Ch. 4 Angles and Parallel Lines Math 10-3

Angles and Parallel Lines Day 1: Angles and Parallel Lines

Terminology Angle Two rays that meet a point called the vertex Right Angle 900 Straight Angle 1800 Ray 1 Vertex Ray 2

Acute Angle An angle less than 900 Obtuse Angle An angle greater than 900 but less than 1800 Reflex Angle An angle greater than 1800 but less than 3600

Full Circle 3600 Quarter Circle 900 Half Circle 1800 *Note: 900 + 900 = 1800 ¾ Circle 2700 *Note: 900 + 900 +900 = 2700

All three angles add up to 1800 Triangle All three angles add up to 1800 A + B + C = 1800 Complementary Angles Two angles that add up to 900 A + B = 900 Supplementary Angles Two angles that add up to 1800 A + B = 1800 C B A A B B A

Parallel Two lines that will never cross; they are always the same distance apart Perpendicular Two lines at right angles (900)

True Bearing (compass) The angle measured clockwise between true north and an intended path or direction, expressed in degrees N NE NW W E SW SE S

Assignment: M103 Angles Day 1 Assignment.doc Game: Angle Basics Bingo (excel document)

Day 2: Estimating and measuring Angles

Estimating Angles One way to estimate angles is to draw dotted lines where 90⁰ and 180⁰ should be ex1: angle 1 is a little less than 90⁰ An estimate of about 70⁰ is good

EX 2: Angle 2 is a bit more than 90⁰ but less than 180⁰. An estimate of about 130⁰ is good.

How to measure Angles using a Protractor Step 1: Position the protractor at the vertex of the angle. Step 2: Line up the straight edge of the protractor (00/1800) along one ray Step 3: Determine where the other ray reaches. This will be your angle.

Ex3. Determine the measure of the following angle:

How to Draw Angles Using a Protractor and Ruler   Step 1: Draw a straight line with your ruler, ~5 cm long Step 2: On one end of your line, position the very center of your protractor on the edge of the line.

Step 3: Using the protractor scale, place a faint mark at the desired degree. Step 4: Remove the protractor, and using your ruler, draw a line that connects the end of your first line with the faint mark you drew in step 3.

Ex 4: Construct an angle of 30⁰

Ex: 5 construct an angle of 250⁰

Assignment: M103 estimating and constructing Angles.doc

Day 3: Describing Angles

Adjacent angles are angles that share a common vertex and a common arm. Complementary angles are angles that add up to 90⁰ Supplementary angles are angles that add up to 180⁰

When describing angles using letter, the middle letter is in the vertex position Ex1: ABC the vertex is at letter B. the angle is between the arms A and C. A B C

When given one angle in a pair of complementary / supplementary angles, we can easily calculate the measure of the second angle We know that  ABC and CBD are complementary (add up to 90⁰) 27⁰ + CBD = 90⁰ CBD = 90⁰ - 56⁰ = 34⁰ A C 27⁰ B D

Ex3: Determine the measure of X and Y We know that y and 56⁰ are complementary. We know that y and 56 are complementary 56 + y = 90 Y= 90 – 56 = 34° We also know that a straight line is equal to 180. therefore, x + 56 + y + 46 = 180 X + 56 + 34 + 46 = 180 x + 136 = 180 180 – 136 = 44° 56⁰ Y X 46⁰

Angles that are opposite to each other have the same measure Angles that are opposite to each other have the same measure. We call these Vertically Opposite Angles. X is vertically opposite 150, x = 150 Y is vertically opposite 30, y = 30 X Y 30° 150°

Example 5 Determine the measure of x and y Y is vertically opposite 25, y = 25 Notice how x and 25 are on a straight line? They are supplementary! X + 25 = 180 X = 180 – 25 = 155° x y 25°

Assignment: M 103 Describing Angles.doc Quiz tomorrow!

