# 2 feet Opposite sides of a rectangle are equal. All sides of a square are equal. 2 X 3 = 6 meters = 18 meters

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1 GEOMETRY Vocabulary 1. Adjacent: Next to each other. Side by side. 2. Angle: A figure formed by two straight line sides that have a common end point. A. Acute angle: Angle that is less than 90 degree. B. Complimentary angles: Two angles that add up to 90 degrees C. Degree: Measure of the size of an angle. D. Obtuse angle: Angle that is greater than 90 degrees and less than 180 degrees. E. Rays: The sides of an angle. F. Reflex angle: An angle that is more than 180 degrees and less than 360 degrees. G. Right angle: Angle that is 90 degrees. H. Straight angle: An angle that is 180 degrees. I. Supplementary angles: Two angles that add up to 180 degrees J. Vertex: The point at which two rays of an angle come together. K. Vertical angles: When two line intersect it is the pairs of nonadjacent angles. 3. Circle: A shape in which all points are an equal distance from a point, the center. A. Circumference: Distance around the outside edge of a circle. B. Diameter: Straight line distance from one side of a circle to the other side, going through the center. C. Radius: Distance from the center to the outside edge of a circle. 4. Congruent: Exactly the same size and shape. 5. Parallel: Two lines that are always the same distance apart. They never intersect, no matter how far they are extended. 6. Parallelogram: Quadrilateral in which the opposite sides are parallel and equal. 7. Perimeter: Distance around the outside edge of a figure. 8. Perpendicular: Two lines intersecting at a 90 degree angle. 9. Polygon: Closed figure made up of line segments. A. Regular polygon: Polygon in which all sides and angles are equal. B. Vertices: Corner point. 10. Polyhedron: Solid figure with flat faces. A. Faces: Flat sides of a figure. B. Edges: The shared surface where two faces meet. C. Prism: A polyhedron with parallel bases. D. Cylinder: A round polyhedron with flat ends, shaped like a can of vegetables. E. Vertice: Corner point. 11. Quadrilateral: A closed four sided figure.

2 12. Reflection: To flip a figure, as in a mirror image. 13. Rhombus: A figure that has four equal sides with opposite sides being parallel. 14. Rotation: Turning of a figure. 15. Semi: Prefix that means half. 16. Shift: Sliding of a figure to a new location. 17. Similar: The same shape, but a different size. 18. Square: A figure which has four equal sides, opposite sides parallel, and adjacent sides perpendicular. 19. Symmetry: A figure in which the two halves are mirror images of each other. 20. Trapezoid: A four sided figure that has one pair of parallel sides. 21. Triangle: Closed three sided figure. A. Scalene triangle: All sides have different lengths. B. Isosceles triangle: Two sides have equal length. C. Equilateral triangle: All three sides have an equal length. D. Right triangle: One of the angles is 90 degrees. Perimeter The perimeter is the distance around the outside edge of a figure. When the figure is a polygon it is the sum of the lengths of the sides. When the figure is curved it is the length of the curved line. The perimeter of a circle is also called the circumference. Square Rectangle 3 meters 6 meters 2 feet Opposite sides of a rectangle are equal. All sides of a square are equal. 2 X 3 = 6 meters = 18 meters Perimeter = 4 X 2 = 8 feet 2 X 6 = 12 meters The perimeter is 18 meters

3 Polygon 1 yd 2 yd. 5 yd. 3 yd 3.5 yd 4 yd = 18.5 yards The perimeter is 18.5 yards. The perimeter/circumference of a circle is found using one of the two following formulas: C = 2 R C = d C is circumference Is 3.14 or 22/7 R is the radius D is the diameter 5cm 14 in C = (2)(3.14)(5) = 31.4 The circumference is 31.4 cm. C = (22/7)(14) = 44 in. The circumference is 44 in.

4 Problems Find the perimeter of the following figures 1. Regular hexagon 8 inches 2. Equilateral triangle 11 centimeters 3. Square 6 meters 4. Rectangle 3 yards 9 yards

5 5. 9mm 7mm 5mm in 6 in 5in 7in 11in 7. 4m 3m 5m 8.Rhombus 10yd

6 8. Parallelogram 29 mm 18mm 9. Trapezoid 17ft 9ft 16ft 25ft

7 10. Find the circumference of each circle. A. E. 5cm = 3.14 = cm B. F. 14cm = 22/7 15cm = 3.14 C. G. 8m = m = 22/7 D. H. 70m = 22/7 21m = 22/7 Find the perimeter of each semicircular shape 11. A. = 22/7 B. 10m 28cm = Find the quarter circle perimeter. 5cm = 3.14

8 13. Find the circumference of the circle that is located inside the square. 10 cm 14. The curve is made up of two semicircles. Find its length. = m 8 m

