Name Period Date GEOMETRY AND MEASURESUREMENT Student Pages for Packet 6: Drawings and Constructions GEO6.1 Geometric Drawings Review geometric notation and vocabulary. Use a compass and a ruler to make geometric drawings. Learn about and understand congruence. 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. GEO6 STUDENT PAGES GEO6.3 Vocabulary, Skill Builders, and Review 15 1 8 Geometry and Measurement Unit (Student Pages) GEO6 SP
altitude WORD BANK (GEO6) Word Definition Example or Picture angle bisector central angle diagonal midpoint perpendicular bisector Measurement and Geometry Unit (Student Pages) GEO6 SP0
6.1 Geometric Drawings Ready (Summary) We will make geometric drawings using a straight edge and a compass. We will review geometric notation and vocabulary. GEOMETRIC DRAWINGS Go (Warmup) Set (Goals) Review geometric notation and vocabulary. Use a compass and a ruler to make geometric drawings. Learn about and understand congruence. Right angles are angles that measure exactly 90 o. These are right angles. These are acute angles. These are obtuse angles. 1. Describe acute angles in your own words. 2. Describe obtuse angles in your own words. GEO6 SP1
6.1 Geometric Drawings LABELING GEOMETRIC FIGURES Name each object. Use the words: angle, line, line segment, ray or point. Object Picture How to Label 1. A dot with a capital letter near it. P 2. 3. 4. 5. 6. Write two names for the ray that has N as an endpoint. E and 7. Write the two names for the segment that A G C has endpoints at M and L. and Two points named on the line with a double arrow line on top. Order does not matter. AB or BA The endpoint and another point on the ray named with a single arrow ray on top. Order does matter. DC only The endpoints named on the line segment with a no- arrow segment on top. Order does not matter. EF or FE 8. Write two names for the angle at x. and K B D F H M Three points. The middle point is always the vertex of the angle. GHK or KHG N x L J Q GEO6 SP2
6.1 Geometric Drawings LABELING GEOMETRIC FIGURES (continued) Name each object. Use the words: congruent triangles or triangle. 9. 10. Object Picture How to Label Three points in any order. Example: LMN Congruent and corresponding parts are named in order. Examples: RST UVW ST VW RST UVW Congruent sides are labeled with the same number of hash marks. Congruent angles are labeled with distinguishing marks as well. The two triangles on the right are congruent because they have the same size and same shape. 11. XYZ 12. ZYX T L R T R 13. CB 14. Draw the appropriate markings on all angles and all sides. S S M W U W U X N V V Y Z A B C GEO6 SP3
6.1 Geometric Drawings DRAWING AN ISOSCELES TRIANGLE 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 other endpoint B. 2. Find the point on AB that divides the segment into two equal or congruent parts. This point 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 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). How long is AD? cm. How long is BD? cm. 5. AD is to BD ( AD BD ). 6. What kinds of angles are AMD and BMD? 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 altitude congruent midpoint right angles A GEO6 SP4
6.1 Geometric Drawings DRAWING A CIRCLE 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?. 2. Draw a horizontal diameter through C. Label the endpoints F and H. Name both radii. and. 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 any circle.. 5. Find and label a point R on the circle. Draw FHR. What kind of angle is FHR?. 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. Name another central angle.. 7. Any segment that has both endpoints on a circle is called a chord. Name both chords in this diagram. and. C GEO6 SP5
6.1 Geometric Drawings DRAWING A TRAPEZOID 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. 3. From point Y, draw YZ to the left of point Y so that it is parallel to WX and is 2 cm. 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 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 this figure. Name this segment 7. Draw an altitude from point Z to WX at a point called V. What kinds of angles are WVZ and XVZ? Diagonal (a segment that is not the side of a polygon, but connects two vertices) Parallel (two lines in the same direction) W GEO6 SP6
6.1 Geometric Drawings YOU WRITE THE DIRECTIONS 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. Word Bank triangle square rectangle parallelogram segment height altitude midpoint congruent parallel perpendicular Your Drawing right angle obtuse angle A Drawing From Your Partner s Directions acute angle GEO6 SP7
