foundations of mathematics 11 ms...

Foundations of Mathematics 11

2.1 Exploring Parallel Lines

Vertically opposite angles are equal

When two lines intersect, the opposite angles are equal.

Supplementary angles add to 180°

Two (or more) adjacent angles on the same side of a line add to 180°.

Complementary angles add to 90°

Two (or more) adjacent angles in a right angle add to 90°.

Example 1. Find the angle measure of angle

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2.1 Exploring Parallel Lines

Vertically opposite angles are equal

When two lines intersect, the opposite angles are equal.

Supplementary angles add to 180°

Two (or more) adjacent angles on the same side of a line add to 180°.

angles add to 90°

Two (or more) adjacent angles in a right angle add to 90°.

Find the angle measure of angle b, using opposite angles.

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, using opposite angles.




Try. Find the angle measure of angle

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Find the angle measure of angle b, using supplementary angles.

Find the angle measure of angle b, using complementary angles.

Find the angle measure of angle b.

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, using supplementary angles.

, using complementary angles.


A transversal is a line that intersects two or more other lines at distinct points.

Explore Parallel Lines Example 4. Draw a transversal that

than 90°. Label every angle formed between intersecting lines with a unique lower-case letter, and measure each angle in

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Draw a transversal that crosses two parallel lines (below) at an angle other every angle formed between intersecting lines with a

case letter, and measure each angle in degrees.

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crosses two parallel lines (below) at an angle other every angle formed between intersecting lines with a

degrees.


When a transversal intersects a pair of nonnot equal.

Example 5. In each diagram, determine whether

know.

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When a transversal intersects a pair of non-parallel lines, the corresponding angles are

In each diagram, determine whether AB is parallel to CD. Explain how you

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lines, the corresponding angles are

. Explain how you


2.2 Angles Formed by Parallel Lines

Any pair of parallel lines makes an

The angles marked a° are equal and are called corresponding angles.

Explore Corresponding Angles

Example 1. On the diagrams below mark three other pairs of angles and label them (Note that the F-shape can be backwards or upside down).

Example 2. Determine the measures of the angles marked by letters.

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2.2 Angles Formed by Parallel Lines

Any pair of parallel lines makes an F-shape with a transversal that crosses them.

The angles marked a° are equal and are called corresponding angles.

Explore Corresponding Angles

On the diagrams below mark three other pairs of F-shaped corresponding them b°, c°, and d°. shape can be backwards or upside down).

Determine the measures of the angles marked by letters.

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crosses them.

shaped corresponding

shape can be backwards or upside down).


Any pair of parallel lines makes a

The angles marked a° are equal and are called alternate

Explore Alternate Interior AnglesExample 3. On the diagrams below mark two pairs of

angles and label them(Note that the Z-shape can be backwards).

Example 4. Determine the measures of the

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Any pair of parallel lines makes a Z-shape with a transversal that crosses them.

° are equal and are called alternate interior angles.

Explore Alternate Interior Angles

On the diagrams below mark two pairs of Z-shaped alternate interior angles and label them b° and c°.

shape can be backwards).

Determine the measures of the angles marked by letters.

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shape with a transversal that crosses them.

interior angles.

shaped alternate interior


Any pair of parallel lines makes a

The angles marked o and x are called coto 180°

Explore Co-Interior Angles Example 5. On the diagrams below mark two pairs of

label them o° and (Note that the C-shape can be backwards).

Example 6. Calculate the measures of the three remaining angles 48° in the

parallelogram.

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Any pair of parallel lines makes a C-shape with a transversal that crosses them.

are called co-interior angles. They are NOT equal. They add

On the diagrams below mark two pairs of C-shaped co-interior angles and x°.

shape can be backwards).

Calculate the measures of the three remaining angles 48° in the

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shape with a transversal that crosses them.

interior angles. They are NOT equal. They add

interior angles and

Calculate the measures of the three remaining angles 48° in the


Use Reasoning to Determine Unknown AnglesExample 7. Determine the measures of

Use Angle Properties to Prove that Lines Are Parallel

Example 8. One side of a cell phone tower will be built as shown. Use the angle measures to prove that

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Determine Unknown Angles Determine the measures of a, b, c, and d.

Use Angle Properties to Prove that Lines Are Parallel

One side of a cell phone tower will be built as shown. Use the angle measures to prove that braces CG, BF, and AE are parallel.

