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Carlisle Math Team Meet #1 – Category 2 M1C2 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Self-study Packet Problem Categories for this Meet: 1. Mystery: ? 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics 5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities For current schedule or information, see http://www.imlem.org M1C1 GEOMETRY ANGLES To name an angle: use three points, one from each ray and the vertex. Vertex letter is always in the middle. Type of Angle Acute Angle Right Angle Obtuse Angle Straight Angle Reflex Angle Description an angle that is less than 90° an angle that is 90° exactly an angle that is greater than 90° but less than 180° an angle that is 180° exactly an angle that is greater than 180° Supplementary Angles Two Angles are Supplementary if they add up to 180 degrees. These two angles (140° and 40°) are Supplementary Angles, beacuse they add up to 180°. Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because 60° + 120° = 180° Qui ck Ti me™and a TIF F (Uncompress ed)dec ompres sor are needed to s ee th i s pi c tu re. M1C1 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. If the two angles add to 180°, we say they "Supplement" each other. GEOMETRY Complementary Angles Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40° and 50°) are Complementary Angles, beacuse they add up to 90°. Quic kTime™ and a TIFF (Unc ompres sed) dec ompres sor are needed to see this pic ture. ANGLE MEASURES IN PLANE FIGURES In this example, a° and b° are vertical angles. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. a° = b° vertical angles are equal: (in fact they are congruent angles) Because b° is vertically opposite 40°, it must also be 40° QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. M1C1 GEOMETRY A full circle is 360°, so that leaves 360° - 2×40° = 280° Angles a° and c° are also vertical angles (and must be equal), which means they are 140° each. Answer: a = 140°, b = 40° and c = 140°. A Transversal is a line that crosses at least two other lines. Transversal crossing two lines Transversal crosses two parallel lines transversal cuts across three lines Alternate Interior Angles. the pairs of angles on opposite sides of the transversal but inside the two lines c and f d and e M1C1 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. GEOMETRY Corresponding angles the angles in matching corners a an d e b and f c and g d and h QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. aa Alternate Exterior Angles the pairs of angles on opposite sides of the transversal but outside the two lines a and h b and g If any pair of: Corresponding Angels are equal, or Alternate Interior Angles are equal, or Alternate Exterior Angles are equal, or Consecutive Interior Angles add up to 180° a=e c=f b=g d + f = 180° ... then the lines are Parallel M1C1 GEOMETRY Interior Angles of Polygons Triangles The Interior Angles of a Triangle add up TO 180° 90° + 60° + 30° = 180° QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Quadrilaterals (Squares, etc) The Interior Angles of a Quadrilateral add up to 360° 80° + 100° + 90° + 90° = 360° QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this pict ure. Pentagon QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. M1C1 made A pentagon has 5 sides, and can be from three triangles, GEOMETRY so you know what... ... its internal angles add up to 3 × 180° = 540° And if it is a regular pentagon (all angles the angle is 540° / 5 = 108° same), then each (Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's internal angles add up to 540°) The General Rule So, each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total: Shape Sides Sum of Internal Angles 3 180° Shape Each Angle Triangle Q ui ck Ti m e ™ an d a T IF F ( Un co m p e r ss ed ) d ec om pr es s or a re ne ed ed t o s ee th i s pi c u t r e. 60° Quadrilateral 4 360° 5 540° 6 720° ... ... Q ui ck Ti m e ™ an d a T IF F ( Un co m p e r ss ed ) d ec om pr es s or a re ne ed ed t o s ee th i s pi c u t r e. 90° Pentagon Q ui ck Ti m e ™ an d a T IF F ( Un co m p e r ss ed ) d ec om pr es s or a re ne ed ed t o s ee th i s pi c u t r e. 108° Hexagon ... Q ui ck Ti m e ™ an d a T IF F ( Un co m p e r ss ed ) d ec om pr es s or a re ne ed ed t o s ee th i s pi c u t r e. ... Any Polygon ... (n-2) × 180° / n n M1C1 120° (n-2) × 180° Q ui ck Ti m e ™ an d a T IF F ( Un co m p re ss ed ) d ec om pr es s or a re ne ed ed t o s ee th i s pi c tu r e. GEOMETRY Category 2 Geometry Meet #1, October 2005 1. 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. Hint: What is the Complement of 48° ? Of 7°? C A 2. In the figure at right, the measure of angle ABC is 60 degrees, the measure of angle DEF is 100 degrees, and the measure of angle DGI is 116 degrees. How many degrees are in the measure of angle FHG? Hint: Write in the angles above, and then write in as B H F E G I D many angles as you can deduce. 