Geometry Made Easy Handbook Common Core Standards Edition

Transcription

1 Geometry Made Easy Handbook ommon ore Standards Edition y: Mary nn asey. S. Mathematics, M. S. Education 2015 Topical Review ook ompany, Inc. ll rights reserved. P. O. ox 328 Onsted, MI This document may not, in whole or in part, be copied, photocopied, reproduced, translated, or reduced to any electronic medium or machine-readable form without prior consent in writing from Topical Review ook orporation or its author.

2 cknowledgments Thank you to everyone who helped put Geometry Made Easy, ommon ore Edition together. That includes Kimberly Knisell, irector of Math and Science in the Hyde Park, NY School istrict. Marin Malgieri who contributed proof reading skills and assistance with several topics. Jennifer riser-eighmy who proof read the grammar and punctuation. llen Miller and Nancy rush for their proofreading skills. Julieen Kane, the graphic designer at Topical Review ook ompany did a fabulous job with the diagrams and drawings. Lastly, but definitely very importantly, Keith Williams, the owner of Topical Review ook ompany deserves a major thank you. We have worked together for almost 20 years now and it is always a pleasure. His company provides excellent materials for students to use at a reasonable price. I used the Little Green Regents Review ooks way back in the old days when I was in high school. The company was already 25 years old at that time. I am honored to have my work published by Topical Review ook ompany and to share in their history. Introduction s the ommon ore Standards are implemented nationally, there are new methods of teaching material that was taught in a more traditional way in the past. The new presentations associated with the standards will help our students to become college and career ready. Geometry Made Easy, ommon ore Edition, is meant to be a reference guide for the mathematical procedures needed to help the student complete their work in using the standards. It is not meant to be a curriculum guide and is not designed to replace any teaching methods that are used in the classroom. It is my hope that this student friendly handbook will help each geometry student to achieve success in completing his/her study of the Geometry ommon ore Standards. Sincerely, Marynn asey,.s. Mathematics, M.S. Education

3 GEOMETRY ME ESY ommon ore Standards Edition Table of ontents UNIT 1: FOUNTIONS pplying Geometric oncepts Geometric Solutions Using Proofs Geometric Terms...13 UNIT 2: ONGRUENE, PROOFS N ONSTRUTIONS Transformational Geometry Terms Rigid Motions Geometric Terms Proving Geometric Theorems onstructing Lines nd ngles Inscribing Polygons in a ircle...54 UNIT 3: SIMILRITY PROOFS, N TRIGONOMETRY Transformation and similarity Similarity and ongruence of Triangles Similarity and ongruence of Polygons Right Triangles and Trigonometric Ratios...93 UNIT 4: EXTENING TO THREE IMENSIONS Volume imensional Figures and Their Properties hanging a 2-imensional Figure To 3-imensioal Figure UNIT 5: ONNETING LGER N GEOMETRY THROUGH OORINTES Graphing asics oordinate or nalytic Proof Example Partitioning a Segment in a Given Ratio Perimeter and rea Parabolas...135

4 GEOMETRY ME ESY ommon ore Standards Edition Table of ontents UNIT 6: IRLES WITH N WITHOUT OORINTES ircumference and rea of a ircle ircles and ngles Similarity of ircles ircles and Their ngles and rcs ngles of Sectors ircles and Segments Equation of a ircle ircles and Polygons ORRELTIONS TO SS INEX...174

5 Unit 1 FOUNTIONS pply geometric concepts. Solve using proofs. Recognize and use geometric terms. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 1

