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**Ch. 4 Angles and Parallel Lines**

Math 10-3

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**Angles and Parallel Lines**

Day 1: Angles and Parallel Lines

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**Terminology Angle Two rays that meet a point called the vertex**

Right Angle 900 Straight Angle 1800 Ray 1 Vertex Ray 2

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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

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

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**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

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

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**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

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**Assignment: M103 Angles Day 1 Assignment.doc**

Game: Angle Basics Bingo (excel document)

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**Day 2: Estimating and measuring Angles**

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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

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

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**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.

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**Ex3. Determine the measure of the following angle:**

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**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.

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**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.

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**Ex 4: Construct an angle of 30⁰**

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**Ex: 5 construct an angle of 250⁰**

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Assignment: M103 estimating and constructing Angles.doc

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**Day 3: Describing Angles**

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**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

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**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

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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

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**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 = Y= 90 – 56 = 34° We also know that a straight line is equal to therefore, x y + 46 = 180 X = x = 180 180 – 136 = 44° 56⁰ Y X 46⁰

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**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°

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**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°

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Assignment: M 103 Describing Angles.doc Quiz tomorrow!

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Day 4: Bisecting Angles

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**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

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**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

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Assignment: M103 Bisecting angles.doc

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**Day 5: replicating Angles**

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**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

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**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

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**Percents of a circle Review: How many degrees in a circle? 360°**

If you shade an entire circle, what percent would this be? 100%

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**Let’s consider the following habits of Mrs. More per month**

Item Amount Percent of total (amount /total) Shoes $ /1750 = 14% Clothes $ % Eating Out $ % Hockey games $ % Gym membership $ % Total $ %

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**How can we change these percents into angles?**

Set up a comparison ratio! Remember, Percent means out of 100! = x Cross multiply and divide to determine approximate degrees!

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

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**We can now construct an accurate pie chart!**

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**Assignment: M103 Replicating Angles.doc**

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**Day 6: Classifying Angles and Lines**

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**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

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**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, , 6 7, 8 1,3 2,4 5,7 6,8

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**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

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Assignment: Classifying lines and angles.doc

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**Day 7: Parallel Lines and transversals**

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**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? Try it!

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**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 ) Vertically opposite angles are equal A complete circle is equal to 3600 ( = 3600)

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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

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g = 105 ( = 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

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

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**Day 8: Calculating angles**

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**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

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**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 from 2160. Angles y and z are corresponding angles of and are thus equal. z x y

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**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°

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Assignment: M103 calculating angles.doc

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