Mini-Lecture 8.1 Lines and Angles

Transcription

1 Mini-Lecture 8.1 Lines and Angles 1. Identify lines, line segments, rays, and angles. 2. Classify angels as acute, right, obtuse, or straight. 3. Identify complementary and supplementary angles. 4. Find measures of angles. 5. Key Vocabulary: space, plane, point, line, line segment, ray, angle, vertex, sides, degrees, straight angle, right angle, acute angle, obtuse angle, complementary angles, supplementary angles, parallel lines, intersecting lines, perpendicular, vertical angles, adjacent angles, transversal, corresponding angles are equal, alternate interior angles are equal. 1. Draw an example of each term. a) line b) ray c) segment d) angle 2. Draw an example of each angle. a) acute b) right c) obtuse d) straight 3. Find each complementary or supplementary angle as indicated. a) Find the complement of a 35º angle. b) Find the complement of a 71º angle. c) Find the supplement of a 152º angle. d) Find the supplement of a 83º angle. 4. Find the measures of angles x, y, and z in each figure. a) 122º b) m n 139º x z x m y y z n Some students are unfamiliar with the vocabulary and need repetition. Refer students to the textbook, Chapter Highlights, for a condensed listing of definitions and concepts. Answers: 1a) d) Answers may vary; 2a d) Answers may vary; 3a) 55, b) 71 c) 28, d) 97 ; 4a) x =58, y=122, z=58, b) x=139, y=139, z=41 M-44 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.

2 Mini-Lecture 8.2 Plane Figures and Solids 1. Identify plane figures: triangles, parallelograms, rectangles, squares, rhombus, trapezoids. 2. Identify solid figures: rectangular solid, cube, pyramid, sphere, cylinder, cone 3. Key Vocabulary: plane, plane figure, polygon, regular polygon, equilateral triangle, isosceles triangle, scalene triangle, right triangle, hypotenuse, legs, quadrilateral, circle, center, radius, diameter, solid. 1. Identify each polygon. a) b) c) d) e) Given a triangle with a = 23, b= 47, find c. f) Given a right triangle and one angle measuring 51º, find the third angle. g) Find the radius of a circle with diameter 14 ft. h) A pizza has a radius of 9 in. What is the diameter of the pizza? 2. Identify each solid. a) b) c) d) The radius of a sphere is 8.3 inches. Find its diameter. e) Find the radius of a sphere if the diameter is 34.5 miles. Some students will have trouble remembering all the different figures. Many students may need to associate the vocabulary with a figure that they are familiar with. Remind students that a right triangle will have the symbol for 90º. Many students will assume a triangle that looks like a right triangle is a right triangle. Answers: 1a) triangle, b) hexagon, c) octagon, d) trapezoid, e) 110, f) 39, g) 7 ft., h) 18 in; 2a) cylinder, b) cube, c) rectangular solid, d) 16.6 in., e) miles. Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. M-45

3 Mini-Lecture 8.3 Perimeter Learning Objectives 1. Use formulas to find perimeter of a rectangle, square, triangle, and other polygons. 2. Use formulas to find circumference. 3. Key Vocabulary: perimeter, circumference, π. 1. Find the perimeter of the following figures. a) 15 m b) 25 ft c) 10 ft 6 m 23 ft 23 ft 8 ft 2 ft 18 ft 7 ft d) A rectangular field measures 441 feet by 108 feet. Find the cost of constructing a fence if fencing costs $29.50 per yd. 2. Find the circumference of each circle. Give the exact and then an approximation by using π = Round to the nearest hundredth. a) radius = 7 ft. b) diameter = 84 m. c) radius = 34.9 ft d) A circular room has a radius of 10.3 feet. Find the distance around the room. e) A circular statue has a base with a diameter of 11 ft. Find the distance around the base of the statue. f) Find the distance around a circular Jacuzzi with a diameter of 8 ft. For this problem, use π = 22 7 Refer students to textbook for formulas of perimeter and circumference. Many students have difficulty understanding the difference between approximation and exact value when working with π. Remind students to read carefully and take note of radius vs. diameter. Answers: 1a) 42 m, b) 89 ft., c) 45 ft., d) $32,391; 2a) 14π, ft, b) 84π, m, c) 69.8π, ft, d) 20.6π, ft, e) 11π, ft., f) 8π, 27 1/7 ft. M-46 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.