6.2 Geometric Constructions GEOMETRIC CONSTRUCTIONS Ready (Summary) 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. Use the figure above to answer the questions. Go (Warmup) Set (Goals) Learn the SSS axiom for triangle congruence. Use a compass and straightedge to make classic Euclidean constructions. Justify constructions by completing proofs. 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 as a right angle. 4. x is degrees. It is also a. 5. y is degrees. A 60 o 6. From the diagram we see that two right angles form a. x y B GEO6 SP8
6.2 Geometric Constructions CREATING A TRIANGLE 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 Your partner s triangle The Side-Side-Side axiom states that if two triangles have corresponding sides that are congruent, then the triangles are congruent to one another. C A 2. Make hash marks on the triangles above to show corresponding, congruent sides. a. CA b. AT c. TC CAT DOG T D O 3. Mark corresponding, congruent angles on the triangles above. a. D b. O c. G G GEO6 SP9
6.2 Geometric Constructions SOME GEOMETRY FACTS Short Name Words Picture Symbols definition of congruent triangles SSS axiom reflexive property of congruence CPCTC radius facts Two triangles are congruent if their corresponding sides are congruent to one another and their corresponding angles are congruent to one another. Two triangles are congruent if their corresponding sides are congruent to one another. In geometry, a figure is congruent to itself. Congruent Parts of Congruent Triangles are Congruent A compass opening defines a length. All arcs drawn with the same compass opening have equal radius lengths. All radii of a given circle have the same length. U T V W GEO6 SP10
6.2 Geometric Constructions CONSTRUCTION 1: BISECT AN ANGLE V Use a straightedge and a compass to divide this angle into two congruent parts (bisect). Directions 1. Draw an arc from V that intersects both rays. Label the points of intersection A and B. 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. What do we know? VA VB, because all from a given circle have the. AX BX, because all drawn with the same compass opening have equal length. 3. Draw VX VX VX property Use the diagram you just constructed to prove that AVX BVX. Statement VA VB ; AX BX ; VX VX AVX BVX Given above axiom Reason AVX BVX GEO6 SP11
6.2 Geometric Constructions CONSTRUCTION 2: DRAW A PERPENDICULAR THROUGH A POINT ON A LINE Use a straightedge and a compass to draw a perpendicular to P. Directions 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. 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. What do we know? TP WP, because all from a given circle have the. TN WN, because all drawn with the same compass opening have equal length. 3. Draw PN. PN PN property Use the diagram you just constructed to prove that Statement TP WP ; TN WN ; PN PN TPN WPN NP TW. Given above axiom Reason TPN WPN TPN and WPN must both be right angles NP TW P 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
6.2 Geometric Constructions C CONSTRUCTION 3: COPY AN ANGLE Use a straightedge and a compass to copy C with the vertex at F and the given ray. Directions 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. 2. Using the same compass setting, draw an arc from F on the given ray. On the horizontal, label the point of intersection H. 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. What do we know? CG, because FH, because GD, because 4. Draw KF. KF, because Use the diagram you just constructed to prove that F. CG FH GD F Statement F Given above. Reason GEO6 SP13
6.2 Geometric Constructions CONSTRUCTION 4: DRAW A PERPENDICULAR SEGMENT FROM A POINT OFF OF A LINE Use a straightedge and a compass to draw a perpendicular from P to the line. Directions 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. 3. Draw PN, which is perpendicular to the line. Label the point of intersection X. P GEO6 SP14
6.3 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (GEO6) Fill in the crossword puzzle using the clues below. Across 1. Bisector of a segment 9 10 6 Down 2. AC of this parallelogram 5. A is 90 3. Forms right angles. 8. An angle that has the center of a circle as its vertex. 4. Line segment or ray in an angle that divides it into two congruent angles. 10. Less than 90 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. 7 1 2 5 8 12 13 9. Lines that never cross and are always the same distance apart. Word Bank proof perpendicular parallel right angle acute angle diagonal theorem congruent midpoint obtuse angle altitude central angle angle bisector height segment 11 14 4 3 A A D A B C B GEO6 SP15
6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. The height of a rectangle measures 62 cm and its length measures 53 cm. Find the area. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 2. The perimeter of a square ballroom is 49.6 cm. What is the area of the ballroom? a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question GEO6 SP16