Day 4: Bisecting Angles

A bisector is a line that divides an angle or line into two equal parts. Method 1: measure the angle with a protractor. Divide the measure by 2. Use protractor and ruler to draw the bisecting line

Ex. 1 Bisect ABC Step 1: measure the angle Step 2: divide by 2 Step 3: draw a bisecting line at 20° using a ruler

Assignment: M103 Bisecting angles.doc

Day 5: replicating Angles

Many people who work in the trades may need to replicate angles Many people who work in the trades may need to replicate angles. Especially carpenters, and construction workers To replicate an angle, use a protractor

Protractor method Using the trapezoid, we will copy CDA Measure the angle with the protractor Draw side AD Use the protractor to mark the correct angle Draw line CD to form CDA A D B C

Percents of a circle Review: How many degrees in a circle? 360° If you shade an entire circle, what percent would this be? 100%

Let’s consider the following habits of Mrs. More per month Item Amount Percent of total (amount /total) Shoes $250 250/1750 = 14% Clothes $800 46% Eating Out $350 20% Hockey games $150 9% Gym membership $200 11% Total $ 1750 100%

How can we change these percents into angles? Set up a comparison ratio! Remember, Percent means out of 100! 14 = x 100 360 Cross multiply and divide to determine approximate degrees!

Item Shoes Clothes Eating Out Hockey games Gym membership Degree 50° 166° 72° 32° 40°

We can now construct an accurate pie chart!

Assignment: M103 Replicating Angles.doc

Day 6: Classifying Angles and Lines

Consider the following rectangle Which sides are parallel? Which sides are perpendicular? Notice how the opposite sides are parallel, and the adjacent sides are perpendicular? Adjacent means “next to” A//C & B //D A,D & D,C & C,B & B,A

Can you name all pairs of adjacent supplementary angles? Many angles are formed by two lines and a transversal – a line that intersects TWO or more lines Can you name all pairs of adjacent supplementary angles? T 1 2 A 4 3 5 6 B 7 8 1,2 3, 4 5, 6 7, 8 1,3 2,4 5,7 6,8

Are other ways to describe adjacent pairs of angles Corresponding angles: two angles formed by two lines and a transversal, located on the same side of the transversal. For example 1 and 5 Opposite angles: non adjacent angles that are formed by two intersecting lines For example  1 and 4 Alternate angles: Two angles formed by two lines and a transversal, located on opposite sides of the transversal For example:  3 and 6 are INTERIOR alternate angles  3 = 6

Assignment: Classifying lines and angles.doc

Day 7: Parallel Lines and transversals

Parallel lines and transversals have some special properties that you may have already notices. How can we determine that lines are parallel? If we draw a perpendicular transversal line between two parallel lines, what will the angles be equal to? 900. Try it!

What will the measure of ALL the other angles formed by the transversal be equal to? 90° This is because of the other rules about angles we already know: Supplementary angles (angles on a straight line) add up to 1800 (900 + 900 = 1800 ) Vertically opposite angles are equal A complete circle is equal to 3600 (900 + 900 + 900 + 900 = 3600)

These properties can help us make more rules about transversal and parallel lines. Consider the following: With just one angle labeled, we can determine the measure of every other angle. b: = 75 Corresponding angles formed from a transversal of parallel lines are equal in measure F = 75 Opposite angles formed from a transversal of parallel lines are equal in measure a b c d e 75° f g

g = 105 (180 -75 = 105) adjacent angles along a transversal line are supplementary c = 75 Alternate interior angles formed from a transversal of parallel lines are equal in measure. a b c d e 75° g f

Assignment: M103 parallel Lines and transversal.doc Quiz tomorrow!!

Day 8: Calculating angles

We can use our knowledge of angles and lines to solve all types of problems. Ex1: Determine the unknown measures in the following diagram. 216° x y

To determine the measure of x, we know that a circle must equal 360 We can then extend the transversal to help create a straight line. z = 216 – 180 = 36° Z can be determined by subtracting 1800 from 2160. Angles y and z are corresponding angles of and are thus equal. z x y

Ex2: Determine the unknown angles in the following diagram: We know m is opposite 87° m =87° 87° and f are supplementary 180 – 97 = 93° We know that a triangle must add up to 180° M + n + 35 = 180° 180° – 87°-35° = 58° n f 87° m 35°

Assignment: M103 calculating angles.doc