GEO6 GEOMETRY AND MEASURESUREMENT Student Pages for Packet 6: Drawings and Constructions
Name ___________________________ Period __________ Date ___________ GEO6 uc e STUDENT PAGES ep r od GEOMETRY AND MEASURESUREMENT Student Pages for Packet 6: Drawings and Constructions ot R GEO6.1 Geometric Drawings Review geometric notation and vocabulary. Use a compass and a ruler to make geometric drawings. Learn about and understand congruence. N GEO6.2 Geometric Constructions Learn the SSS axiom for triangle congruence. Use a compass and straightedge to make classic Euclidean constructions. Justify constructions by completing proofs. 8 15 Sa m pl e: D o GEO6.3 Vocabulary, Skill Builders, and Review 1 Geometry and Measurement Unit (Student Pages) GEO6 – SP Drawings and Constructions WORD BANK (GEO6) Definition Example or Picture uc e Word ep r od altitude ot R angle bisector pl e: D diagonal o N central angle m midpoint Sa perpendicular bisector Measurement and Geometry Unit (Student Pages) GEO6 – SP0 Drawings and Constructions 6.1 Geometric Drawings uc e GEOMETRIC DRAWINGS Ready (Summary) Set (Goals) We will make geometric drawings using a straight edge and a compass. We will review geometric notation and vocabulary. Go (Warmup) ep r od Review geometric notation and vocabulary. Use a compass and a ruler to make geometric drawings. Learn about and understand congruence. These are obtuse angles. pl e: D o These are acute angles. N ot R Right angles are angles that measure exactly 90o. These are right angles. 2. Describe obtuse angles in your own words. Sa m 1. Describe acute angles in your own words. Geometry and Measurement (Student Pages) GEO6 – SP1 Drawings and Constructions 6.1 Geometric Drawings LABELING GEOMETRIC FIGURES Object uc e Name each object. Use the words: angle, line, line segment, ray or point. Picture 1. How to Label A dot with a capital letter near it. od P Two points named on the line with a “double arrow line” on top. Order does not matter. ep r A 2. B AB or BA ot R The endpoint and another point on the ray named with a “single arrow ray” on top. Order does matter. D 3. C DC only The endpoints named on the line segment with a “no- arrow segment” on top. Order does not matter. N E 4. F o EF or FE Three points. The middle point is always the vertex of the angle. 5. pl e: D K G GHK or KHG H 6. Write two names for the ray that has m N as an endpoint. N _____ and _____ Sa 7. Write the two names for the segment that has endpoints at M and L . _____ and _____ 8. Write two names for the angle at x. Geometry and Measurement (Student Pages) M x L J Q _____ and _____ GEO6 – SP2 Drawings and Constructions 6.1 Geometric Drawings LABELING GEOMETRIC FIGURES (continued) Picture How to Label M Three points in any order. 9. L Example: N T LMN od Object uc e Name each object. Use the words: congruent triangles or triangle. Congruent and corresponding parts are named in order. ep r W RST UVW Examples: S R 10. T ot R ST VW U V W N Congruent sides are labeled with the same number of “hash marks.” U Congruent angles are labeled with distinguishing marks as well. V o S pl e: D R RST UVW The two triangles on the right are congruent because they have the same size and same shape. 11. Z C X CB __________ A Sa 13. B ZYX __________ m 12. Y XYZ __________ 14. Draw the appropriate markings on all angles and all sides. Geometry and Measurement (Student Pages) GEO6 – SP3 Drawings and Constructions 6.1 Geometric Drawings DRAWING AN ISOSCELES TRIANGLE uc e An isosceles triangle is a triangle with two equal sides. Use a centimeter ruler and follow the directions to draw an isosceles triangle. Use the words in the box to fill in the blanks. 1. Draw a 6-cm horizontal line segment from and extending to the right of point A. Label the od other endpoint B. 2. Find the point on AB that divides the segment into two equal or congruent parts. This point ep r is known as the ____________________ of AB . Label it point M. 3. Locate point M on AB and draw a 4-cm line segment, DM , that is perpendicular ( ) to ot R AB at point M. Label the other endpoint D. ( AB _____ MD ) 4. Connect point A with point D and point B with point D to make triangle ABD (∆ABD). N How long is AD ? _____ cm. How long is BD ? ______ cm. 5. AD is ____________________ to BD ( AD _____ BD ). o 6. What kinds of angles are AMD and BMD? ____________________ pl e: D What kinds of angles are MDA and MDB? ____________________ What kinds of angles are MAD and MBD? ____________________ 7. MD is called the height, or ____________________ of ∆ABD. acute angles m altitude Sa congruent midpoint right angles Geometry and Measurement (Student Pages) A GEO6 – SP4 Drawings and Constructions 6.1 Geometric Drawings DRAWING A CIRCLE uc e Use a centimeter ruler and a compass. Follow the given directions to draw a circle. 