4 Mini-Lecture 8.4 Area 1. Find the area of geometric figures: rectangle, square, triangle, parallelogram, trapezoid, circle. 2. Key Vocabulary: area, square units. 1. Find the area of the following. Use π = Round to the nearest hundredth. 2. a) 7 yd b) c) 3 yd 8 ft d = 24 miles d) 13.4 cm Parallelogram 3.8 ft 8.2 cm e) A small rug is in the shape of a trapezoid. The bases measure 19.8 in and 22.4 in. and the height is 8 in. Find the area of the rug. f) A side of a square towel measures 7.5 in. Find how many square inches of material is needed to make 8 towels. g) 10 km 5.5 km 5.5 km 12 km Refer students to textbook for Area Formulas of Common Geometric Figures and Area Formula of a Circle. Many students have difficulty when the height lies outside of a triangle, parallelogram, or trapezoid. Answers: 1a) 21 yd 2, b) 15.2 ft 2, c) mi 2, d) cm 2, e) in 2, f) 450 in 2, g) km 2 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. M-47

5 Mini-Lecture 8.5 Volume 1. Find the volume of a rectangular solid and cube. 2. Find the volume of a sphere and circular cylinder. 3. Find the volume of a cone and square-based pyramid. 4. Key Vocabulary: volume, cubic units. 1. Find the volume of each rectangular solid or cube. Round to the nearest tenth. a) Rectangular solid: length = 8.3 in.; width = 3 in.; height = 9 in. b) Cube: side = 7.5 cm c) Find the volume of a box in the shape of a cube that is in. 2. Find the volume of each sphere or circular cylinder. Give the exact volume and an approximate. Use π = Round to the nearest hundredth. a) Sphere: diameter = 12 in. b) Circular cylinder: radius = 8 feet; height = 3 feet c) A birdbath is made in the shape of a hemisphere (half-sphere). If it has a diameter of 15 in, approximate the volume. 3. Find the volume of each cone or square-based pyramid. If necessary, give the exact volume and an approximate. Use π = Round to the nearest hundredth. a) Cone: height = 6 m; diameter = 5 m. b) Square-based pyramid: height = 11 cm; edge of base = 10.5 cm c) An ice cream cone with a 5-centimeter diameter and 9-centimeter depth is filled exactly level with the top of the cone. Approximate how much ice cream ( in cubic centimeters) is in the cone. Refer students to the textbook for Volume Formulas of Common Solids. Some students need to be able to visualize the solids. Encourage students to seek out actual items in their environment of these common solids. Remind students that volume is measured in cubic units. Answers: 1a) in 3, b) cm 3, c) in 3 ; 2a) 288π, in 3, b) 192π, ft 3, c) π, in 3 25 ; 3a) 2 π, m3, b) cm 3, c) 75 π, 58.9 cm3 4 M-48 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.

6 Mini-Lecture 8.6 Square Roots and the Pythagorean Theorem 1. Find the square root of a number (perfect square). 2. Approximate square roots with a calculator or chart. 3. Use the Pythagorean Theorem. 4. Key Vocabulary: square root, radical sign, perfect square, right triangle, hypotenuse, legs. 1. Find the square root. a) 16 b) 121 c) d) Use Appendix E or a calculator to approximate each square root. Round the square root to the nearest thousandth. a) 7 b) 53 c) 102 d) Using Pythagorean theorem, find the unknown length in each right triangle. If necessary, approximate the length to the nearest thousandth. a) b) 24 m 12 ft? 28 m? 9 ft Find the length of the side not given. If necessary, approximate the length to the nearest thousandth. c) leg = 5 ft, leg = 14 ft. d) leg = 3 km; hypotenuse = 6 km Refer students to Pythagorean Theorem: ( leg ) + ( other leg ) = ( hypotenuse) Many students need practice in identifying leg, other leg, and hypotenuse. Some students will get exact answer (in radical form) and approximate (using a calculator) confused. Answers: 1a) 4, b) 11, c) 3/13, d) 5; 2a) 2.646, b) 7.28, c) 10.1, d) ; 3a) 15 ft., b) , c) ft, d) km Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. M-49

7 Mini-Lecture 8.7 Congruent and Similar Triangles 1. Determine whether two triangles are congruent. 2. Find the ratio of corresponding sides in similar triangles. 3. Find unknown length of sides in similar triangles. 4. Key Vocabulary: congruent, Angle-Side-Angle (ASA), Side-Side-Side (SSS), Side-Angle- Side (SAS), similar. 1. Determine whether each pair of triangles is congruent. 80º a) 7 in 5 in 7 in b) 63º 12 in 8 in 8 in 5 in 12 in. c) 12 m 135º 4 m 12 m 63º 80º 135º 4 m 2. Find each ratio of the corresponding sides of the given similar triangles. 1.6 m a) b) 50 cm 1.2m 25 cm 1.8 m 2.4 m 24 cm 48 cm 3. Solve. Round to the nearest tenth. a) A tree casts a shadow of 26 ft. Nearby, a 5-ft pole casts a shadow of 3 ft. What is the height of the tree? b) A rock climber is 6 feet tall and his shadow measures 9 feet long. The rock she wants to climb casts a shadow of 580 feet. How tall is the rock? Some students need a review on solving proportions. Encourage students to mentally visualize applied problems and draw and label a diagram before solving. Answers: 1a) yes, b) yes, c) yes; 2a) ½, b) 4/3; 3a) 43 1/3 ft., b) 386 2/3 ft M-50 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.