Park Forest Math Team

Transcription

1 Park Forest Math Team Meet #1 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. : ngle measures in plane figures including supplements and complements J. Number Theory: 1v1s1btl1Ly rules, factors, primes, composites 4. rithmetic: Order of operations; mean, median, mode; rounding; statistics 5. lgebra: Simplifying and evaluating cxpress!ons, solvin g equations with 1 unknown 1nclud1ng 1dcnt1ties

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3 Category 2 Meet #1 - October, 2015 For all problems below, angles are marked in degrees as shown. Figures are not necessarily drawn to scale. 1) ngle measures 112 degrees. ngle has the same measure as angle C. How many degrees are in the measure of angle C if it is less than 180 degrees? C 2) Regular hexagon EUCLI has diagonals EC and C. How many degrees are in the measure of angle EC? U C E L I 3) In the diagram below, is parallel to. ngle C is complementary to angle C. ngle EFG is supplementary to angle HJ. How many degrees are in the measure of angle? 1) nswers E F G C 2) 3) H J

4 Solutions to Category 2 Meet #1 - October, ) Since the base angles of the triangle are equal, nswers angle C = half of ( ) = 34 degrees. The exterior angle C = , or 146 degrees. 1) 146 2) Each angle of the regular hexagon has a measure of 2) degrees. The diagonal C bisects angle C, making angle CL = 60 degrees. Triangle EUC is 3) 65 isosceles, with UE = UC, so the base angles UEC and UCE = half of ( ), or 30 degrees. Therefore, the measure of the required angle, EC, is ( ), or 30 degrees. 3) Since angle C is complementary to angle C, then angle C measures (90-69), or 21 degrees. ngle H is 90 degrees, being adjacent to the collinear 90-degree angle. ngle HJ corresponds to angle C and = 69 degrees. ngle HJ is the supplement of angle EFG and = 44 degrees. Therefore, angle H measures (69-44), or 25 degrees. Now two angles of triangle are known, so the required angle = ( ), or 65 degrees.

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6 Category 2 Meet #1 - October, ) Two angles are "poplimentary" if the sum of their degree measures is 20. Two angles are "cuplimentary" if the product of their angle measures is 96. ngle X and angle W are both poplimentary and cuplimentary. X < W. How many degrees are in the measure of angle W? 2) E C Line segment is parallel to line segment E. egree measures of angles are as written on the diagram. What is the measure of angle C if it is less than 180 degrees? 3) Line 1 is parallel to line 2. ngle MW measures 115 degrees. ngle MW is supplementary to angle G. How many degrees are in the measure of angle GP? G nswers P line 1 1) 2) M W line 2 3)

7 Solutions to Category 2 Meet #1 - October, 2013 nswers 1) lgebraically, the solution boils down to X + W = 20 and XW = 96. 1) 12 Guessing and checking would yield a pair of numbers that solves both conditions: 8 and 12. 2) 53 Since the question asks for the larger solution, W = 12. 3) 130 2) One strategy is to draw a line through C that is parallel to the other two parallel lines, thus creating pairs of angles that are either congruent or supplementary, as shown in this diagram: E C The three angles at the bottom of the diagram form a straight angle, so their sum is 180 degrees C + 70 = 180, so C = 53. 3) The parallel lines again create pairs of angles that are either congruent or supplementary, as shown in the diagram below: G P line W line 2 Since angle G is supplementary to angle MW, it measures 65 degrees. ngle GP measures, therefore, 130 degrees, because the degree measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles.

8 Meet #1 October 2011 Category 2 1. The supplementary angle to a given measures 4 times as much as s complementary angle. How many degrees are there in the measure of? 2. In the drawing below, the measure of one angle is given as degrees, and the measure of the angle is half as much as the measure of the angle. How many degrees are there in the measure of the angle? x z y The angle between the hour-hand and the minute-hand on a clock at o clock is degrees. What is it minutes earlier? (Measure from the hour hand to the minute-hand counter-clockwise). nswers 1. degrees 2. degrees 3. degrees

9 Meet #1 October 2011 Solutions to Category 2 nswers 1. If we write this information algebraically, then we know that: ( ). This we can simplify to: and rearrange to: to find. lternatively, we can engage in a little trial and error to find out the value of The missing angle in the triangle equals (opposite angles), and so we know that as the three angles in a triangle. Given that is half as much as y, we can rewrite this as to find that degrees. is the supplement to and so equals degrees. 3. It takes the minute hand an hour to complete a revolution, so minutes earlier, it was degrees back (pointing at the numeral ). The hour hand progresses at a pace of only degrees every hour, so in minutes it only progresses degrees. The answer then is degrees.