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? Hint: Write an equation and solve it. (90 – x) + (180 – x) = … Answers 1. _______________ 2. _______________ 3. _______________ M1C1 GEOMETRY Solutions to Category 2 Geometry Meet #1, October 2005 Answers 1. 50 2. 24 3. 28 1. The measures of two complementary angles add up to 90 degrees. Since 27 + 48 = 75, the unknown amount x must have been 90 – 75 = 15 degrees. Likewise, since 48 + 7 = 55, y must be 90 – 55 = 35. The value of x + y is thus 15 + 35 = 50. 2. 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. C A B F H E G I D 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. 270 2x 8x 10 10 10 280 2x 8x 2x 2x 280 10 x x 28 M1C1 GEOMETRY Category 2 Geometry Meet #1, October 2004 1. Line l and line m are parallel. The measure of angle DBA is 150°, and the measure of angle CEF is 30°. How many degrees are in the measure of angle GAB? Hint:CEF is the same as AGB. What is GBA? Now you k l .G . m . .D . E . . F B .A C n can find GAB. 2. The eight-pointed star in the figure at right was created by placing equilateral triangles, such as A, along the inside edges of a regular octagon. How many degrees are in the angle measure of a point on the star? A Hint: What is the exterior angle of an Octagon? 360/8. Now what is the interior angle of an octagon? What is the interior angle of an equilateral triangle? 3. If the supplement of angle x is five times the complement of angle x, how many degrees are in the measure of angle x? Give your answer to the nearest tenth of a degree. Answers 1. _______________ 2. _______________ 3. _______________ M1C1 Hint: Make an equation and solve. GEOMETRY Solutions to Category 2 Geometry Meet #1, October 2004 Answers 1. 120 2. 15 1. Angle GBA is supplementary to angle DBA, whose measure is 150°, so the measure of angle GBA is 180° – 150° = 30°. Angle BGA and angle CEF are corresponding angles, so the measure of angle BGA must be equal to that of CEF, which is also 30°. Triangle GAB must have a total of 180°, so the measure of angle GAB is 180° – 2 30° = 120°. 3. 67.5 135° Interior angle 2. The interior angle of the regular octagon can be found in several ways. One way is to subdivide the octagon into six triangles, each of which has an angle sum of 180 degrees. The total interior angle is thus 6 180 = 1080 degrees. In a regular octagon, this total is shared equally among the eight interior angles, so each of them has an angle of 1080 ÷ 8 = 135 degrees. Each equilateral triangle has three 60 degree angles. The vertices of two triangles meet at each vertex of the octagon, occupying 2 60 = 120 degrees of that angle. The angle measure of a point on the star is the rest of the interior angle, or 135 – 120 = 15 degrees. 3. The supplement of angle x is 180 – x, and the complement of angle x is 90 – x. Translating the statement to algebra, we get 180 x 590 x. Solving for x, we get 180 x 450 5x 180 x 5x 450 5x 5x 180 4 x 450 180 180 4 x 450 180 22.5 4 x 270 112.5 4 x 270 67.5 4 4 x 67.5 M1C1 GEOMETRY Category 2 Geometry Meet #1, October 2003 1. Lines TP, BG, and DM intersect at point O. mBOT 47 degrees and mMOG 29 degrees. How many degrees are in the measure of angle DOP? Hint: Find vertical angle to BOT. Add to MOG. Find supplement. B K I m Q n R S G O P D J H M T 2. Lines m and n are parallel. mHIJ 148 degrees and mQRS 133 degrees. How many degrees are in the measure of angle IJK if it is degrees? less than 180 angles on the diagram. Hint: Write known Find angle JIK. Find IKR. 3. The sum of the supplement of angle A and the complement of angle A measures sixteen degrees more than a straight angle. How many degrees are in the measure of angle A? Answers 1. _______________ 2. _______________ 3. _______________ M1C1 Hint: 180-A = Supplement. What is the Complement? Write equation & solve. GEOMETRY Solutions to Category 2 Geometry Meet #1, October 2003 Answers 1. 104 2. 101 1. The measures of angles BOT, TOM, and MOG must add up to 180 degrees since O is a point on line BG. Thus the measure of angle TOM must be 180 – 47 – 29 = 104 degrees. Angles TOM and DOP are verticle angles and therefore have the same measure. The measure of angle DOP is 104 degrees. 3. 37 2. Angle HIJ measures 148 degrees, so angle JIK must measure 180 – 148 = 32 degrees. Angle QRS and angle IKR are corresponding angles, so they have the same measure. This means angle JKI must measure 180 – 133 = 47. The total angle sum of triangle IJK has to be 180 degrees, so angle IJK must measure 180 – 32 – 47 = 101 degrees. 3. The supplement of angle A measures 180 – A. The complement of angle A measures 90 – A. Their sum is 180 A 90 A 270 2A . If this amount is sixteen degrees more than a straight angle, then we can write the equation 270 2A 180 16 and solve for A. 270 2A 180 16 270 2A 196 270 196 2A 270 196 2A 74 2A A 37 M1C1 GEOMETRY Category 2 Geometry Meet #1, October, 2002 1. The measure of a certain angle a is 39 degrees. Let s be the supplement of this angle a and let c be the complement of angle a. How many degrees are there in the measure of angle (s + c)? 