1. With a radius of 3 cm, draw a circle with center at point C. How many degrees is a full turn of a circle? _____. od 2. Draw a horizontal diameter through C. Label the endpoints F and H. Name both radii. _____ and _____ . ep r 3. Name the diameter. _____ How long is this diameter? _____ cm. 4. Write an equation that relates the length of the diameter (d) to the length of the radius (r) of ot R any circle. ____________________ . 5. Find and label a point R on the circle. Draw FHR. What kind of angle is FHR? ____________________ . N 6. Find and label a point N on the circle so that FCN is an obtuse angle. We call this a central angle, because its vertex is at the center of the circle. o Name another central angle. pl e: D ____________________. 7. Any segment that has both endpoints on a circle is called a chord. Name both chords • C Sa m in this diagram. _____ and _____ . Geometry and Measurement (Student Pages) GEO6 – SP5 Drawings and Constructions 6.1 Geometric Drawings DRAWING A TRAPEZOID uc e Use a centimeter ruler and follow the given directions to draw a trapezoid. 1. Starting at point W, draw a 4 cm horizontal line segment. Label the other endpoint X. 2. From point X, draw XY so that it is perpendicular to WX and is 3 cm. od 3. From point Y, draw YZ to the left of point Y so that it is parallel to WX and is 2 cm. ep r 4. Draw in ZW . What is the approximate length of this segment? _____ cm 5. Refer to trapezoid WXYZ. Are there any right angles? _____ Name them if any exist __________________ ot R Are there any acute angles? _____ Name them if any exist __________________ Are there any obtuse angles? _____ Name them if any exist __________________ 6. Draw a line segment from point Z to point X. This is called a diagonal. Draw another one in N this figure. Name this segment _____ o 7. Draw an altitude from point Z to WX at a point called V. What kinds of angles are WVZ pl e: D and XVZ? __________ m Diagonal (a segment that is not the side of a polygon, but connects two vertices) Sa Parallel (two lines in the same direction) Geometry and Measurement (Student Pages) W GEO6 – SP6 Drawings and Constructions 6.1 Geometric Drawings YOU WRITE THE DIRECTIONS square rectangle parallelogram segment midpoint congruent parallel perpendicular right angle height altitude obtuse angle acute angle ep r triangle od Word Bank uc e Make a geometric drawing here. Write detailed directions of your drawing on a separate piece of paper. Give the directions to your partner and see if your partner can duplicate your drawing using only your written directions. You must use at least six words from the word bank. A Drawing From Your Partner’s Directions Sa m pl e: D o N ot R Your Drawing Geometry and Measurement (Student Pages) GEO6 – SP7 Drawings and Constructions 6.2 Geometric Constructions uc e GEOMETRIC CONSTRUCTIONS Ready (Summary) Set (Goals) od Learn the SSS axiom for triangle congruence. Use a compass and straightedge to make classic Euclidean constructions. Justify constructions by completing proofs. ot R Go (Warmup) ep r We will learn about the Side-Side-Side axiom (SSS) for triangle congruence. We will use SSS to complete proofs that justify three classic Euclidean compassstraightedge constructions. x A B y N 60o o Use the figure above to answer the questions. pl e: D 1. Starting at point A and tracing clockwise, the rotation around the full circle (from A back to A) is _____ degrees. 2. Starting at point A and tracing clockwise, the rotation around half of the circle (from A to B) is _____ degrees. The is referred to as a straight angle. 3. The small square in the diagram means that the angle is _____ degrees. This is referred to m as a right angle. Sa 4. x is _____ degrees. It is also a _____________________. 5. y is _____ degrees. 6. From the diagram we see that two right angles form a ______________________. Geometry and Measurement (Student Pages) GEO6 – SP8 Drawings and Constructions 6.2 Geometric Constructions CREATING A TRIANGLE uc e Your teacher will give you three pieces of spaghetti in different lengths. Tape the three pieces together to make a triangle. 