9 15. Find the circumference of the following circles. = 3.14 A. Radius is 4 centimeter. D. Diameter is 18 inches. B. Radius is 12 meters. E. Diameter is 16 feet C. Radius is 5.5 miles. F. Diameter is 6.6 km 16. The length of the three sides of a triangle are 3.cm, 5.2cm, and 8.4 cm. Find the perimeter. 17. The length of three sides of a triangle are 7.5 in, 6.1in, and 4.9in. Find the perimeter. 18. The lengths of two sides of an isosceles triangle are each 2 1/2 cm. The third side is 3 cm. Find the perimeter. 19. The length of each side of an equilateral triangle is 4 1/2 in. Find the perimeter. 20. A rectangle has a length of 8.5 meters and a width of 3.5 meter. Find the perimeter. 21. Find the perimeter of a rectangle with length of 5 1/2 ft. and a width of 4 ft. 22. Find the perimeter of a regular pentagon that measures 3.5 in. on each side. 23. The length of each side of a square is 12.2cm. Find the perimeter. 24. Find the perimeter of a square that is 0.5m on each side. 25. What is the circumference of a circle with a diameter of 1.5 inches. Pi = The diameter of a circle is 4.2 ft. What is the circumference? Pi = The radius of a circle is 36cm. Find the circumference. Pi = Find the circumference of a circle that has a radius of 2.5 m. 29. How many feet of fencing should be purchased for a rectangular garden that is 18 ft long and 12 ft wide? 30. How many meters of binding are required to bind the edge of a rectangular quilt that measures 3.5m by 8.5 m. 31. The length of a rectangular park is 55yd. The width is 47yd. How many feet of fencing are needed to surround the park? 32. The perimeter of a rectangular playground is 440ft. If the width is 100ft, what is the length of the playground? 33. A rectangular vegetable garden has a perimeter of 64ft. The length of the garden is 20 ft. What is the width of the garden? 34. Each of two sides of a triangular banner measures 18in. If the perimeter of the banner is 46 in, what is the length of the third side of the banner? 35. The perimeter of an equilateral triangle is 13.2cm. What is the length of each side? 36. The perimeter of a square picture frame is 48in. What is the length of each side? 37. A square rug has a perimeter of 32ft. What is the length of each side? 38. Find the length of molding needed to put around a circular table that is 4.4ft in diameter. Pi = A bicycle has a tire diameter of 24 inches. How many feet does the bicycle travel when the wheel makes eight revolutions? Pi = Bias binding is sewed around the edge of a rectangular tablecloth measuring 72in by 45in. If the bias comes in packages containing 15ft of binding, how many packages of bias are needed for the tablecloth?

10 Area The formula for the area of a rectangular shaped figure is A=LW, where L is the length and W is the width. (In a square the length and width are the same.)(a square unit and a unit squared mean the same thing.) 2ft 4m 5ft A =5 X 2 = 10 square feet or 10 feet squared 4m A =4 X 4 = 16 square meters or 16 meters squared The formula for the area of a circular shaped figure is A= R 2, where R is the radius of the circle. Is represented by 3.14 or 22/7 3ft 7m A = 3.14 X 3 2 =28.26 square feet Or feet squared A = 22/7 X 7 2 = 154 square meters or 154 meters squared The formula for a triangular shaped figure is A = 1/2 bh, where b is the base of a triangle and h is the height of a triangle. 8in 7ft 15in A = (1/2)(15)(8) = 60 square inches or 60 inches squared 6ft. A = (1/2)(6)(7) = 21 square feet or 21 feet squared

11 The formula for the area of a parallelogram is A =bh, where b is the base and h is the height 5in A = 10 X 5 = 50 square inches 0r 50 inches squared 10in The formula for the area of a trapezoid is A = 1/2 (b 1 + b 2 )h where b 1 and b 2 are the two bases (the two parallel sides) and h is the height. 10cm A = (1/2 )( )(8) = 112 sq. cm. 8cm or 112 cm. sq. 18cm Area Problems 1. 6ft rectangle 2ft 2. 7in triangle 8in 3. 7yd 3yd rectangle

12 4. triangle 22cm 17cm 5. 12m 12m in rectangle 5in 7. 7cm square 8. 19ft 27ft rectangle

13 9. 2m triangle 21 m ft triangle 17ft 11. 9ft trapezoid 6ft 16ft 12. 6m 8m trapezoid 10m 13. 4yd 3yd trapezoid 9yd

14 Find the area of the circles below. = 3.14 Find the area of the circles below. = 22/ cm 7cm m 14m cm 14cm m 7om 22. We have a circle inside a square. What is the area of the circle? = cm

15 23.Find the area of the quarter circle. = cm 24.Find the area of each semicircle. = 22/7 A. B. 28m 7m 25. The following figures are made up of semicircles and quarter circles. Find the area of each figure. =22/7 A. B. 7cm 7cm 7cm 7cm 7cm C. 7cm 7cm 7cm

