Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON The telephone company is installing telephone lines for ten buildings. Each building is to be connected to each of the other buildings with one line. How many telephone lines are needed? 45

Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 (For help, go to Lesson 2-8.) Describe the number-line graph of each inequality. 1.a 32.a 0 3.a 54.a –2 > – < – < – > – Check Skills You’ll Need 9-1

Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 Solutions 1.The graph is a line that starts at 3 and extends to the right without end. 2.The graph is a line that starts at 0 and extends to the left without end. 3.The graph is a line that starts at 5 and extends to the left without end. 4.The graph is a line that starts at –2 and extends to the right without end. 9-1

Use the figure to name each of the following. H, I, J, and K Name a point with a capital letter. Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 a. four points b. four different segments c. five other names for KI d. five different rays, andOI, IO, OK, KO HOName a segment by its endpoints., HJ, KI, and OI IK Horizontal line KI has several names. The first letter names the endpoint.HO, OJ, KI, OK, and JH Quick Check 9-1

MP, OP, QT, ST MQ, NR, OS MN, NO, QR, RS You are looking directly down into a wooden crate. Name each of the following. Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 a. four segments that intersect PT b. three segments parallel to PT c. four segments skew to PT Quick Check 9-1

Draw two intersecting lines. Then draw a segment that is parallel to one of the intersecting lines. Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 Use the lines on notebook or graph paper. First draw two lines that intersect. Then draw a segment that is parallel to one of the lines. Quick Check 9-1

Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 Use the figure. Name each of the following. 1.four points 2.another name for EA 3.three different rays 4.three segments that are parallel to HG 5.four segments that are skew to CG 6.four segments that intersect AE A, B, C, D EQ AQ, AE, GR EF, DC, AB AB, AD, EF, EH 9-1

Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON The Jackson County Bird Sanctuary has three times as many owls as hawks. It has 40 hawks and owls in all. How many of each are in the sanctuary? 30 owls, 10 hawks

Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 (For help, go to Lesson 7-5.) Solve. 1.n + 45 = x = y = 2y a + 15 = a + 45 Check Skills You’ll Need 9-2

Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 Solutions 1. n + 45 = x = 90 n + 45 – 45 = 180 – 4575 – 75 + x = 90 – 75 n = 135 x = y = 2y a + 15 = a y – 2y = 2y – 2y + 902a – a + 15 = a – a + 45 y = 90 a + 15 = 45 a + 15 – 15 = 45 – 15 a =

Find the measure of 3 if m 4 = 110°. Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 Replace m 4 with 110°.m ° = 180° Solve for m 3.m ° – 110° = 180° – 110° m 3 = 70° 3 and 4 are supplementary. m 3 + m 4 = 180° Quick Check 9-2

1 3, 2 4, 5 7, , 6 3 In the diagram, p || q. Identify each of the following. Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 a. congruent corresponding angles b. congruent alternate interior angles Quick Check 9-2

Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON ° In the diagram, d e. 1.Find the m 5 if m 8 is 35°. 2.Name the congruent corresponding angles. 3.Name the congruent alternate interior angles. 1 5, 2 6, 4 8, ,

Classifying Polygons PRE-ALGEBRA LESSON Draw an example of each kind of angle and describe its properties. a. acute angle A b. right angle R c. obtuse angle O Check students’ drawings.

Classifying Polygons PRE-ALGEBRA LESSON 9-3 (For help, go to Lesson 9-2.) For the angle measures given, classify the angle as acute, right, or obtuse. 1.85°2.95°3.160° 4.90°5.36°6.127° Check Skills You’ll Need 9-3

Classifying Polygons PRE-ALGEBRA LESSON 9-3 Solutions 1.acute2.obtuse3.obtuse 4.right5.acute6.obtuse 9-3

Classify the triangle by its sides and angles. Classifying Polygons PRE-ALGEBRA LESSON 9-3 The triangle has no congruent sides and one obtuse angle. The triangle is a scalene obtuse triangle. Quick Check 9-3

Name the types of quadrilaterals that have at least one pair of parallel sides. Classifying Polygons PRE-ALGEBRA LESSON 9-3 All parallelograms and trapezoids have at least one pair of parallel sides. Parallelograms include rectangles, rhombuses, and squares. Quick Check 9-3