6 1.3 GEOMETRI TERMS OMMON GEOMETRI TERMS, EFINITIONS, N SYMOLS The tables that follow contain informal summaries and short cuts for many of the symbols, terms, and definitions that are used in geometry. Some can be used as reasons in a proof and others are descriptive terms. For more formal definitions, use your text or a math dictionary. lways listen to the directives of your teacher in using symbols or short cuts. SYMOLS USE IN PROLEMS N PROOFS escription Symbol Example Parallel m n Perpendicular m n ongruent pproximately equal to Similar ~ ~ EF Maps to R45 R 45 R or or 45 R or R or ircle P R45 Geometric Terms ONGRUENT VS SIMILR ongruent: Geometric figures are congruent if they have corresponding sides that are equal in measure and corresponding angles that are equal in measure. Rigid motion transformations result in figures that are congruent. EF E E EF F E F Similar: Geometric figures are similar if they have corresponding angles that are equal in measure and corresponding sides that are proportional. The ratio of the proportion of the corresponding sides is called the constant of proportionality. Transformations involving dilations result in similar figures. E 6 H ~ EFGH E = 2 = 2 3 F EF GH F 6 F 4 G G = 2 = 2 H FG EH Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 13

7 1.3 OMMONLY USE TERMS N EFINITIONS Geometric Terms Word, Term, and iagram and Labels rief efinition Mathematical Symbol Point P location in space. Has no length, width or depth. Line Plane Segment ollinear Points Ray ngle < or Has infinite length, no width. Has infinite length and width. No depth. Part of line between 2 points. Points that are on the same line /line segment. partial line that starts at a point & goes in one direction. Formed when 2 rays meet at a point or when 2 lines intersect. Label with the vertex (point at the center. The size of the opening between the rays is measured in degrees or radians. isect line bisector-figure 1 angle bisector-fig. 2 Perpendicular ^ Parallel l Fig. 1 Fig. 2 ( ( To cut in half: bisector cuts a line segment into 2 parts, an angle into 2 angles. 2 lines that intersect at right angles. 2 lines in a plane that never meet. Right ngle Measures 90. Geometry Made Easy ommon ore Standards Edition 14 opyright 2015 Topical Review ook Inc. ll rights reserved.

8 Straight ngle Measures 180. cute ngle Measures more than 0, less than 90. Obtuse ngle Measures more than 90, R less than 180. Reflex ngle S Measures more than 180, less than 360. Equiangular ll angles are congruent. 1.3 Equilateral ll sides are congruent. Geometric Terms Scalene Isosceles vertex angle and base angles in an isosceles Regular Polygon djacent ngles 1 and 2 are adjacent s. Linear Pair Opposite ngles 2 and 4 are opposite. Vertex iagonal onsecutive ngles /Sides ì and ü and ï ï and ï and ï í ý ì and ü ï and ï ï ï ï ï ï and ï î and þ í ý ï and ï ï ï î and þ h 1 2 Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. ll sides of a figure are different lengths. 2 sides are congruent: is isosceles. and is the base. is the vertex angle, and are the base angles. polygon with equal sides & equal angles. 5, 6, 8, 10 and 12 sided polygons are often used. Next to each other. ngles which share only one side and a vertex but have no interior points in common. 2 adjacent angles formed by the intersection of 2 lines. They are supplementary angles. cross from each other. Not sharing a side or a vertex. The point of an angle, the corner of a polygon. onnects 2 opposite vertices (corners in a geometric figure. Sides or angles that are one after the other. 15

9 Geometric Terms π is an irrational number. The ratio of circumference The symbol π should be to the diameter of a circle. included in your answer Use the π button on your if the problem says calculator to solve a problem leave in terms of Pi. involving π. Supplementary ngles 2 angles whose sum is180. m 1 + m 2 = 180 (Need not be adjacent. omplementary ngles 2 angles whose sum is 90. m 1 + m 2 = 90 (Need not be adjacent. ase In a formula requiring a is the base in height measurement, the parallelogram and base of the polygon is the E. and are side that the altitude is both bases in trapezoid drawn to. The formula for and h is the altitude. the area of a trapezoid uses both bases. ltitude and Height ltitude EF is the segment EF is the altitude in drawn from a vertex that is the figure and measures to the opposite side. the height. ircle ll the points in a plane at a given distance from a given P point called the center. circle is named by the center. Part of a curve between rc two points. iameter line segment whose endpoints are on the circle and passes through the center. Radius istance from the center of a circle to any point on the circle (the edge. Interior ngle of a polygon n angle formed on the inside of a polygon where the sides intersect. Exterior ngle of a polygon n outside angle formed is exterior. Vector Pi π,, E, & F are all interior s. Geometry Made Easy ommon ore Standards Edition 16 opyright 2015 Topical Review ook Inc. ll rights reserved. F E F h 2 1 E h 1 P F E F h h by extending the side of the figure. directed line segment. It has magnitude represented by its length and direction shown by an arrow.

