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Transcript
Unit 1: Essentials of Geometry Lesson 1.1: Points, Lines, and Planes Objective Understand and use the basic undefined terms and defined terms in geometry such as point, line, plane, collinear, coplanar. USING UNDEFINED TERMS AND DEFINTIONS Point Line ________________ Name Plane ______________________ Names _____________________ Names Example 1 a. Name three points that are collinear: ___________________________ b. Name four points that are coplanar: ____________________________ c. Name three points that are not collinear: ________________________ Line Segment Ray Example 2 Give another name for GH . _______________ Name all the rays with endpoint J. ______________________ Which of those rays are opposite rays? ___________________ INTERSECTIONS The intersection of two lines: _____________________ The intersection of two planes: ____________________ Opposite Rays Example 3 Sketch a plane and a line that is in the plane. Sketch a plane and a line that does not intersect the plane. Sketch a plane and a line that intersects at one point. Example 4 Name the intersection of PQ and line k. ________________ Name the intersection of plane A and plane B. ________________ Name the intersection of plane A and line k. _________________ Example 5 You are given an equation of a line and a point. Use substitution to determine whether the point is on the line. y = x – 4; A (5, 1) _______________________ y = x + 1; A (1, 0) _______________________ Example 6 Graph the inequality on a number line. Tell whether the graph is a segment, a ray or rays, a point, or a line. x<3 _______________________ -7 < x < 4 _______________________ Unit 1: Essentials of Geometry Lesson 1.2: Use Segments and Congruence Objective Find the distance between two points using the Ruler Postulate and Segment Addition Postulate Use the Distance Formula to find the distance between two points in the coordinate plane. Understand and apply the definition of congruence (congruent segments). USING SEGMENT POSTULATES Ruler Postulate Example 1 Measure the length of the segment to the nearest millimeter. When three points are collinear, one point is ____________________ the other two. For instance, point _____________ is between points A and C. Segment Addition Postulate Example 2 Use the map to find the distance between Lubbock and St. Louis. __________________. Example 3 Use the diagram to find GH . GH = ____________ CONGRUENT SEGMENTS Example 4: Plot J(-3, 4), K(2, 4), L(1, 3), and M(1, -2) in a coordinate plane. Then determine whether JK and LM are congruent. Example 5: Use the number line to find the indicated distance. VW = ___________ XY = ___________ XZ = ___________ VX= ___________ Unit 1: Essentials of Geometry Lesson 1.3 Apply Pythagorean Theorem Lesson 7.1 from Geometry Textbook Objectives Use the Pythagorean Theorem to find the length of the unknown side of a right triangle and the area of the triangle. Determine which numbers form a Pythagorean Triple RIGHT TRIANGLES Pythagorean Theorem _____________________________ Example 1 Find the length of the sides of each of the given triangles. x = ___________ x = ___________ Example 2 Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. HINT: A = ½(base of triangle)(height of triangle). Example 3 A = __________________ A = __________________ PYTHAGOREAN TRIPLES __________________________________________ Example 3 Complete the table of Pythagorean Triples and their multiples. Multiple of 2 3, 4, 5 6, 8, 10 5, 12, 13 8, 15, 17 7, 24, 25 14, 28, 50 15, 36, 39 40, 75, 85 Multiple of x Example 4 Ladder A 20 foot ladder is resting against the side of a house. The base of the ladder is 4 feet away from the house. Approximately how high above the ground does the ladder touch the house? Example 5 Real Estate An investor owns a triangular plot of land as shown in the diagram. Find the perimeter of the plot of land. One acre of land is equivalent to 43,560 square feet. How many acres are in this plot of land? Round to two decimal places. The investor is planning on selling the land. The market rate in this area is $5000 per acre. How much should the investor ask for the land? Unit 1: Essentials of Geometry Lesson 1.4 Using the Midpoint and Distance Formulas Lesson 1.3 from Geometry Textbook Objectives Use midpoint and distance formula to find the distance between two points and their midpoint. MIDPOINTS AND SEGMENT BISECTORS Example 1 The figure shows a gate with diagonal braces. MO bisects NP at Q. If PQ is 22.6, find PN. PN = ___________________ Example 2 Point M is the midpoint of VW . Find x and the length of VM . VM = _________________________ COORDINATE PLANE The Midpoint Formula ________________________________ Example 3 Find the coordinates of the midpoint M. Find the coordinates of the endpoint K. M = ___________________ K = ____________________ Example 4 Find the length of the segment and then find the coordinate of the midpoint of the segment. Length = ____________ Midpoint _____________ DISTANCE FORMULA The Distance Formula If A(x1, y1) and B(x2, y2) are points in the coordinate plane, then the distance between A and B is ___________________________________________ Example 5 Find the length of RS . RS = ______________ Example 6 The coordinates of two segments are given. Find each segment length. Tell whether the segments are congruent. AB = ________________ CD = ___________________ AB CD ? ___________________ Unit 1: Essentials of Geometry Lesson 1.5: Measure and Classify Angles Lesson 1.4 from Geometry Textbook Objective Find the measure of an angle using different postulates such as the Protractor Postulate and Angle Addition Postulate. Classify angles as acute, right, obtuse, and straight. Use a