6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 Draw a picture, write an appropriate formula, and substitute to solve each problem. 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. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 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 b. Write an appropriate formula c. Substitute and solve d. Answer the question GEO6 SP17
6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. Find the area of a circle whose diameter is 24 cm. e. Sketch the figure f. Write an appropriate formula g. Substitute and solve h. Answer the question Express each ratio as a fraction in simplest form. 3. 12 : 36 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. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question 4. 18 out of 20 students are UCLA basketball fans. 5. 25 of the 35 students received an A grade on the last quiz. GEO6 SP18
6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 The scale drawing below of a floor plan was created using a 2 cm = 9 ft scale. BATH1 BED1 Living Room Kitchen BED2 Office Use the ratio strip to answer the questions. Scale 2 cm = 9 ft 1. If the length of the living room on the scale drawing is 7 cm, what is the actual length of the living room? 2. Jo has an 8ft by 10ft area rug. Will this rug fit in the living room? Explain. 3. What is the actual area of the dining room? Backyard 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 4 cm 8 cm 10 cm 2 cm = 9 ft 9 ft 27 ft 63 ft Dining Room GEO6 SP19
6.3 Vocabulary, Skill Builders, and Review Convert each measurement. SKILL BUILDER 5 1. 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) 3. How many cups are in 2 gallons? (HINT: 4 cups = 1 quart, 4 quarts = 1 gallon) Determine the unit rate. 4. 60 kilometer in 6 hours 5. $25 in 2 hours 6. 6 cups of fruit for 4 milkshakes. 7. 7 books in 5 weeks GEO6 SP20
6.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Use a straightedge and follow the directions to draw each figure. 1. Equilateral triangle a. Draw a horizontal base of 8 cm and label it PR. b. Find the midpoint and label it M. c. Draw the perpendicular bisector to PR at M. d. Find point Q so that MQ has a length of 3 cm. e. What is the approximate length of PQ? cm RQ? cm f. What kind of triangle is PQR? 2. Parallelogram a. Draw a horizontal base of 6 cm. Label it GH. b. Find the midpoint of GH. Label it N. c. Draw NK 3 cm long so that it is perpendicular to GH at N. Draw GK. 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 P G GEO6 SP21
6.3 Vocabulary, Skill Builders, and Review Compute. 2 1 1. 5 7 3 8 4. 1 3 - - 4 8 SKILL BUILDER 7 1 1 2. 6 4 3. 5. 10 1 6. 4 Write each fraction or mixed number as a decimal. 7. 9 20 Simplify each expression. 10. 42 7 3 8. 11. 4 3 9. 5 72 28 6 2 3 7-8 12 3 5 1-2 4 8 36-24 12. 7 2 3 6 GEO6 SP22
6.3 Vocabulary, Skill Builders, and Review TEST PREPARATION (GEO6) Show your work on a separate sheet of paper and choose the best answer. 1. Which of the following shows CD? A. B. C D C D C. D. C D 2. Which of the following correctly labels this drawing? A. EF B. EF C. EF D. EF 3. Which line segment is perpendicular to AB? A. CA B. CM C. AB D. CB 4. Refer to the figure. Which of the following statement is true? A. UW TU B. TW TU C. TUV TVW D. TVU TVW 5. Two triangles are congruent if the corresponding sides are congruent to one another. A. The statement is always true. B. The statement is never true. C. The statement is sometimes true. C D. The statement is true ONLY if the angles are congruent. E A C F M U T V B W GEO6 SP23
6.3 Vocabulary, Skill Builders, and Review This page is intentionally left blank for notes. GEO6 SP24
6.3 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (GEO6) Show your work on a separate sheet of paper and write your answers on this page. 6.1 Geometric Drawings 1. Draw and label line segment BD. 2. Find and label a point R on the circle so that ACR is an acute angle. A C D 6.2 Geometric Constructions 3. Find and label the congruent angles of TUW. 4. Find and label the congruent sides of TUW. 5. What do you know about TVU and TVW? U T V W GEO6 SP25
Home-School Connection (GEO6) Here are some questions to review with your young mathematician. 1. Write two names for the ray that has Q as an endpoint. Q R T 2. Draw a 2-cm line segment, PQ, that is perpendicular to MN. Label the other endpoint Q. (MN PQ ) Parent (or Guardian) signature MG 5.2.1 MG 6.2.3 MG 7.3.1 MG 7.3.4 MR 7.1.2 MR 7.2.6 MR 7.3.2 M P N Selected California Mathematics Content Standards 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 FIRST PRINTING DO NOT DUPLICATE 2009 Measurement and Geometry Unit (Student Pages) GEO6 SP26