Grade 7/8 Math Circles Winter 2013 Angles
1University of WaterlooFaculty of MathematicsGrade 7/8 Math CirclesWinter 2013AnglesWhat is an angle?An angle is the measure of space within two, or three, or more lines that diverges from a commonpoint.AZComplementary Angles: angles that add up to 90 30B60CGSupplementary Angles: angles that add up to 180 60D120ECAOpposite Angles: equal angles when two lines intersect.F1404040140BD
2ExerciseFind the value of x.AGD3x 1i. B7x - 16H20ii.C77FETraversal: is a line that intersects two or more parallel lines.From traversal lines we get alternate angle and corresponding angles.BDACGEHFAll pairs of alternate angles are equal to each other, and all pairs of corresponding are also equalto each other.To easily see alternate angles, draw the letter ”Z” on the traversal.BAEBDCAHGEFDCHGF
3All the angles on the Z are alternate angles.On the left the alternate angles are ADH and DHG.On the right the alternate angles are CDH and DHE.To easily see corresponding angles, draw the letter ”F” on the traversal.BAEBDCAHDGEFCHGFAll the angles on the F are corresponding angles.On the left the corresponding angles are CDH and GHF , and BDC and DHG.On the right the corresponding angles are ADH and EHF , and ADB and EHD.Exercise1) What other angles, of any kind, can you find in the traversal?
42) Find all the missing values and angles.FBAH826y 23x - 298SGFii.2w - 76573xyxz 150iii.C80EIi.x - 55yzyiv.Pool Table ExerciseThe goal of pool is to shoot the billiard balls on the table into the pockets of the pool table, usinga stick, called the cue, and white ball, called the cue ball. Angles are formed when the cue hits thecue ball, which then hits the other billiard balls on the table, or when the cue ball, or any of theother billiard balls, hit the side of the table.Below is a pool table with billiard balls positioned through out. To start, estimate the angle thatis needed for the cue ball to hit ball 1. The cue ball will then be in the position of ball 1. Estimatethe following angles needed for the cue ball in position of ball 1, to hit ball 2, etc. The goal is tohit ball 8 into a pocket.1.5.2.6.3.7.4.8.
5900 /36018027015372684Angles in TrianglesThere are 3 kinds of typical triangles that we speak of :, and,.60An equilateral triangle is.606040An isosceles triangle is.7070
660.A right triangle is9030Sum of Interior AnglesThe sum of the interior the angles in a triangle is 180 .180Using this fact, we can then find the sum of interior angles in any polygon. Let’s look at a squarefor example.180180If you start at one corner, and draw a line to other cornerof the square, we see the inside of a square is composedof two triangles. So 180 180 360 . Meaning thesum of interior angles in a square will always add up to360 .180180180Again, start at one corner of the pentagon, and draw aline to every corner of the shape. The pentagon is madeup of 3 triangles. So 3 180 540 .
7ExerciseFind the sum of interior angles for the hexagon below.What pattern do you see?As the shapes grow in size by one side, the number of triangles that can fit into the shape also growsby one triangle. Can you figure out what the formula for the sum of interior angles of a polygon?
8RadiansA radian is.A radian is given by the angle formed between two radiicoming from the center of a circle, to an arc on thecircumference of a circle, which has a length of the radius.ai.e. a r57.417We denote a radian with θ If C the circumference of a circle then,rC π d 2πrSo if we want to calculate the radians of an entire circlewe have,2πr 2πrIf the radians of an entire circle is 2π, and we know the degrees of circle is 360 , that means,θ 2π 360 .ExerciseWhat do the following radian measure equal to in degrees?a) π c)π 4π 2d)3π 2b)Convert the degrees into radians.a) 60 c) 135 b) 225 d) 315
9Problem Set1. In the diagram find the degrees of all the angles.y uzx40wv322. What is the angle the hour and minute hand of a clock make when the time is 12:20 p.m.?(Assume the hour hand stays at the hour.)3. Which of the following set of angles would form an isosceles triangle?a) 30 , 60 , 90 b) 50 , 30 , 100 c) 15 , 150 , 15 d) 10 , 75 , 95 e) 20 , 20 , 140 4. Find the value of 90x60E d)f)E14xR11268AE
105. A semicircle is split into 9 equal parts. Find the degree of CAT .CTAB 1206. In the diagram, what is the degree of Y RD?YIR1607. Change the degrees into radians, or vice versa.15 30 240 300 5πe)2a)b)c)d)f)8π2g)9π4h)2π38. Find the sum of interior angles of the star.9. What is the sum of interior angles of a decagon?10. What is the sum of interior angles of a 54 sides figure?11. How many degrees are there in a circle?140D
1112. As Sammy is driving down the 401, the road becomes very dark and he can’t see anything! Whatshould the angle of the light from the lamppost tothe ground be so that Sammy can see?13. There are 16 spokes equally spaced around a wheel.What is the angle between any two spokes in the wheel?14. Below is another pool table. This time however, the grey balls without numbers are stationary.Meaning they will not move. What are the angles needed to get ball 8 into a pocket? (Hint:You can bounce off objects if you need to.) Approximate answer are fine.900 /36018027051642738AD15. In the diagram, ABC is 90 . CBD is 50 larger than ABD. What is the degree of ABD?BC
1216. BAKC is a square and 4BEC is an equilateral triangle. What is the degree of AEK?KAECB17. AIL is 70 and GRAI and LAKE are equal squares. What is the obtuse angle in RAK?RKAEGIL18. In 4DOG, O is 4 degrees larger than D and G is 2 times D. What is the degree of D?19. Two angles are complementary and one angle is 38 larger than the other one. What is thedegree of the smaller angle?20. The degree of A, is twice the degree of B, which is twice the degree of C. If A and Care supplementary, what is the degree of B?21. PERY is a rhomus. If PY PL PA LA, what is the degree of Y P E?PEAYRL22. Given the diagram below, can you prove why the sum of all the angles in a triangle equal180 ?xybac
Grade 7/8 Math Circles Winter 2013 Angles What is an angle? An angle is the measure of space within two, or three, or more lines that diverges from a common point. Complementary Angles: angles that add up to 90 A B C Z 60 30 Supplementary Angles: angles that add up to 180 D 60 120 F E G Oppos
Two circles are congruent circles if and only if they have congruent radii. A B if AC BD . AC BD if A B. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles . Internally tangent circles Externally tangent circles Pairs of Circles
Math Course Progression 7th Grade Math 6th Grade Math 5th Grade Math 8th Grade Math Algebra I ELEMENTARY 6th Grade Year 7th Grade Year 8th Grade Year Algebra I 9 th Grade Year Honors 7th Grade Adv. Math 6th Grade Adv. Math 5th Grade Math 6th Grade Year 7th Grade Year 8th Grade Year th Grade Year ELEMENTARY Geome
Coplanar Circles and Common Tangents In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common tangent. A
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 16 / 150
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 15 / 150
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 15 / 149
circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common internal tangent A common external tangent two circles. Notes: intersects the segment that joins the centers of the two circles .
Teacher of Grade 7 Maths What do you know about a student in your class? . Grade 7 Maths. University Grade 12 Grade 11 Grade 10 Grade 9 Grade 8 Grade 7 Grade 6 Grade 5 Grade 4 Grade 3 Grade 2 Grade 1 Primary. University Grade 12 Grade 11 Grade 10 Grade 9 Grade 8 Grade 7 Grade 6 Grade 5 . Learning Skill