1. Trace your triangle and your partner’s triangle below. Your triangle ot R ep r od Your partner’s triangle o N The Side-Side-Side axiom states that if two triangles have corresponding sides that are congruent, then the triangles are congruent to one another. pl e: D ∆CAT ∆DOG C A D T m 2. Make hash marks on the triangles above to show corresponding, congruent sides. Sa a. CA _____ b. AT _____ c. TC _____ Geometry and Measurement (Student Pages) O G 3. Mark corresponding, congruent angles on the triangles above. a. D _____ b. O _____ c. G _____ GEO6 – SP9 Drawings and Constructions 6.2 Geometric Constructions SOME GEOMETRY FACTS Two triangles are congruent if their corresponding sides are congruent to one another and their corresponding angles are congruent to one another. reflexive property of congruence In geometry, a figure is congruent to itself. U V W N Congruent Parts of Congruent Triangles are Congruent pl e: D o CPCTC T ot R SSS axiom Two triangles are congruent if their corresponding sides are congruent to one another. Symbols uc e Picture od definition of congruent triangles Words ep r Short Name A compass opening defines a length. Sa m radius facts All arcs drawn with the same compass opening have equal radius lengths. All radii of a given circle have the same length. Geometry and Measurement (Student Pages) GEO6 – SP10 Drawings and Constructions 6.2 Geometric Constructions od uc e CONSTRUCTION 1: BISECT AN ANGLE ep r V Use a straightedge and a compass to divide this angle into two congruent parts (bisect). What do we know? ot R Directions 1. Draw an arc from V that intersects both rays. Label the points of intersection A and B. VA VB , because all ________________ from a given circle have the ____________. drawn with the same compass opening have 3. Draw VX VX VX _______________ property pl e: D o N 2. With the same compass setting, draw an arc from A and an arc from B into the interior of the angle, long enough so they intersect. Label the point of intersection X. Draw AX and BX . AX BX , because all _______________ equal _____________length. Use the diagram you just constructed to prove that AVX BVX . m Statement AX BX ; VX VX Sa VA VB ; Reason Given above ∆AVX ∆BVX _____ axiom AVX BVX ___ ___ ___ ___ ___ Geometry and Measurement (Student Pages) GEO6 – SP11 Drawings and Constructions 6.2 Geometric Constructions od uc e CONSTRUCTION 2: DRAW A PERPENDICULAR THROUGH A POINT ON A LINE ep r P Use a straightedge and a compass to draw a perpendicular to P. Directions What do we know? TP WP , because all _______________ ot R 1. From P, draw an arc that intersects the line to the left and to the right. Label the points of intersection T and W. from a given circle have the ___________. 2. Draw an arc from T and an arc from W with the same compass setting (but longer than in step 1) so that they intersect above P. Label the point of intersection N. Draw TN and WN . drawn with the same compass opening have 3. Draw PN . PN PN __________________ property TN WN , because all _______________ o N equal _____________length. pl e: D Use the diagram you just constructed to prove that NP TW . Statement Reason Given above ∆TPN ∆WPN _____ axiom TPN WPN ___ ___ ___ ___ ___ m TP WP ; TN WN ; PN PN Sa TPN and WPN must both be right angles NP TW Geometry and Measurement (Student Pages) The sum of TPN and WPN must be ______ (a straight angle), and if they are congruent, they must each be _______. Two lines that form right angles are ____________________ to one another. GEO6 – SP12 Drawings and Constructions 6.2 Geometric Constructions F od C uc e CONSTRUCTION 3: COPY AN ANGLE Use a straightedge and a compass to copy C with the vertex at F and the given ray. What do we know? CG _____, because ot R 1. Draw an arc from C that intersects both rays. On the horizontal, label the point of intersection G. Label the other point of intersection D. ep r Directions 2. Using the same compass setting, draw an arc from F on the given ray. On the horizontal, label the point of intersection H. GD _____, because pl e: D o N 3. Draw GD . Then place the sharp point your compass on G and the pencil tip on D to “measure” GD . Using this compass setting, draw an arc from H that intersects the arc above it. Label this point of intersection K. Draw HK . FH _____, because 4. Draw KF . KF _____, because Use the diagram you just constructed to prove that F ____ . Statement Given above. m CG _____ Reason FH _____ Sa GD _____ ∆______ ∆______ F ____ Geometry and Measurement (Student Pages) GEO6 – SP13 Drawings and Constructions 6.2 Geometric Constructions uc e CONSTRUCTION 4: DRAW A PERPENDICULAR SEGMENT FROM A POINT OFF OF A LINE N ot R ep