Download 2.1 Explorin Vertically opposite angles are equal When two lines

Foundations of Mathematics 11
Ms Moon
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 b,, using opposite angles.
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Foundations of Mathematics 11
Ms Moon
Example 2. Find the angle measure of angle b,, using supplementary angles.
Example 3. Find the angle measure of angle b,, using complementary angles.
Try.
Find the angle measure of angle b.
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Foundations of Mathematics 11
Ms Moon
A transversal is a line that intersects two or more other lines at distinct points.
Explore Parallel Lines
Example 4. Draw a transversal that crosses two parallel lines (below) at an angle other
than 90°. Label every angle formed between intersecting lines with a
unique lower-case
case letter, and measure each angle in degrees.
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Foundations of Mathematics 11
Ms Moon
When a transversal intersects a pair of non
non-parallel lines, the corresponding angles are
not equal.
Example 5. In each diagram, determine whether AB is parallel to CD.. Explain how you
know.
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Foundations of Mathematics 11
Ms Moon
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
Example 1. On the diagrams below mark three other pairs of F-shaped
shaped corresponding
angles and label them b°, c°, and d°.
(Note that the F-shape
shape can be backwards or upside down).
Example 2. Determine the measures of the angles marked by letters.
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Foundations of Mathematics 11
Ms Moon
Any pair of parallel lines makes a Z-shape
shape with a transversal that crosses them.
The angles marked a°° are equal and are called alternate interior angles.
Explore Alternate Interior Angles
Example 3. On the diagrams below mark two pairs of Z-shaped
shaped alternate interior
angles and label them b° and c°.
(Note that the Z-shape
shape can be backwards).
Example 4. Determine the measures of the angles marked by letters.
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Foundations of Mathematics 11
Ms Moon
Any pair of parallel lines makes a C-shape
shape with a transversal that crosses them.
The angles marked o and x are called co
co-interior
interior angles. They are NOT equal. They add
to 180°
Explore Co-Interior Angles
Example 5. On the diagrams below mark two pairs of C-shaped co-interior
interior angles and
label them o° and x°.
(Note that the C-shape
shape can be backwards).
Example 6. Calculate the measures of the three remaining angles 48° in the
parallelogram.
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Foundations of Mathematics 11
Ms Moon
Use Reasoning to Determine Unknown Angles
Example 7. Determine the measures of a, b, c, and d.
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 braces CG, BF, and AE are parallel.
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Foundations of Mathematics 11
Ms Moon
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 angle.
Use Reasoning to Determine the Relationship between the Exterior and Interior
Angles of a Triangle
Example 2. Determine the relationship between an exterior angle of a triangle and its
non-adjacent interior angles
angles.
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Foundations of Mathematics 11
Try.
Ms Moon
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
adjacent interior angles.
Use Exterior Angle to Determine Angle Measure
Example 3. Determine the measure of each indicated angle.
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Foundations of Mathematics 11
Try.
Ms Moon
Determine the measure of each indicated angle.
Use Reasoning to Determine Solve Problems
Example 4. Determine the measures of
, and
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.
Foundations of Mathematics 11
Ms Moon
2.4 Angles Properties in Polygons
Convex Polygon
A convex polygon is any polygon in which each interior angle measures less than 180°.
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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Foundations of Mathematics 11
Ms Moon
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 ((n – 2) triangle(s) to cover it.
Therefore, the sum of the interior angles is 180° (n – 2).
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 interior angles of a polygon with the
given sides.
a)
12
b)
15
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Foundations of Mathematics 11
Ms Moon
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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Foundations of Mathematics 11
Ms Moon
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 measures
360°
The sum of the measures of the exterior angles of an n-sided regular polygon is 360°
Try.
a)
Determine the number of sides of a regular polygon whose exterior angle
measures
15°
b)
40°
c)
1°
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