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Ch. 4 Angles and Parallel Lines Math 10-3 Slide 2 Day 1: Angles and Parallel Lines Slide 3 Terminology AngleTwo rays that meet a point called the vertex Right Angle90 0 Straight Angle180 0 Ray 1 Ray 2Vertex Slide 4 Acute AngleAn angle less than 90 0 Obtuse AngleAn angle greater than 90 0 but less than 180 0 Reflex AngleAn angle greater than 180 0 but less than 360 0 Slide 5 Full Circle360 0 Quarter Circle90 0 Half Circle180 0 *Note: 90 0 + 90 0 = 180 0 Circle270 0 *Note: 90 0 + 90 0 +90 0 = 270 0 Slide 6 TriangleAll three angles add up to 180 0 A + B + C = 180 0 Complementary Angles Two angles that add up to 90 0 A + B = 90 0 Supplementary Angles Two angles that add up to 180 0 A + B = 180 0 A B C A B A B Slide 7 ParallelTwo lines that will never cross; they are always the same distance apart PerpendicularTwo lines at right angles (90 0 ) Slide 8 True Bearing (compass) The angle measured clockwise between true north and an intended path or direction, expressed in degrees N E S W NE SE SW NW Slide 9 Assignment: M103 Angles Day 1 Assignment.doc Game: Angle Basics Bingo (excel document) Slide 10 Day 2: Estimating and measuring Angles Slide 11 Estimating Angles One way to estimate angles is to draw dotted lines where 90 and 180 should be ex1: angle 1 is a little less than 90 An estimate of about 70 is good Slide 12 EX 2: Angle 2 is a bit more than 90 but less than 180 . An estimate of about 130 is good. Slide 13 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 (0 0 /180 0 ) along one ray Step 3: Determine where the other ray reaches. This will be your angle. Slide 14 Ex3. Determine the measure of the following angle: Slide 15 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. Slide 16 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. Slide 17 Ex 4: Construct an angle of 30 Slide 18 Ex: 5 construct an angle of 250 Slide 19 Assignment: M103 estimating and constructing Angles.doc Slide 20 Day 3: Describing Angles Slide 21 Adjacent angles are angles that share a common vertex and a common arm. Complementary angles are angles that add up to 90 Supplementary angles are angles that add up to 180 Slide 22 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 Slide 23 When given one angle in a pair of complementary / supplementary angles, we can easily calculate the measure of the second angle A B C D We know that ABC and CBD are complementary (add up to 90 ) 27 + CBD = 90 CBD = 90 - 56 = 34 27 Slide 24 Ex3: Determine the measure of X and Y We know that y and 56 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 56 46 X Y Slide 25 Angles that are opposite to each other have the same measure. We call these Vertically Opposite Angles. 30 150 X Y X is vertically opposite 150, x = 150 Y is vertically opposite 30, y = 30 Slide 26 Example 5 Determine the measure of x and y 25 x 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 Slide 27 Assignment: M 103 Describing Angles.doc Quiz tomorrow! Slide 28 Day 4: Bisecting Angles Slide 29 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 Slide 30 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 Slide 31 Assignment: M103 Bisecting angles.doc Slide 32 Day 5: replicating Angles Slide 33 Many people who work in the trades may need to replicate angles. Especially carpenters, and construction workers To replicate an angle, use a protractor Slide 34 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 B C D Slide 35 Percents of a circle Review: How many degrees in a circle? 360 If you shade an entire circle, what percent would this be? 100% Slide 36 Lets consider the following habits of Mrs. More per month ItemAmountPercent 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% Slide 37 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! Slide 38 Item Shoes Clothes Eating Out Hockey games Gym membership Degree 50 166 72 32 40 Slide 39 We can now construct an accurate pie chart! Slide 40 Assignment: M103 Replicating Angles.doc Slide 41 Day 6: Classifying Angles and Lines Slide 42 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 Slide 43 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? 3 4 56 T A B 12 7 8 1,2 3, 4 5, 6 7, 8 1,3 2,4 5,7 6,8 Slide 44 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 Slide 45 Assignment: Classifying lines and angles.doc Slide 46 Day 7: Parallel Lines and transversals Slide 47 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? 90 0. Try it! Slide 48 Slide 49 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 180 0 (90 0 + 90 0 = 180 0 ) Vertically opposite angles are equal A complete circle is equal to 360 0 (90 0 + 90 0 + 90 0 + 90 0 = 360 0 ) Slide 50 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 ab c d e fg 75 Slide 51 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. ab cd e f g 75 Slide 52 Assignment: M103 parallel Lines and transversal.doc Quiz tomorrow!! Slide 53 Day 8: Calculating angles Slide 54 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 Slide 55 To determine the measure of x, we know that a circle must equal 360 X = 144 We can then extend the transversal to help create a straight line. z = 216 180 = 36 Z can be determined by subtracting 180 0 from 216 0. Angles y and z are corresponding angles of and are thus equal. z x y Slide 56 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 35 m f 87 Slide 57 Assignment: M103 calculating angles.doc