10 Category 2 - Meet #1, October E C In the above diagram, right triangle C and equilateral triangle C intersect at point E. Given the angle measurements above, how many degrees in the measure of EC? 2. The supplement to an angle x is two-and-a-half times its complement. How many degrees in the measure of x? 3. How many degrees in the Z measure of angle Z? nswers

11 Solutions to Category 2 - Meet #1, October 2009 nswers C=60 degrees, therefore =30 degrees, therefore E =105 degrees, and so EC equals 75 degrees. nother way to see this: EC=60 degrees, CE=45 degrees, therefore CE=75 degrees. 2. We can write the equation 180 x = 2.5 (90 x) and solve: 1.5 x = and so x = 30 degrees. 3. y completing triangle CO to 180 degrees we get OC=24 degrees. y completing around O to 360 degrees we get O=46 degrees. Complete triangle CO to 180 to get OC=54 degrees. Finally Z supplements it, so is 126 degrees. Z C O 290

12 Category 2 Meet #1, October 2007 T M a 35 g 48 N b c d 61 O P 1. In the diagram at the left, the four segments : MQ, NR, OS, and PT all intersect at the same point creating 8 angles labeled with either their angle measures or a variable name. What is the value of a + b + c + d + g? S Q R 2. Two angles have a sum of 170 degrees. The complement of one of the angles is equal to the supplement of the other. How many degrees are in the measure of the larger angle? 3. In the diagram to the right, C FE and C E GH K. The measure of angle C is 73 degrees and the measure of angle FGH is 111 degrees. What is the measure of angle EFG? ( the symbol means "is parallel to") E F C G H K nswers

13 Solutions to Category 2 Meet #1, October 2007 nswers Since there is a total of 360 degrees around a point, all the angles in the diagram must add up to 360 degrees. So a + b + c + d + g = 360g = 360g=216. Therefore the sum of a, b, c, d & g = 216 degrees. 2. The complement of an angle x is 90 x and the supplement of an angle y is 180 y. If the complement of one equals the supplement of the other 90 x = 180 y y x = 90. That means the difference of the two angles is 90 degrees. Since we know the sum is 170 and one of them is 90 more than the other, simple algebra or guess and check can find that 130 & 40 fit both those requirements and 130 is the larger number. 3. Since and C are parallel, the same side interior angles(c and C) have a sum of 180 degrees. So angle C has a measure of 107 degrees. ngle C is alternate interior to EC which is alternate interior to EF, so angle C has the same measure as EF(107 degrees). ngle FE and EF are same side interior angles they have a sum of 180 degrees and angle FE has a measure of 73. Since FG is alternate interior with FGH, FG has a measure of 111 degrees. Since angles FE and EFG form angle FG, their sum is 111 degrees. So : FE + EFG = EFG = 111 EFG = 38 F E 38 K C G H

14 Category 2 Meet #1, October Tim added x degrees to a 27-degree angle. The complement of this new angle was 48 degrees. He then added y degrees to this 48-degree angle. The complement of this new angle was 7 degrees. Find the value of x + y. C 2. In the figure at right, the measure of angle C is 60 degrees, the measure of angle EF is 100 degrees, and the measure of angle GI is 116 degrees. How many degrees are in the measure of angle FHG? E F G I H 3. The sum of the complement of angle x and the supplement of angle x is 10 degrees less than eight times the angle x. How many degrees are in the measure of angle x? nswers

15 Solutions to Category 2 Meet #1, October 2005 nswers The measures of two complementary angles add up to 90 degrees. Since = 75, the unknown amount x must have been = 15 degrees. Likewise, since = 55, y must be = 35. The value of x + y is thus = Vertical angles are congruent, straight angles have a sum of 180 degrees, and triangles have an angle sum of 180 degrees. Using these three facts, the angles of the two small triangular regions can be determined from the angle measures given. The measure of angle FHG is 24 degrees. 60 C E F G I H 3. The complement of angle x is 90 x and the supplement of angle x is 180 x. Their sum is (90 x) + (180 x) = 270 2x. We know that this sum is equal to ten less than eight times angle x, or 8x 10. Now we can write an equation and solve for x x = 8x = x = 8x +2x = +2x 280 = 10x x = 28