10 Unit 2 ONGRUENE, PROOFS N ONSTRUTIONS Experiment with transformations in the plane. Understand congruence in terms of rigid motions. Prove geometric theorems. Make geometric constructions. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 17

11 onstructions 2.5 onstruction: drawing done in geometry using only a compass and a straight edge. Protractors, rulers, and graph paper or other devices for measuring are not permitted. No parts of a construction may be sketched. Each part of a construction must be drawn with the compass and/or a straight edge. The lines and arcs used to make the construction are to be left on the paper. O NOT ERSE the construction lines. In constructions, we are often duplicating the measure of something into a new drawing or we are using the compass to measure equal distances. If the compass should be changed from a previous step, it will be indicated. lso, the demonstration shown may not be the only way to do a construction. s we have seen throughout this geometry book, there are often several ways to reach the same result. Provided that acceptable construction methods are used with valid logical reasoning, other steps may be used to do a construction. Justify or explain the construction: This means to discuss why a particular method works. Follow your teacher s instructions about specifics that may be required. Examples onstructions onstruct a line segment congruent to a given line segment. Steps: ONSTRUTING LINES N NGLES Given: onstruct: 1 raw a line, m, in a different location than. 2 Put the compass point on and the pencil point on. This measures the length of. 3 Move the compass point to any point on line m and label the point. 4 Swing an arc through line m. 5 Label the point of intersection of m and the arc,. 6 iscussion: y using the compass to measure the length of the given segment, we can mark off an equal segment on another line. m Geometry Made Easy ommon ore Standards Edition 46 opyright 2015 Topical Review ook Inc. ll rights reserved.

12 onstruct an angle congruent to a given angle. Given: onstruct: EF Steps: 1 raw a line m and place a point, E on the line. 2 Put the compass point on in the given angle. Swing an arc through each side of. Label the points where the arcs intersect the sides of the angle with R and S. 3 Move the compass point to point E on line m. 4 Swing an arc that intersects m and label the point of intersection. 5 With the point still at E, swing another arc in approximately the place that the other side of the angle will be. 6 Put the compass point on R and the pencil point on S to measure that distance. 7 Move the point of the compass to point. 8 Swing an arc from point to intersect with the arc from step 5. Label that point F. 9 onnect point F and point E. 10 EF 2.5 R S F E m onstructions iscussion: If two angles are congruent, the measure between their sides at a specific distance from the vertex is equal. y marking off two equal segments on the sides of the given angle and measuring between the endpoints of the segments, we can determine the measures of corresponding parts of the congruent angle. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 47

13 Unit 3 SIMILRITY, PROOFS, N TRIGONOMETRY Understand similarity in terms of similarity transformations. Prove theorems involving similarity. efine trigonometric ratios and solve problems involving right triangles. pply geometric concepts in modeling situations. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 57

14 3.2 ISOSELES TRINGLES This information is included with the general triangle information, but it is specific to isosceles triangles. Known: triangle is isosceles (given, or already proven In an isosceles triangle, the angles opposite the congruent sides (base angles are congruent. In an isosceles triangle, two sides are congruent. The altitude of an isosceles triangle drawn from the base divides the triangle into two congruent right triangles. Note: The altitude of an isosceles triangle can often be found using the Pythagorean Theorem. See page 76 for Isosceles triangles and the Pythagorean Theorem. RIGHT TRINGLES (See also page 75 Hypotenuse - Leg Theorem: In two right triangles, if the hypotenuse and a leg of one are congruent to the corresponding parts of the other, the triangles are congruent. It must be stated that the triangles are right triangles first. dditional Right Triangle Information: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. The two acute angles in a right triangle are complementary. Pythagorean Theorem: c is the hypotenuse and a and b are the legs. Side c is the longest side. If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. s a formula it is written: c 2 = a 2 + b 2. It is also commonly written as a 2 + b 2 = c 2. Note: The Pythagorean Theorem is used to find a missing side of a triangle, and it also can be used to prove whether or not a triangle is a right triangle. If the sides of a given triangle can be substituted into the Pythagorean Theorem, using the longest side as c and the numbers check, then the triangle is a right triangle. b E EF a M c F Similarity and ongruence of Triangles Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 67