r od P o Use a straightedge and a compass to draw a perpendicular from P to the line. Directions pl e: D 1. From P, draw an arc that intersects the line to the left and to the right. Label the points of intersection T and W. 2. From T, draw an arc below the line (be sure that it is somewhere underneath P). From W, draw an arc with the same compass opening that intersects the previous arc below the line. Label the point of intersection N. Sa m 3. Draw PN , which is perpendicular to the line. Label the point of intersection X. Geometry and Measurement (Student Pages) GEO6 – SP14 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (GEO6) 1 2 3 4 od 5 6 8 7 uc e Fill in the crossword puzzle using the clues below. 9 ep r 10 11 ot R 12 13 14 Down 2. AC of this parallelogram N Across 1. Bisector of a segment 3. Forms right angles. o 5. A _______ is 90 D C A B A 4. Line segment or ray in an angle that divides it into two congruent angles. 6. Greater than 90 and less than 180 11. Another word for altitude 7. A proven mathematical statement 12.A line with two end points 8. Squares A and B are _____. 13. The perpendicular distance from a vertex to the opposite side of a plane figure. 14. Convincing argument to justify and mathematical statement. 9. Lines that never cross and are always the same distance apart. A B Sa m pl e: D 8. An angle that has the center of a circle as its vertex. 10. Less than 90 proof diagonal altitude perpendicular theorem central angle Geometry and Measurement (Student Pages) Word Bank parallel congruent angle bisector right angle midpoint height acute angle obtuse angle segment GEO6 – SP15 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 uc e Draw a picture, write an appropriate formula, and substitute to solve each problem. 2. The perimeter of a square ballroom is 49.6 cm. What is the area of the ballroom? a. Sketch the figure a. Sketch the figure c. Substitute and solve m pl e: D o c. Substitute and solve b. Write an appropriate formula N b. Write an appropriate formula ot R ep r od 1. The height of a rectangle measures 62 cm and its length measures 53 cm. Find the area. d. Answer the question Sa d. Answer the question Geometry and Measurement (Student Pages) GEO6 – SP16 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 uc e Draw a picture, write an appropriate formula, and substitute to solve each problem. 2. A triangular lawn has an area of 59.2 square meters. The base of the lawn is 14.8 meters wide. Find the height of the lawn. a. Sketch the figure a. Sketch the figure c. Substitute and solve m pl e: D o c. Substitute and solve b. Write an appropriate formula N b. Write an appropriate formula ot R ep r od 1. The area of a trapezoid is 40.25 sq. inches and with bases that measures 7 inches and 4.5 inches. Find the height of the trapezoid. d. Answer the question Sa d. Answer the question Geometry and Measurement (Student Pages) GEO6 – SP17 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 uc e Draw a picture, write an appropriate formula, and substitute to solve each problem. 2. The length of cereal box is 21 cm. The width 6.5 cm. The height is 6 cm more than the length of the box. Find the volume. e. Sketch the figure a. Sketch the figure f. Write an appropriate formula b. Write an appropriate formula c. Substitute and solve pl e: D o N g. Substitute and solve ot R ep r od 1. Find the area of a circle whose diameter is 24 cm. h. Answer the question d. Answer the question m Express each ratio as a fraction in simplest form. Sa 3. 12 : 36 _________________ 4. 18 out of 20 students are UCLA basketball fans. _______________ 5. 25 of the 35 students received an A grade on the last quiz. ________________ Geometry and Measurement (Student Pages) GEO6 – SP18 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Backyard Office od BED2 Dining Room Kitchen Kitchen Living Room Scale 2 cm = 9 ft ep r BATH1 BED1 uc e The scale drawing below of a floor plan was created using a 2 cm = 9 ft scale. Use the ratio strip to answer the questions. ot R 1. If the length of the living room on the scale drawing is 7 cm, what is the actual length of the living room? o N 2. Jo has an 8ft by 10ft area rug. Will this rug fit in the living room? Explain. pl e: D 3. What is the actual area of the dining room? Sa m 4. Red built a round dining table that has a diameter of 1 m. Will the dining table fit in the dining room? 2 cm 8 cm 4 cm 10 cm 2 cm = 9 ft 9 ft Geometry and Measurement (Student Pages) 27 ft 63 ft GEO6 – SP19 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Convert each measurement. A map (scale drawing) of a city shows 1 in = 0.6 miles. On the map, the distance from the library to the park is 4 inches. What is the actual distance from the library to the park? 2. A half marathon is 13.1 miles. Convert this distance to kilometers and to meters. (HINT: 1 kilometer 0.6 miles) ot R ep r od uc e 1. pl e: D o N 3. How many cups are in 2 gallons? (HINT: 4 cups = 1 quart, 4 quarts = 1 gallon) Determine the unit rate. 5. $25 in 2 hours 6. 