Foundations of Mathematics 11 2. Vertically opposite angles are When two lines intersect, the o Supplementary angles add to Two (or more) adjacent angles Complementary angles add t Two (or more) adjacent angles Example 1. Find the angle mea 20 .1 Exploring Parallel Lines e equal opposite angles are equal. o 180° on the same side of a line add to 180°. to 90° in a right angle add to 90°. asure of angle b, using opposite angles. Ms Moon .
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Foundations of Mathematics 11 Ms Moonmskmoon.weebly.com/uploads/1/7/1/8/17186300/chapter_2_notes.pdfExplore Co-Interior Angles Example 5. On the diagrams below mark two pairs of label
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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.
Ms Moon
, using opposite angles.
Foundations of Mathematics 11
Example 2. Find the angle measure of angle
Example 3. Find the angle measure of angle
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.
Ms Moon
, using supplementary angles.
, using complementary angles.
Foundations of Mathematics 11
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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A transversal is a line that intersects two or more other lines at distinct points.
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.
Ms Moon
A transversal is a line that intersects two or more other lines at distinct points.
crosses two parallel lines (below) at an angle other every angle formed between intersecting lines with a
degrees.
Foundations of Mathematics 11
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
Ms Moon
lines, the corresponding angles are
. Explain how you
Foundations of Mathematics 11
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.
Ms Moon
crosses them.
shaped corresponding
shape can be backwards or upside down).
Foundations of Mathematics 11
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.
Ms Moon
shape with a transversal that crosses them.
interior angles.
shaped alternate interior
Foundations of Mathematics 11
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
Ms Moon
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
Foundations of Mathematics 11
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.
Ms Moon
One side of a cell phone tower will be built as shown. Use the angle are parallel.
Foundations of Mathematics 11
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
In any triangle, the sum of the measures of the interior angles is proven to be 180°.
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.
Ms Moon
In any triangle, the sum of the measures of the interior angles is proven to be 180°.
the Exterior and Interior
Determine the relationship between an exterior angle of a triangle and its
Foundations of Mathematics 11
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.
Ms Moon
The measure of any exterior angle of a triangle is proven to be equal to the sum of the
Foundations of Mathematics 11
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
Ms Moon
.
Foundations of Mathematics 11
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
Foundations of Mathematics 11
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
Ms Moon
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
Foundations of Mathematics 11 Ms Moon
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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 ���°����
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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)?
Foundations of Mathematics 11 Ms Moon
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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°
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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