15 Theorem: In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Examples Given: Figure 1, right triangle, with hypotenuse c, legs a and b. b Prove: a 2 + b 2 = c Figure 1 Figure 2 x c c b z y Similarity and ongruence of Triangles Statement a Reason 1. Right 1. Given. 2. raw altitude from right 2. n altitude of a triangle is drawn from a angle perpendicular to vertex of a triangle and is perpendicular at. Label as shown to the side opposite the vertex. in Figure 2. (Labels are for convenience Perpendicular lines form right angles and all right angles are congruent. 4. ; 4. Reflexive Property. 5. ; 5. ll right angles are congruent. 6. ~ ; 6. If two angles of a triangle are congruent to ~ two angles in another triangle, the triangles are similar. similarity criterion. 7. In & : a y = 7. In similar triangles, corresponding sides c a are proportional. In & : b = x c b 8. a 2 = cy ; b 2 = cx 8. In a proportion, the product of the means equals the product of the extremes. 9. y + x = c 9. Segment addition. 10. a 2 + b 2 = cy + cx 10. ddition property of equality. 11. a 2 + b 2 = c(y + x 11. Factor the right side of the equation. 12. a 2 + b 2 = c Substitution. Note: There are many proofs of this theorem. Geometry Made Easy ommon ore Standards Edition 68 opyright 2015 Topical Review ook Inc. ll rights reserved. a

16 3.2 lternate method to prove the Pythagorean Theorem using similarity. This is a less formal proof than the statement-reason proof. In it, the technique of redrawing parts of a diagram separately is used, so they are more easily identified. Given: Right triangle with altitude drawn to the hypotenuse. Prove: Using similarity, prove that the sum of the legs squared is equal to the square of the hypotenuse. x c b z y b a Paragraph Proof: raw the three triangles separately and label appropriately. Each of the smaller right triangles shares one angle with the large triangle making each one similar to the large triangle,, using similarity. Using the transitive property, they are similar to each other. Since they are similar, their sides are proportional. x z a c y z a y b x z Similarity and ongruence of Triangles Triangles & Triangles & = = ( 2 = ( 2 = ( 2 + ( 2 = + ( 2 + ( 2 = ( + ( 2 + ( 2 = ( ( 2 + ( 2 = ( 2 onclusion: The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 69

17 Unit 4 EXTENING TO THREE IMENSIONS Explain volume formulas and use them to solve problems. Visualize relationships between two-dimensional and three-dimensional objects. pply geometric concepts in modeling situations. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 107

18 Unit 5 ONNETING LGER N GEOMETRY THROUGH OORINTES oordinate or nalytic Geometry is the work involved in associating a graph with algebraic processes. Formulas and equations are used along with logical reasoning skills to determine the classification of a figure, compare figures, find the dimensions of polygons or segments, and prove the congruency or similarity of figures drawn on a graph. Use coordinates to prove simple geometric theorems algebraically. Parallel and perpendicular lines Prove geometric figures algebraically Partitioning a segment Perimeter and area Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 117

