Complementary angles (two angles whose sum is 90 ) and supplementary angles (two angles whose sum is 180. A straight line = 180. In the figure below and to the left, angle EFH and angle HFG form a straight line so they are supplementary angles. In the figure below and to the right, angle ABD and angle DBC form a right angle to they are complementary. Sample questions: 1. In the figure below, angle EFH is 120, what is the measure of angle HFG? 2. In the figure below, angle ABC is a right angle. If the measure of angle ABD is 70, what is the measure of angle DBC? 3. Points E, F, and G are collinear. Point H is a point not on EG. If the measure of angle GFH is 64, then what is the measure of angle EFH? (Hint 1: collinear means they're on the same line. Hint 2: draw a figure) 1
Congruent Angles have the same angle (in degrees or radians). That is all. They don't have to point in the same direction. They don't have to be on similar sized lines. 4. In the figure below, angle ABD and angle DBC are congruent angles. If the measure of angle ABD is 40, what is the measure of angle DBC? Bisect: "Bisect" means to divide into two equal parts. You can bisect lines, angles, and more. The dividing line is called the "bisector." 5. Angle ACB is bisected by line CD. If the measure of ACD is 38 what is the measure of angle ACB? Intersecting lines: When two lines intersect, adjacent angles are supplementary (they make a line and add up to 180, and vertical angles (angles across from each other) are equal. 2
Parallel lines, transversals, and corresponding angles: A line that cuts through two parallel lines is called a transversal. The angles in matching corners are called corresponding angles. In this figure angle a corresponds with angle e. Angle b corresponds with angle f and so on. Corresponding angles are equal (congruent). And since vertical angles are equal, that means that angles a, c, e, and g are all equal. Likewise angles b, d, f, and h are also equal. And since angle a and angle b form a line, the sum of the angles is 180. 6. In the figure below, CD is parallel to AB, and PQ intersects CD at R and AB at T. If the measure of angle CRP = 110, then what is the measure of angle BTQ? 7. In the figure to the right. Lines l and m are parallel. Find the measures of angles, 1, 2, 3, 4, 5, 6, and 7. 3
8. In parallelogram VWXY shown, points U, V, Y, and Z form a straight line. Given the angle measures as shown in the figure, what is the measure of angle WVY? What is the measure of angle WXY? What is the measure of angle XYZ? The Proportional Segments Theorem: Three or more parallel lines cut any two transversals into proportional segments. For example, in the following figure BC = PQ CD QR 9. In the figure below, the three horizontal lines are parallel. Find the length of segment x. 4
Similarly, if a segment with endpoints on two sides of a triangle is parallel to the third side, it divides the two sides into proportional segments. 10. In the figure below, the two horizontal lines are parallel. Solve for x. Interior angles of a triangle The three interior angles of any triangle add up to 180 11. In triangle DEF below, the measure of angle EDF is 35 and the measure of DFE is 65. What is the measure of DEF? (label the angles) 12. In triangle ABC, angle A is a right angle and angle B measures 30. What is the measure of angle C? (Draw a picture and label the angles. Remember that a right angle is 90 ). 5
13. In the following figure, what is the value of X? (remember that supplementary angles = 180 ) Right triangles and complementary angles. In the triangle below angle A is a right angle. Since the sum of the angles of a triangle is 180 and angle A is 90, the sum of angle B and angle C must be 90. That means that angle B and C are complementary. Perimeter and Area of quadrilaterals and other polygons Perimeter. Perimeter is the outline of a physical area. From Latin, meaning around (peri)and measure (metron), a perimeter is basically a boundary of any kind, measuring around the shape. In mathematics, perimeter refers to the length of this boundary. You might be asked to calculate the perimeter of a polygon, which is the sum of the length of each side. Area of a rectangle. From Latin: area - "level ground, an open space,". The number of square units it takes to completely fill a rectangle. Formula: Width Height. Area of a square. The same as for a rectangle, but the width and height are the same so you can also say s 2 where s is a side of the square. Area of a parallelogram. width X height or (Base X Height) Area of a triangle. The formula for area of a triangle is 1 of the base times the height. 2 6
Area of a trapezoid. areas. Or a trapezoid can be divided into 2 triangles, then add the two 14. A homeowner wants to put a wallpaper border on the top edge of all the walls of his kitchen. The kitchen measures 6.5 meters by 4 meters. What is the required length, in meters, of the border? 15. What is the area, in square units, of the rectangle, shown in the standard (X,Y) coordinate plane below? 16. What is the perimeter, in feet, of a rectangle with width 12 feet and length 20 feet? 17. What is the area of parallelogram ABCD? 18. In the figure to the right, sides a, b, c, d are 8 inches, 12 inches, 10 inches, and 9 inches respectively. The height is 8 inches. What is the area of the trapezoid? 7
19. A rectangular rug has an area of 35 square feet, and its width is exactly 2 feet shorter than its length. What is the length, in feet, of the rug? 20. What is the perimeter, in meters, of the figure to the right? 21. What is the area, in square units, of the triangle shown in the standard (X,Y) coordinate plane shown? 22. Andy made a model of the DPC Candy Store which has a side entrance accessed by a ramp. The dimensions of the ramp on his model are 7 cm high and 24 cm wide. Andy decides to decorate the ramp with candy tiles which are 1 cm square. How many candy tiles are needed to cover the ramp? 8
