs based on theorems and

postulates about congruent triangles solving problems.

3. Special Angles in a Triangle

Take a Blank piece of paper and draw a triangle of any type and cut it out.

Next rip out the 3 angles.

Now place those 3 angles together so there vertex's are touching.

What is the result? What does this give evidence to?

3. Special Angles in a Triangle

Can we prove that the sum of the interior angles of any triangle is 180°. (p.86)

Example 1

Let’s determine the relationship between an exterior angle of a triangle and its non-adjacent angles

The measure of an exterior angle of a triangle is equal to the sum of the measure of the 2 non-adjacent interior angles

Example 2

Example 3

Summary p. 90

Ex. 2.3 (p.90) #1-12#5-19

Practice

FM20.4 Demonstrate understanding of properties of

angles and triangles including: deriving proofs based on theorems and

postulates about congruent triangles solving problems.

4. Polygons have Many Angles

The sum of the interior angles of any convex polygon has to do with the number of triangles formed when you connect one vertex in the polygon to every other vertex.

4. Polygons have Many Angles

Lets look at the following quadrilateral.

How many triangles are formed? What is sum of the measures of interior

angles of a triangle? So is there are two triangles is a quad what

is the sum of the interior angles of a quad?

This is true because all quadrilaterals have a sum of 360° for their interior angles.

With partner and with this information in mind complete the chart on p. 94.

With this chart in mind make a conjecture about the relationship between the sum of the measures of the interior angles of a polygon, S, and the number of sides of a polygon, n

Use this conjecture to predict sum of all interior angles of a dodecagon.

What is the sum of all the exterior angles of a pentagon?

This will hold true for all regular convex polygons.

The sum of all exterior angles of any convex polygons is 360°

Example 1

This leads us to the formula to find the measure of 1 interior angle of a regular polygon.

𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑜𝑓 1 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒=180(𝑛−2)

𝑛

Example 2

Example 3

Summary p.99

Ex. 2.4 (p.99) #1-14#5-21

Practice