6 cups of fruit for 4 milkshakes. 7. 7 books in 5 weeks Sa m 4. 60 kilometer in 6 hours Geometry and Measurement (Student Pages) GEO6 – SP20 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Use a straightedge and follow the directions to draw each figure. uc e 1. Equilateral triangle a. Draw a horizontal base of 8 cm and label it PR . od b. Find the midpoint and label it M. ep r c. Draw the perpendicular bisector to PR at M. d. Find point Q so that MQ has a length of 3 cm. f. What kind of triangle is ∆PQR? ot R e. What is the approximate length of PQ ? _____ cm RQ ? _____ cm pl e: D 2. Parallelogram o N P a. Draw a horizontal base of 6 cm. Label it GH . b. Find the midpoint of GH . Label it N. m c. Draw NK 3 cm long so that it is perpendicular to GH at N. Draw GK . Sa d. Complete parallelogram HGKL by drawing KL and HL . (Recall that opposite sides must be congruent and parallel.) e. What is the approximate length of GK ? _____ cm Geometry and Measurement (Student Pages) G GEO6 – SP21 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 5 2 1 7 3 8 2. 1 1 6 4 4. 1 3 - 4 - 8 5. 10 3 7 - 8 12 3. 6. 3 5 1 -2 4 8 N ot R 1 4 ep r od 1. uc e Compute. 9 20 8. pl e: D 7. o Write each fraction or mixed number as a decimal. 3 4 5 36 24 9. - 12. 7 2 3 6 Simplify each expression. 42 7 3 11. 72 28 62 Sa m 10. Geometry and Measurement (Student Pages) GEO6 – SP22 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review TEST PREPARATION (GEO6) 1. Which of the following shows CD ? C D. D C 2. Which of the following correctly labels this drawing? EF B. C. EF E B. CM A C. AB pl e: D UW TU B. TW TU C. TUV TVW D. TVU TVW B M D. 4. Refer to the figure. Which of the following statement is true? A. EF C o CA N 3. Which line segment is perpendicular to AB ? A. F D. EF ot R A. D C od C. B. D C ep r A. uc e Show your work on a separate sheet of paper and choose the best answer. CB T U U V W m 5. Two triangles are congruent if the corresponding sides are congruent to one another. Sa A. C. The statement is always true. The statement is sometimes true. Geometry and Measurement (Student Pages) B. The statement is never true. D. The statement is true ONLY if the angles are congruent. GEO6 – SP23 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review Sa m pl e: D o N ot R ep r od uc e This page is intentionally left blank for notes. Geometry and Measurement (Student Pages) GEO6 – SP24 Drawings and Constructions 6.3 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (GEO6) uc e Show your work on a separate sheet of paper and write your answers on this page. 6.1 Geometric Drawings od 1. Draw and label line segment BD. C D 6.2 Geometric Constructions N A ot R ep r 2. Find and label a point R on the circle so that ACR is an acute angle. T pl e: D o 3. Find and label the congruent angles of TUW . 4. Find and label the congruent sides of TUW . U U V W Sa m 5. What do you know about TVU and TVW ? Geometry and Measurement (Student Pages) GEO6 – SP25 Drawings and Constructions Home-School Connection (GEO6) uc e Here are some questions to review with your young mathematician. R Q T P N ot R M ep r 2. Draw a 2-cm line segment, PQ , that is perpendicular to MN . Label the other endpoint Q. ( MN ____ PQ ) od 1. Write two names for the ray that has Q as an endpoint. N Parent (or Guardian) signature ____________________________ Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g. straightedge, ruler, compass, protractor, drawing software). Draw quadrilaterals and triangles from given information about them (e.g. a quadrilateral having equal sides but no right angles, a right isosceles triangle). Identify and construct basic elements of geometric figures (e.g. altitudes, mid-points, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems pl e: D MG 5.2.1 o Selected California Mathematics Content Standards MG 6.2.3 MG 7.3.1 MG 7.3.4 m MR 7.1.2 MR 7.2.6 Sa MR 7.3.2 FIRST PRINTING Measurement and Geometry Unit (Student Pages) DO NOT DUPLICATE © 2009 GEO6 – SP26
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