A contractor is framing the wooden deck shown below in the shape of a regular dodecagon (12 sides). Write a formula to find the perimeter of the deck. Evaluate the formula for a side length of 3 ft. Classifying Polygons PRE-ALGEBRA LESSON 9-3 To write a formula, let x = the length of each side. The perimeter of the regular dodecagon is x + x + x + x + x + x + x + x + x + x + x + x. Therefore a formula for the perimeter is P = 12x. P = 12xWrite the formula. = 12(3) Substitute 3 for x. = 36Simplify. For a side length of 3 ft, the perimeter is 36 ft. Quick Check 9-3

Classifying Polygons PRE-ALGEBRA LESSON 9-3 Name the following. 1.a type of triangle that has at least two congruent sides and one right angle 2.a type of quadrilateral that can have opposite sides parallel and no right angles 3.Write a formula for the perimeter of a regular heptagon (7 sides). Evaluate for a side of 12 in. P = 7x; 84 in. isosceles right triangle parallelogram, rhombus 9-3

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON Draw several different quadrilaterals. Connect the midpoints of the sides of each figure. Write a sentence explaining in what way the figures inside the quadrilaterals are alike. They are all parallelograms. Sample answer

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 (For help, go to Lesson 9-3.) Sketch each figure. 1.equilateral triangle2.rectangle3.pentagon 4.hexagon5.octagon Check Skills You’ll Need 9-4

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 Solutions

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 How many diagonals does a nonagon have? AH, AG, AF, AE, AD, and AC are some of the diagonals. One strategy for solving this problem is to draw a diagram and count the diagonals. A nonagon has nine sides. You can draw six diagonals from one vertex of a nonagon. 9-4

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 (continued) A nonagon has 27 diagonals. You can organize your results as you count the diagonals. Do not count a diagonal twice. (The diagonal from A to C is the same as the one from C to A.) Then find the sum of the numbers of diagonals. VertexNumber of Diagonals A B C D E F G H I Total27 Quick Check 9-4

Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 Solve. 1.How many diagonals does a quadrilateral have? 2.How many triangles can you form if you draw all the diagonals from one vertex of a pentagon? 3.How many triangles can you form if you draw all the diagonals of a rectangle? 8 triangles 2 diagonals 3 triangles 9-4

Congruence PRE-ALGEBRA LESSON Replace the question marks with the correct digits. a = b –. 4 2 = ???? ??? 9, 9, 7 6, 4, 9, 8

Congruence PRE-ALGEBRA LESSON 9-5 (For help, go to Lesson 6-3.) ABC ~ XYZ. For the given part of ABC, find the corresponding part of XYZ. 1. A2. C 3.AB4.CA Check Skills You’ll Need 9-5

Congruence PRE-ALGEBRA LESSON 9-5 Solutions 1. X2. Z 3.XY4.ZX 9-5

V X, TW, TUVWUX TV WXWU, TUXUXU, VU Congruence PRE-ALGEBRA LESSON 9-5 In the figure, TUV WUX. a. Name the corresponding congruent angles. b. Name the corresponding congruent sides. c. Find the length of WX. WX,TV, Since TV = 300 m, WX = 300 m.and Quick Check 9-5

Congruence PRE-ALGEBRA LESSON 9-5 List the congruent corresponding parts of each pair of triangles. Write a congruence statement for the triangles. ACBECDAngle ACEC Side CABCEDAngle ACBECD by ASA. a. 9-5

Congruence PRE-ALGEBRA LESSON 9-5 (continued) by SAS. MKJLJK MKLJSide MKJLJK Angle JK Side b. Quick Check 9-5

Congruence PRE-ALGEBRA LESSON 9-5 Given that JKL MNO, complete the following. 1. L2.JK3.JL 4.If two sides and the angle between those sides of one triangle are congruent to two sides and the angle between those sides of another triangle, why can you conclude that the two triangles are congruent? OMNMO SAS 9-5

Circles PRE-ALGEBRA LESSON Solve the proportion: = 4545 n n = 21

Circles PRE-ALGEBRA LESSON 9-6 (For help, go to Lesson 6-2.) Solve each proportion. Round to the nearest whole number where necessary. 1. =2. = 3. =4. = x x x x 360 Check Skills You’ll Need 9-6

