2 Angles and Parallel Lines Day 1:Angles and Parallel Lines
3 Terminology Angle Two rays that meet a point called the vertex Right Angle900Straight Angle1800Ray 1VertexRay 2
4 Acute AngleAn angle less than 900Obtuse AngleAn angle greater than 900 but less than 1800Reflex AngleAn angle greater than 1800 but less than 3600
5 Full Circle3600Quarter Circle900Half Circle1800*Note: = 1800¾ Circle2700*Note: = 2700
6 All three angles add up to 1800 TriangleAll three angles add up to 1800A + B + C = 1800Complementary AnglesTwo angles that add up to 900A + B = 900Supplementary AnglesTwo angles that add up to 1800A + B = 1800CBAABBA
7 ParallelTwo lines that will never cross; they are always the same distance apartPerpendicularTwo lines at right angles (900)
8 True Bearing (compass) The angle measured clockwise between true north and an intended path or direction, expressed in degreesNNENWWESWSES
11 Estimating AnglesOne way to estimate angles is to draw dotted lines where 90⁰ and 180⁰ should beex1: angle 1 is a little less than 90⁰An estimate of about 70⁰ is good
12 EX 2:Angle 2 is a bit more than 90⁰ but less than 180⁰.An estimate of about 130⁰ is good.
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 (00/1800) along one rayStep 3: Determine where the other ray reaches. This will be your angle.
14 Ex3. Determine the measure of the following angle:
15 How to Draw Angles Using a Protractor and Ruler Step 1: Draw a straight line with your ruler, ~5 cm longStep 2: On one end of your line, position the very center of your protractor on the edge of the line.
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.
21 Adjacent angles are angles that share a common vertex and a common arm. Complementary angles are angles that add up to 90⁰Supplementary angles are angles that add up to 180⁰
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 AB C
23 When given one angle in a pair of complementary / supplementary angles, we can easily calculate the measure of the second angleWe know that ABC and CBD are complementary (add up to 90⁰)27⁰ + CBD = 90⁰CBD = 90⁰ - 56⁰ = 34⁰AC27⁰BD
24 Ex3: Determine the measure of X and Y We know that y and 56⁰ are complementary.We know that y and 56 are complementary56 + y = Y= 90 – 56 = 34°We also know that a straight line is equal to therefore, x y + 46 = 180X = x = 180180 – 136 = 44°56⁰YX46⁰
25 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 = 150Y is vertically opposite 30,y = 30XY30°150°
26 Example 5 Determine the measure of x and y Y is vertically opposite 25, y = 25Notice how x and 25 are on a straight line?They are supplementary!X + 25 = 180X = 180 – 25 = 155°xy25°
33 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 workersTo replicate an angle, use a protractor
34 Protractor method Using the trapezoid, we will copy CDA Measure the angle with the protractorDraw side ADUse the protractor to mark the correct angleDraw line CD to form CDAADBC
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%
36 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 $ %
37 How can we change these percents into angles? Set up a comparison ratio!Remember, Percent means out of 100!= xCross multiply and divide to determine approximate degrees!
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
43 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 linesCan you name all pairs of adjacent supplementary angles?T12A4356B781,2 3, , 6 7, 8 1,3 2,4 5,7 6,8
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 5Opposite angles: non adjacent angles that are formed by two intersecting lines For example 1 and 4Alternate 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
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? Try it!
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 ( = 1800 )Vertically opposite angles are equalA complete circle is equal to 3600 ( = 3600)
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 measureF = 75Opposite angles formed from a transversal of parallel lines are equal in measureabcde75°fg
51 g = 105( = 105) adjacent angles along a transversal line are supplementaryc = 75Alternate interior angles formed from a transversal of parallel lines are equal in measure.abcde75°gf
52 Assignment:M103 parallel Lines and transversal.docQuiz tomorrow!!
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°xy
55 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.zxy
56 Ex2: Determine the unknown angles in the following diagram: We know m is opposite 87° m =87°87° and f are supplementary180 – 97 = 93°We know that a triangle must add up to 180°M + n + 35 = 180°180° – 87°-35° = 58°nf87°m35°