protractor to construct and find the measure of an angle. USING ANGLE POSTULATES The angle that has sides AB and AC is denoted _______________________. The point A is the ____________ of the angle. Example 1 Name the angles in the figure. ____________________________________________ Protractor Postulate Words: ____________________________________ Symbols: __________________________________ CLASSIFYING ANGLES ________________ ________________ ________________ _______________ Example 2 Use the diagram to find the measure of the indicated angle. Then classify the angle. a) KHJ b) GHK _______________ ________________ _______________ ________________ c) GHJ d) GHL ________________ ________________ _______________ ________________ Angle Addition Postulate Example 3 Given that mLKN 145 , find mLKM and mMKN . mLKM = _______________ mMKN = ________________ CLASSIFYING ANGLES Angles that have the same measure are called _____________________. MEASURES ARE EQUAL ANGLES ARE CONGRUENT ______________________ ______________________ Example 4 The figure shows angles formed by the legs of an ironing board. Identify the congruent angles. If mHGI 40 , what is mGJK ? ______________________________ ________________ Angle Bisectors MN is the ______________________ of PMQ . PMN QMN Example 5 In the diagram at the right, YW bisects XYZ , and mXYW 18. Find mXYZ . mXYZ _________________________ Unit 1: Basics of Geometry Lesson 1.6 Describe Angle Pair Relationships Lesson 1.5 from Geometry Textbook Objective Classify pairs of angles as vertical, supplementary, complementary, and a linear pair. Apply understanding of angle pair relationship to find the measures of given angles. COMPLEMENTARY AND SUPPLEMENTARY ANGLES Example 1 In the figure, name a pair of complementary angles and supplementary angles, and a pair of adjacent angles. Example 2 Given that <1 is a complement of <2 and m<1 = 68o, find m<2. ______________ Given that <3 is a supplement of <4 and m<4 = 56o, find m<3. ______________ Example 3 Assume that <A is supplementary to <B and complementary to <C. Determine m<A, m<B, and m<C. m<A = x°, m<B = (x – 20)° and m<C = (x + 30)° ______________________________ VERTICAL ANGLES AND LINEAR PAIRS ______________________________ ______________________________ Example 4 Identify all of the linear pairs and all of the vertical angles in the figure at right. Vertical angles ____________________________ Linear pairs ______________________________ Example 5 Two angles form a linear pair. One angle is 5 times larger than the other. Find the measures of the two angles. ______________ _______________ Example 6 Solve for x and y. Then find the angle measures. x = ________ y = __________ Unit 1: Essentials of Geometry Lesson 1.7 Classify Polygons Lesson 1.6 from Geometry Textbook Objectives Classify polygons as concave, convex, and regular. Classify polygons by the number of their sides. IDENTIFY POLYGONS ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ CONCAVE AND CONVEX POLYGONS Convex polygon _________________________________ Concave polygon ________________________________ CLASSIFYING POLYGONS A polygon is named by its number of sides. # of sides 3 4 5 6 8 9 10 12 7 Type of polygon # of sides Type of polygon EQUILATERAL AND EQUIANGULAR Equilateral polygon ______________________ Equiangular polygon _____________________ Regular polygon _________________________ n Example 1 Classify the polygon by the number of its sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. _________________________ ________________________ _____________________ Example 2 Use your ruler and draw a concave, regular heptagon. _________________________ Example 3 A racked of billiard balls is shaped like an equilateral triangle. The expressions shown represent side lengths of the rack. Find the lengths of each side. x = ___________ side length ____________________ Example 4 Three vertices of a regular quadrilateral are A( 0, 4) , B(0, -4) , and C (4, 4). What would the other vertex of the figure be? Example 5 The figure is a regular polygon. Expressions are given for two side lengths. Find the value of x. _____________________________ _____________________ Unit 1: Essentials of Geometry Lesson 1.8 Perimeter, Circumference, and Area Lesson 1.7 from Geometry Textbook Objective Use various formulas to find the perimeter, circumference, and area of closed plane figures such as a quadrilateral, triangle, and circle. Develop and utilize a problem solving plan. REVIEWING PERIMETER, CIRCUMFERENCE, AND AREA Formulas for Perimeter, Area, and Circumference Square Rectangle Side length s. length l and width w P = ________ P = _____________ A = ________ A = _____________ Triangle Circle Side lengths a, b, and c. Base b and height h. radius r P = ________ P = _____________ A = ________ A = _____________ Example 1 Example 2 Find the area and perimeter of a rectangle length 12 inches and width 5 inches. Find the radius, circumference, and area of a circle with a diameter of length 8 cm. A = ___________ r = ____________ P = ___________ C = ____________ Example 3 Find the area and perimeter of the triangle defined by D(1,3), E(8, 3) and F(4, 7) A = _____________ USING A PROBLEM SOLVING PLAN Example 4 You have a part-time job at a school. You need to buy enough grass seed to cover the school’s soccer field. The field is 50 yards wide and 100 yards long. The instructions on the seed bags say that one bag will cover 5000 square feet. How many bags do you need? Example 5 You are planning a deck along two sides of a pool. The pool measures 18 feet by 12 feet. The deck is to be 8 feet wide. What is the area of the deck? Example 6 You are making a triangular flag with a base of 24 inches and an area of 360 square inches. How long should it be?