Grade 7/8 Math Circles October 14/15, 2014 Angles

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Faculty of MathematicsWaterloo, Ontario N2L 3G1Centre for Education inMathematics and ComputingGrade 7/8 Math CirclesOctober 14/15, 2014AnglesAnglesAn angle is the space between two lines that intersect each other.Terms and angles you should know:An acute angle isA right angle isAn obtuse angle isA straight angle isClassify these angles as on of the above:1

a)b)c)d)Two angles are complementary ifFind the complementary angle for the given angles(find a):a)b)c)d)Two angles are supplementary if2e)

Find the supplementary angle for the given angles(find a):a)b)c)d)Trick to remember:An opposite angle isCan we explain this using our knowledge of supplementary angles?3

A transversal isCorresponding angles areIn the picture above there are 2 pairs of corresponding angles: D H and B F.Alternate angles are4

In the picture above there are 2 pairs of alternate angles: C F and A H.Interior angles areIn the picture above D and F are interior angles. D F 180Find the missing angles.5

Triangles and MoreFirst, lets talk about the 3 types of triangles.Equilateral:Isosceles:Scalene:6

What is the sum of the interior angles of a triangle? 180. Any triangle, regardless of sizehas 3 interior angles that add up to 180.Using our knowledge of angles (interior, alternate, supplementary . . . ) can we show why thisis true? Why is is that with any given triangle the angles always add up to 180.7

What is the sum of angles in a parallelogram? Can you show this using triangles? The sumof the angles in a parallelogram is equal to 360 and we can show this using triangles. Drawa square, and from one corner draw a line to the opposite corner.What about a pentagon? Start with a pentagon, chose a corner and draw a line to eachcorner that is not directly beside it.RadiansUntil now, all the angle we have seen were in degrees. However, angles like many other measurements can have different unit (similar to kilometers and miles). For angles, a commonlyused unit is the radian. One radian is the angle formed when the arc length on a circle isequal to the radius. How many radians in one full circle?Looking at the arc length formulana 360 2πr here n is angle made by the arc length.8

In this formula we have the number 360. This is the number of degrees in a circle. Toshow how many radians we have in a circle lets replace that with a variable rad. Giving usna rad 2πrNow n is the number of degrees, or in our case radians, and we defined 1 radian as thenangle when the arc length is the same as the radius. In other words when a r. For ease ofcalculation lets make that be 1, a r 1. Our formula is now:1 2π 11 radrad 2πMultiply both sides by radTherefore there are 2π radians in a circle. This lead the equality 2π radians 360 ConversionWe can use this equality to convert from degrees to radians and back. For example, convert60 to radians. If we take the formula and divide by 360 on each side we get2π 13602π 60 3602π 606π 60360Then we multiply each side by the desired anglesimplifySo 60 degrees is equal toπ3radians.Going from radians to degrees we can do something similar. For example, convertdegrees. We start by dividing each side by 2 to getπ 180Now we want π4 so we multiply each side by 149π4to

π4π4 1804 45and simplifyExamples Convert the following to either degrees or radians1. π2. 120 4π3.34. 300 5.12π66.π187. 20 10

PROBLEMS1. Identify the type of angles.a)b)c)d)e)2. What is a complementary angle? What is a supplementary angle?3. (a) What is the complementary angle of 30 ?(b) What is the complementary angle of 17 ?(c) What is the supplementary angle of 25 ?(d) What is the supplementary angle of 111 ?(e) What is the supplementary angle of 90 ?(f) If angle w and x are complementary and y and z are complementary what are xand (w y z)?(g) If x is supplementary to 42 and y is complementary to 42 what is x y?4. List all pairs of angles that are supplementary to each other.11

5. Find the missing angles.a)b)c)d)6. Find the missing angles.12

a)b)c)d)7. Find x and y in the following hexagon:8. Two angles are supplementary to each other. One is 56 degrees larger then the other.What is the smallest angle.9. You order a pizza and it is cut into 10 even pieces. If you were to measure the angleof a slice, what would it be in radians and in degrees?10. Convert(a) 3π(e)8π94π(c)6(f)π18(d) 270 (g) 20 (b) 190 11. Find a pattern in the sum of angles inside a triangle, square, pentagon . . . Using thispattern What is the sum of interior angles in a 92 faced figure?12. he A is three times the size B and B is twice the size of C. If A and two times C are supplementary what is B?13

Angles Angles An angle is the space between two lines that intersect each other. Terms and angles you should know: An acute angle is A right angle is An obtuse angle is A straight angle is Classify these angles

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