19 5.4 PERIMETER N RE The distance formula was presented on page 126. It can be used to find the perimeter or area of polygons that are drawn on a coordinate plane. hoose the sides of the polygon that are needed to calculate the answer to the problem and use the formula to find their lengths. Expressing Geometric Properties with Equations SW SW Examples 2 d = ( x x + ( y y Lincoln Hotel is planning to reseed the grass for the rectangular lawn in front of the hotel. The corners of the land to be reseeded are shown on the following sketch. Grass seed is purchased by determining how many square feet of area it will cover. If each unit on the graph represents 50 feet, find the area that needs to be reseeded. y Scale Factor: 1 unit = 50 ft N(7, 7 (1, 5 W E (9, 3 S (3, 1 Since this is a rectangle, the formula for area is = lw. Find the length and the width of the rectangle and multiply each by is the scale factor used on the diagram above. The area of the lawn is the product of those two measurements. = ( ( = = 20 Length of lawn = SE SE = ( ( 3 1 = = 40 Width oflawn = = lw x = ( 50 20( = , 711sqft The area of the lawn to be reseeded is about 70,711 square feet. Geometry Made Easy ommon ore Standards Edition 132 opyright 2015 Topical Review ook Inc. ll rights reserved.

20 5.4 Lincoln Hotel, in Example 1 above, wants to make a flower border all the way around the lawn. The flower border will be 1 foot wide on all sides of the lawn. How many square feet of flowers will be needed? W 1 ft 1 ft l = ft S w= ft E The perimeter of the lawn is needed. The length and width are already known from example 1 above. The formula for the perimeter of a rectangle is P = 2l + 2w. on t forget the corners of the flower border as they are not part of the lawn measurement. Each corner is 1 ft 1 ft. P lawn = 2( (316 P = 1080 ft. The flower border is 1 foot wide square feet of border is needed to edge the grass, plus the 4 square feet for the corners. onclusion: bout 1084 square feet of flower border is needed. Expressing Geometric Properties with Equations Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 133

21 Find the area and perimeter of triangle with vertices at ( 3, 1, ( 1, 3 and (4, y ( 1, 3 ( 2, 1 ( 3, 1 (4, 2 x nalysis: Use the distance formula to find the length of each side and add them for the perimeter. Expressing Geometric Properties with Equations d d 2 2 = ( ( 2 3 = = 50 = = ( ( 2+ 1 = 49+ 1= 50 = d = ( ( = = 20 = 2 5 P = = To find the area, the altitude is needed. s shown in the perimeter calculations, this is an isosceles triangle. =, is the base. The length of the altitude drawn from to is needed. ltitude: Since this is an isosceles triangle the altitude drawn from the vertex opposite the base bisects the base. Find the midpoint,, of base. raw and find its length using the distance formula. x2 x1 y2 y1 Midpoint Formula: M = + +, 2 2 Midpoint of : M rea of : M = 3 + ( 1 +, = ( 21, = 15 = ( ( onclusion: The perimeter of is units and the area is 15 sq units. 2 Geometry Made Easy ommon ore Standards Edition 134 opyright 2015 Topical Review ook Inc. ll rights reserved.

22 Unit 6 IRLES WITH N WITHOUT OORINTES Understand formulas for circles. Understand and apply theorems about circles. Find arc lengths and areas of sectors of circles. Translate between the geometric description and the equation for a conic section. Use coordinates to prove simple geometric theorems algebraically. pply geometric concepts in modeling situations. Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 139

23 ORRELTIONS TO OMMON ORE STTE STNRS ommon ore State Standards Unit #. Section # ongruence, Proof, and onstruction (G.O G.O G.O G.O G.O G.O G.O G.O G.O G.O , 2.4 G.O G.O G.O G.O Similarity, Proof, and Trigonometry (G.SRT G.SRT G.SRT G.SRT G.SRT G.SRT G.SRT G.SRT , 3.3, 3.4 G.SRT Extending To Three imensions (G.GM G.GM G.GM G.GM G.GM orrelations onnecting lgebra and Geometry Through oordinates (G.GPE G.GPE G.GPE.2* G.GPE G.GPE G.GPE G.GPE Geometry Made Easy ommon ore Standards Edition 172 opyright 2015 Topical Review ook Inc. ll rights reserved.

24 ORRELTIONS TO OMMON ORE STTE STNRS ircles With and Without oordinates (G. G G G Modeling (G.MG G.MG.1 G.MG.2 G.MG.3 Throughout the units * G.GPE.2 is not included in all of the content frameworks. It is included in the the Geometry Standards. orrelations Geometry Made Easy ommon ore Standards Edition opyright 2015 Topical Review ook Inc. ll rights reserved. 173