23. In triangle XYZ shown, XS and SZ are 3 and 12 units, respectively. If the area of triangle XYZ is 45 square units, how many units long is altitude YS? Pythagorean theorem For all right triangles (triangles with a 90 angle), the square of the hypotenuse is equal to the sum of the squares of the two sides. 24. What is the length of the hypotenuse of the triangle below? 25. What is the length of side X in the triangle below? 9
26. What is the length, in inches, of the diagonal of a rectangle whose dimensions are 16 inches by 30 inches? 27. In the square ABCD shown, the length of AB = 4. What is the length of AC? 28. A zipline is installed from a tower 20 m high crossing a river 50 m wide. What is the length of the zipline wire? 29. A 15-foot supporting wire is attached to a telephone pole 12 feet from the ground. The wire is then anchored to the ground. The telephone pole stands perpendicular to the ground. How far, in feet, is the anchor of the supporting wire from the base of the telephone pole? (Hint: if they don't give you a diagram, draw one) 10
Circumference of a circle The perimeter of a circle is called the circumference. It is equal to the diameter times pi. Area of a circle The area of a circle is the radius squared multiplied by pi. 30. The circle in the following figure is inscribed in a square with a perimeter of 24 inches. What is the area of the shaded region in square inches? 31. Barbara's living room is a rectangle with the dimensions of 14 feet by 12 feet. If the room is hardwood floor partially covered by a circular throw rug with a diameter of 8 feet, what is the approximate area of hardwood floor, in square feet, that remains exposed? 11
32. In the figure below, the circumference of circle X is 12 π and the circumference of circle Y is 6 π. What is the length of XY? This figure is for questions 33-34. In the following figure, the circle centered at P is tangent to the circle centered at Q. Point Q is on the circumference of circle P. The length of QP = 6 inches. 33. What is the circumference of circle P? 34. What is the area, in square inches, of circle Q? 12
The figure below is for questions 35-37. In the figure below, O is the center of the circle, C and D are points on the circle, and C, O, and D are collinear. The length of CD is 16 inches. 35. What is the circumference, in inches, of the circle? 36. What is the length of radius, CO, in inches? 37. What is the area, in square inches, of the circle? 38. Purdy the guinea pig is running on her exercise wheel when, due to a manufacturing error, the wheel breaks free of its axis. Purdy remains in her wheel, running in a straight line. After the wheel rotated exactly 10 times the wheel hit the wall and stopped. If the diameter of the wheel is 10 inches, how far, in inches, did Purdy travel? 39. A pipe of radius 6 cm sends water to two smaller pipes of equal size. If each of the smaller pipes allows exactly half as much water to flow as the larger pipe, what is the radius of one of the smaller pipes? 13
Volume of a rectangular solid The volume of a rectangular solid is length times width times height, in other words, the area of the base times the height. Volume of a cube is also length X width X height, but all the dimensions are the same so it can also be written V = s 3 40. What is the volume, in cubic inches, of the rectangular solid at right? 41. The recreation center has swimming pool that is 50 meters long, 25 meters wise, and 2 meters deep. The pool is surrounded by special non-slip tiles, as shown in the figure below. What is the volume of water in the pool, in cubic meters, if the pool is only filled half-way? 14
42. A rectangular back yard pool is 10 meters long, 6 meters wide, and holds 120 cubic meters of water. If the pool is the same depth in all parts, about how many meters deep is the water in the pool? Volume of a cone = 1 3 πr2 h 43. What is the volume of a cone, in cubic inches, of a cone with a 3 inch radius and a height of 6 inches? 44. What is the volume of a cone, with radius of 7 cm and an outside edge of 18 cm? More on Parallelograms A parallelogram has two pairs of parallel sides. Opposite sides are equal. Opposite angles are equal. consecutive angles (angles adjacent to each other) add up to 180. 45. In the parallelogram shown at the right, angle A = 35. What are the measures of angles B, C, and D? 15
Diagonals: A polygon's diagonals are line segments from one corner to another. Diagonals of certain quadrilateral have special properties. Quadrilateral Mutually bisecting diagonals Perpendicular diagonals Perpendicular bisecting diagonals Square Yes Yes Yes Rhombus Yes Yes Yes Rectangle Yes No No Parallelogram Yes No No Kite No Yes No Trapezoid No No No 46. In parallelogram ABCD below, BD and AC are diagonals which intersect at point E. Length AC is 12 cm and length BD 10 cm. What is the length of BE? 16
47. Diagonals of rhombus ABCD intersect at point O in the figure below. What is the measure of angle AOB? 48. In the rhombus below, diagonal AC = 6 and diagonal BD = 8. What is the length of each of the four sides? 17
Answers: 1. 60 2. 20 3. 116 4. 40 5. 76 6. 110 7. 1) 135, 2) 45, 3) 135, 4) 45, 5) 45, 6) 45, 7) 135 8. angle WVY 30, angle WXY 30, angle XYZ 30 9. 9 10. 3 11. 80 12. 60 13. 115 14. 21m 15. 12 16. 64 ft 17. 240 18. 80 19. 7 feet 20. 82 m 21. 8 units squared 22. 84 23. 6 units 24. 17 25. 6 6 or 14.69693846 26. 34" 27. 4 2 or 5.65685... 28. 10 29 or 53.85... 29. 9 feet 30. 36-9π or 7.74 31. 168-16π or 117.73 32. 9 33. 12π or 37.68 34. 144π or 452.16 35. 16π or 50.24 36. 8 inches 37. 64π or 200.96 38. 100π or 314 39. 3 cm 40. 120 41. 1250 42. 2m 43. 18π or 56.52 44. 850 45. B) 145, C) 35, D) 145 46. 5 cm 47. 90 48. 5 18