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One side of a cell phone tower will be built as shown. Use the angle are parallel.


2.3 Angles Properties in Triangles

In any triangle, the sum of the measures of the interior angles is proven to be 180°.

Use Angle Sums to Determine Angle Measures

Example 1. Determine the measure of each indicated

Use Reasoning to Determine the Relationship Angles of a Triangle

Example 2. Determine the relationship between an exterior angle of a triangle and its non-adjacent interior angles

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2.3 Angles Properties in Triangles


Use Angle Sums to Determine Angle Measures

Determine the measure of each indicated angle.

Use Reasoning to Determine the Relationship between the Exterior and Interior

Determine the relationship between an exterior angle of a triangle and its interior angles.

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the Exterior and Interior

Determine the relationship between an exterior angle of a triangle and its


Try. Prove

The measure of any exterior angle of a triangle is proven to be equal to the sum of the measures of the two non-adjacent interior angles.

Use Exterior Angle to Determine Angle MeasureExample 3. Determine the measure of each indicated angle.

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The measure of any exterior angle of a triangle is proven to be equal to the sum of the adjacent interior angles.

Use Exterior Angle to Determine Angle Measure Determine the measure of each indicated angle.

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The measure of any exterior angle of a triangle is proven to be equal to the sum of the


Try. Determine the measure of each indicated angle.

Use Reasoning to Determine Solve Problems

Example 4. Determine the measures of

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Determine the measure of each indicated angle.

Use Reasoning to Determine Solve Problems

Determine the measures of , and

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.


2.4 Angles Properties in Polygons

Convex Polygon

A convex polygon is any polygon in which each interior

Concave Polygon

A concave polygon is a polygon in which one or more interior angle measures more than 180°.

Determine Convex and Concave Polygons

Example 1. State if each polygon is concave or convex.

Try. State if each polygon is concave or convex.

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2.4 Angles Properties in Polygons

A convex polygon is any polygon in which each interior angle measures less than 180°.

A concave polygon is a polygon in which one or more interior angle measures more

Determine Convex and Concave Polygons

State if each polygon is concave or convex.

if each polygon is concave or convex.

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angle measures less than 180°.

A concave polygon is a polygon in which one or more interior angle measures more


Determine Properties of Angles in Polygons

Example 2. Use the following table to determine how to find the sum of the interior angles of a polygon from the number of sides.

For a polygon with n sides, it takes a minimum of (Therefore, the sum of the interior angles is 180°

Try. Calculate the sum of the measures of the interior angles of a polygon with 9

sides. Try. Calculate the sum of the measures of the

given sides. a) 12

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Determine Properties of Angles in Polygons

Use the following table to determine how to find the sum of the interior polygon from the number of sides.

takes a minimum of (n – 2) triangle(s) to cover it. the interior angles is 180° (n – 2).

Calculate the sum of the measures of the interior angles of a polygon with 9

Calculate the sum of the measures of the interior angles of a polygon with the

b) 15

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Use the following table to determine how to find the sum of the interior

to cover it.

Calculate the sum of the measures of the interior angles of a polygon with 9

interior angles of a polygon with the

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Example 3. Determine the number of sides of a polygon whose interior angle sum equals 4140°

Try. Determine the number of sides of a polygon whose interior angle sum equals:

a) 720° b) 1260° c) 2880°

For a regular polygon with n sides, the measure of each interior angle is ��°��

�

Example 4. A regular polygon is a polygon with all sides equal and all angles equal.

How can you find the measure of each interior angles of a regular pentagon?

Try. What is the measure of each interior angle in a regular decagon (10-sided)?

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Example 5. Determine the number of sides of a regular polygon whose interior angle measures 170°.

Try. Determine the number of sides of a regular polygon whose interior angle

measures; a) 156° b) 175° c) 171°

An exterior angle is formed between a ray formed by extending the sides of polygon in one direction and the next side of the polygon (adjacent to the ray).

Example 6. Determine the measure of each exterior angle of a regular pentagon?

The exterior angle of an n-sided regular polygon measures360°

�

The sum of the measures of the exterior angles of an n-sided regular polygon is 360°

Try. Determine the number of sides of a regular polygon whose exterior angle

measures a) 15° b) 40° c) 1°

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