16 26.The following figures are made up of semicircles and quarter circles. Find the area of each figure. = 3.14 A. B. 10 in 10in 4in 4in C. 2cm 27. Find the area of the striped parts. A. Figure shows a square and a circle. = 22/7 B. Figure shows a square and 4 quarter circles 14 ft. = cm 10cm C. Figure shows a square and two semicircles = yd

17 28. The figure is made up of a rectangle and two semicircles. = What is its area? 10yd 12yd 29. The figure is made up of a triangle ad a semicircle. = 3.14 What is its area? 12cm 10cm 30. The figure shows a rectangle and a semicircle. = 22/7 7m 11m What is the area of the striped portion of the figure? 31. The figure shows a square and two semicircles. = in 2in What is the area of the striped portion of the figure?

18 32. The rectangle shows a rectangle and two semicircles. = cm 35cm What is the area of the striped portion of the figure? 33. The figure is made up of a square, a semicircle, and a triangle. = in 8 in What is the area and the perimeter of the figure? 34. The figure shows a square, a semicircle, and a quarter circle. = ft. What is the area and perimeter of the striped portion of the figure?

19 Volume The volume of a rectangular box is determined by the formula V = LWH. V represents volume. L represents length. W represents width. H represents height. 2ft 3ft 4ft V = 4 X 3 X 2 = 24 cubic feet /feet cubed The volume of a cylinder is determined by the formula V = R 2 h V represents volume Is represented by 3.14 or 22/7 R is the radius of the end. h is the height of the cylinder 4in V = (3.14)(4 2 )(6) V = cubic inches/inches cubed 6in

20 Problems Find the volume of the following figures cm 4m 5m 5cm 3m 3. 15yd 12yd 4. 9in 5in

21

22

23

24

25

26

27

28 Angles 1. Angle: A figure formed by two straight lines that have a common end point. A. Acute angle B. Circle: contains 360 degrees C. Complementary angles: Two angles that add up to 90 degrees D. Degree: The measure of the size of an angle. E. Obtuse angle: An angle that is greater than 90 degrees and less than 180 degrees. F. Rays: The sides of an angle G. Reflex angle: An angle that is more than 180 degrees and less than 360 degrees. H. Right angle: Angle that is 90 degrees I. Straight angle: An angle that is 180 degrees J. Supplementary angles: Two angles that add up to 180 degrees. K. Vertex: The point at which the two rays of an angle come together. L. Vertical angles: When two lines intersect it is the pairs of nonadjacent angles. acute angle right angle obtuse angle straight angle reflex angle Angles are typically named in one of two different ways x 1 C The angle is labeled by a single letter or number. A B The angle is named using three points with the vertex always being in the middle. This is called angle CAB or angle BAC

29 The size of an angle is usually designated by one or more curved lines. 70 o 315 o 130 o 70 o Be careful when a diagram has Multiple curved lines. Be aware where the angle starts and ends. The following are symbols that are commonly used. ABC This reads This is a symbol Symbol for two lines Parallel Triangle angle ABC for a right angle that are perpendicular, form right angles. Parts with the same number of slash marks are same size. Could be more than one slash mark. D C C A B D A B Complementary angles Supplementary angles BAC and CAD add up to 90 o BAC and CAD add up to 180 o

30 Assignments 1. How many degrees are in a right angle? 2. How many degrees are in a straight angle? 3. What is the complement of a 62 degree angle? 4. What is the complement of a 13 degree angle? 5. What is the supplement of a 48 degree angle? 6. What is the supplement of a 106 degree angle? 7. What is the complement of a 7 degree angle? 8. What is the complement of a 76 degree angle? 9. What is the supplement of an 89 degree angle? 10. What is the supplement of a 21 degree angle? 11. What is the complement of a 69 degree angle? 12. What is the complement of a 31 degree angle? 13. What is the supplement of a 162 degree angle? 14. What is the supplement of a 72 degree angle? G 15. C 16. A 48 o B 79 o o D E F angle AOB is straight angle angle DEF is straight angle How big is angle AOC? How big is angle FEG? 16. X W 17. P R 51 o 29 o Z Y O N How big is angle ZYW? How big is angle NOR?

31 18. C D 19. G F 30 o 40 o 80 o 38 o B A E D How big is angle ABC? How big is angle DEG? 20. Z 21. R V 94 o 22 o 59 o A Y X 195 o F How big is angle VYX? Going counterclockwise K from point A, how big is Angle AFK? C B 68 o A 210 o E D How big is the unlabeled angle CBA F How big is the unlabeled angle DEF? 24. What kind of angles are the following? A. B. C.