2. Line l is parallel to line m. Find the measure of angle ø in degrees. ø 33° 68° 3. In the figure shown at right, angles CAE, GFE, and CDB are right angles and angle ACE measures 27 degrees. How many degrees are in the measure of the angle DGF? Answers 1. _______________ 2. _______________ 3. _______________ M1C1 Hint: Knowing the angle at C, write the angles at B. Now you can find the angles at G. GEOMETRY Solutions to Category 2 Geometry Meet #1, October, 2002 Answers 1. 192 1. If s is the supplement of a 39 degree angle, then s 180 39 141. If c is the complement of angle a, then c 90 39 51. Thus the value of s + c is 141 + 51 = 192. 2. 35 3. 117 2. The obtuse angle in the triangle is the supplement of 68 degrees, or 180 – 68 = 112 degrees. Since every triangle has a total of 180 degrees, ø must be 180 – 112 – 33 = 35 degrees. A shorter way to arrive at this result is 68 – 33 = 35, since an exterior angle of a triangle is equal to the sum of the two non-adjacent angles. 3. If the measure of angle ACE is 27 degrees, then the measure of angle AEC must be 90 – 27 = 63 degrees. The angle sum in quadrilateral DEFG must be 360 degrees. Angles EFG and EDG are right angles and AEC is 63 degrees. Thus the measure of angle DGF is 360 – 90 – 90 – 63 = 117 degrees. Another approach: Exterior angle ABD must equal the sum of the two interior angles ACE =27° and right angle CDB, so ADB=117°. AB and FG are parallel (since both are perpendicular to the same line) and so GBA=DGF=117°. M1C1 GEOMETRY Category 2 Geometry Meet #1, October, 2001 1. In the figure to the right, angles GRM and MRT are complementary. Angles MRT and TRY are supplementary. The measure of angle TRY is 127 degrees. How many degrees are in the measure of angle GRY? G M R Y T A 2. In the figure to the left, angles ABC and ADE are right angles. FED and FBC are straight lines. The measure of angle ACB is 55 degrees. How many degrees are in the measure of angle EFB? D E F B C 3. In the figure to the right, regular pentagon AGHIF sits inside regular hexagon ABCDEF so that the two shapes share base AF. How many degrees are in the measure of angle GAB? D C H B I G E Hint: Find the exterior angle for each shape and subtract. A F Answers 1. _____________ 2. _____________ 3. _____________ M1C1 GEOMETRY Solutions to Category 2 Geometry Meet #1, October, 2001 Answers 1. We know that angle TRY measures 127, so its supplement, angle MRT, must be 180 127 53 degrees. Angle GRM, the complement to angle MRT, must be 90 53 37 degrees. Angle GRY is the supplement to GRM, so it must be 180 37 143 degrees. Alternatively, we might notice that angles GRY and TRY must have a sum of 270 degrees (GRM and MRT together make the other 90 degrees in the 360 degrees around point R) and 270 127 143 degrees. 1. 143 2. 35 3. 12 G 37 M 143 R Y 53 127 T A 35 55 E 55 35 F 90 D 90 125 90 90 B 55 C 2. The figure contains four similar right triangles: ABC, ADE, FBE, and FDC. All four of these triangles have an angle that measures 90 degrees and an angle that measures 55 degrees. The third angle must equal 35 degrees since the total angle sum of any triangle is 180 degrees and 90 55 35 180 . In particular, triangle FDC contains a right angle at D and a 55 degree angle at C, so the angle at F must be 35 degrees. 3. Regular hexagons have interior angles of 120 degrees and regular pentagons have interior angles of 108 degrees. This can be determined by partitioning the polygon into triangles, each containing 180 degrees. Thus, the measure of angle GAB is 120 108 12 degrees. M1C1 GEOMETRY Category 2 Geometry Meet #1, October, 2000 1. A regular octagon and a regular hexagon share a common side. What is the number of degrees in the measure of the exterior angle formed where they meet? 2. In the figure, rays AB and DE are parallel. Angle C is a right angle and angle B measures 52 degrees. Find the measure of angle D if it is less than 180 degrees. A B E C D 3. How many degrees are in an exterior angle of a regular 18-gon? Answers 1. _____________ 2. _____________ 3. _____________ M1C1 GEOMETRY Solutions to Category 2 Geometry Meet #1, October, 2000 Answers 1. The interior angle measures of the regular octagon and the regular hexagon are 135 degrees and 120 degrees respectively. 135 120 255 , so the exterior angle between the shapes must account for the remaining 105 degrees of a full circle. 1. 105 degrees 2. 142 degrees 3. 20 degrees 2. By extending ray AB and creating a line parallel to segment BC, we can see that angle D is composed of a 90 degree angle and the same 52 degrees that is found at B. Thus, angle D is 142 degrees. B A E C D 3. A regular 18-gon can be divided into 16 triangles each with an angle sum of 180 degrees. The total angle sum of the 18-gon is 16 180 2880 . Since the 18-gon is regular, each interior angle is one eighteenth of 2880, or 160 degrees. An exterior angle is the supplement of the interior angle, which is 20 degrees in this case. Alternatively, some students will know that the sum of all the exterior angles of a polygon is always 360. Since the 18-gon is regular, we simply divide as follows: 360 18 20 . M1C1 GEOMETRY M1C1 GEOMETRY M1C1 GEOMETRY