32 D. E. F. Solve for X X 140 o 3X 160 o X X 145 o X + 18 X 74 o X 42 o X 74 o 29 o A B C Angle ABC is a straight angle

33 X X 67 o 3X 2X 172 o A B C ABC is a straight angle X X 3X 4X 4X A B C 5X ABC is a straight angle 6X

34 Following are a number of properties that can help us determine the size of angles 1. Vertical angles are equal. Vertical angles are the nonadjacent angles formed by the intersection of two lines. C A B D A and B are vertical angles C and are vertical angles 2. Parallel lines that are intersected by a transversal form multiple pairs of equal angles. Parallel lines are always the same distance apart. AB and CD below are parallel. Transversals of parallel lines intersect both parallel lines. EF below is a transversal. A G B E C H D F A. Alternate interior angles formed by parallel lines and a transversal are equal. Alternate interior angles are on opposite sides of a transversal. AGH and DHG are alternate interior angles CHG and BGH are alternate interior angles B. Corresponding angles formed by parallel lines and a transversal are equal. Corresponding angles are in the same relative position in relation to separate parallel lines. DHG and BGE are corresponding angles. DHF and BGH are corresponding angles. CHF and AGH are corresponding angles. CHG and AGE are corresponding angles.

35 3. The sum of the interior angles of any n-gon is (n 2)(180 o ). N is the number of sides of the polygon. On the surface this may appear to be a confusing statement. Closer examination shows that it is actually quite simple. A. Let s look at the triangle shape. It has three sides. Therefore n = 3. (3 2)(180 o ) = 180 o. All triangles have 180 interior degrees. B. Let s look at the quadrilateral shape. A quadrilateral has four sides. Therefore n = 4. (4 2)(180 o ) = 360 o. All quadrilaterals have a total of 360 interior degrees. C. The same formula holds true regardless of the number of sides. 4. All the angles in an equilateral triangle are the same size. All the sides in an equilateral triangle are the same size. 5. The angles opposite the equal sides of an isosceles triangle are the same size. An isosceles triangle has two sides that are the same size. A B C Since AB = AC ABC = ACB 6. The opposite angles of a parallelogram are equal. A B D C AD BC and AB DC, therefore ABCD is a parallelogram. Therefore, ABC = ADC and BAD = BCD.

36 Assignments o 2. X X 45 o What is angle X? What is angle X? o L1 3X X A L2 B L1 and L2 are parallel lines. How big are angles A and B? o L1 6. L1 A 136 o B A B L2 L2 L1 and L2 are parallel lines. L1 and L2 are parallel lines. How big are angles A and B How big are angles A and B? X + 39 o 5X L1 L1 4X 2X L2 L1 and L2 are parallel lines. L2 L1 and L2 are parallel lines.

37 What is X? What is X? 9. One angle in a triangle is a right angle and one angle is 30 degrees. What is the third angle? 10. Two angles of a triangle are 42 degree and 103 degrees. What is the third angle? C 12. A Y 13. X B 134 o 116 o What is angle Y? This is an isosceles triangle. What is angle X? A B X 125 o D 38 o C X ABCD is a trapezoid What is angle X? AB and DC are parallel o What is angle X? S X X R P 112 o PQRS is a rhombus. Q This figure is a square.

38 What is angle X? What is angle X? 18. S 19. A B P 118 o X X R D C Q 25 o ABCD is a trapezoid. T AB and DC are parallel. PQRS is a parallelogram. What is angle X? What is angle X? 20. A B 21. A 50 o X 104 o X D B E C D ABCD is a parallelogram. C ABCD is a rhombus. Angle EDA is 60 o What is angle X? What is angle X? E 23. A B 24. X 46 o D A X D 35 o 112 o C C B ABCD is a parallelogram. ABCD is a parallelogram. Angle BAC is a right

39 What is angle X? angle. What is angle X? 25. A 26. A X D D 64 o 48 o X B B C E Angle DCB is 26 degrees. C What is angle X? Angle CBD is 18 degrees. What is angle X? 27. F 28. J M A E X P N 50 o 116 o X B C D ABDE is a trapezoid. K L What is angle X? JKLM is a square. 29. A D 30. Angle KPM is 116 o. What is angle X? C F F X B X C 38 o E D E G ABCD is a rectangle. FCD is an equilateral CDEF is a parallelogram.

40 triangle. What is angle X? What is angle X? 31. F E 32. F 120 o E A D D C X 110 o A B B C ABCD is a parallelogram. ABCD and ADEF are parallelograms Angle ADC is 110 o. AB = BF What is angle FAB? What is angle X? 33. D 34. A X E A 40 o D X 32 o B C B C 40 o ABCD is a parallelogram. E ABCD is a rhombus. What is angle X? What is angle X? 35. D C 36. E D X F X 40 0 C 28 o A B A B ABDE is a rectangle. ABCD is a parallelogram. BCDF is a rhombus. What is angle X? What is angle X?

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