Circles PRE-ALGEBRA LESSON 9-6 Solutions

Circles PRE-ALGEBRA LESSON 9-6 Find the circumference of the circle. C =dWrite the formula. = 37.68Simplify. C(3.14)(2)6Replace with 3.14 and d with (2)6. The circumference of the circle is about in. Quick Check 9-6

Circles PRE-ALGEBRA LESSON 9-6 Make a circle graph for Jackie’s weekly budget. Jackie’s Weekly Budget Entertainment (e) Food (f) Transportation (t) Savings (s) 20% 10% 50% Use proportions to find the measures of the central angles. = = e = 72° t = 36° t e = f = 72° = s = 180° s 360 f

Use a compass to draw a circle. Draw the central angles with a protractor. Circles PRE-ALGEBRA LESSON 9-6 (continued) Add a title. Jackie’s Weekly Budget Label each section. Savings Entertainment Food Transportation Quick Check 9-6

Circles PRE-ALGEBRA LESSON 9-6 Draw a circle graph of the data. First add to find the total number of students = 440 p 67° s 83° Use proportions to find the measures of the central angles. f 98° j 112° = j 360 = s 360 = p = f 360 Spring Dance Attendance Freshmen (f) Sophomores (p) Juniors (j) Seniors (s)

Circles PRE-ALGEBRA LESSON 9-6 (continued) Use a compass to draw a circle. Draw the central angles with a protractor. Seniors Freshmen Juniors Sophomores Label each section. Spring Dance Attendance Add a title. Quick Check 9-6

Circles PRE-ALGEBRA LESSON 9-6 Solve. 1.Find the circumference of a circle with a diameter of 2.5 in. 2.Ten out of 22 students surveyed prefer milk with their breakfast. Find the measure of the central angle to represent this data in a circle graph. 3.Draw a circle graph of the data. about 7.85 in. about 164° After-School Number of Activities Students (for one class) Band 5 Basketball 8 Baby-sitting 10 Library 7 9-6

Constructions PRE-ALGEBRA LESSON A rectangular field is three times as long as it is wide. What are its width and length if the perimeter is 600 yd? width: 75 yd; length: 225 yd

Constructions PRE-ALGEBRA LESSON 9-7 (For help, go to Lesson 9-1.) State the meaning of each symbol. 1.B2.AB3.AB4.AB Check Skills You’ll Need 9-7

Constructions PRE-ALGEBRA LESSON 9-7 Solutions 1.point B 2.a line segment with endpoints A and B 3.a ray with endpoint A and containing point B 4.a line containing points A and B 9-7

Constructions PRE-ALGEBRA LESSON 9-7 Construct a segment congruent to WX. Step 1 Draw a ray with endpoint G. Step 2 Open the compass to the length of WX. 9-7

Constructions PRE-ALGEBRA LESSON 9-7 Step 3 With the same compass setting, put the compass tip on G. Draw an arc that intersects the ray. Label the intersection H. (continued) GH WX Quick Check 9-7

Constructions PRE-ALGEBRA LESSON 9-7 Construct an angle congruent to W. Step 2 With the compass point at W, draw an arc that intersects the sides of W. Label the intersection points M and N. Step 1 Draw a ray with endpoint A. 9-7

Constructions PRE-ALGEBRA LESSON 9-7 (continued) Step 3With the same compass setting, put the compass tip on A. Draw an arc that intersects the ray at point B. Step 4 Open the compass to the length of MN. Using this setting, put the compass tip at B. Draw an arc to determine the point C. Draw AC. CAB NWM Quick Check 9-7

Constructions PRE-ALGEBRA LESSON 9-7 Construct the perpendicular bisector of WY. Step 1Open the compass to more than half the length of WY. Put the compass tip at W. Draw an arc intersecting WY. With the same compass setting, repeat from point Y. 9-7

PRE-ALGEBRA LESSON 9-7 Step 2Label the points of intersection S and T. Draw ST. Label the intersection of ST and WY point M. ST is perpendicular to WY and ST bisects WY. (continued) Constructions Quick Check 9-7

PRE-ALGEBRA LESSON 9-7 Construct the bisector of W. Step 1Put the compass tip at W. Draw an arc that intersects the sides of W. Label the points of intersection S and T. Constructions 9-7

PRE-ALGEBRA LESSON 9-7 (continued) WZ bisects W. Step 2 Put the compass tip at S. Draw an arc. With the same compass setting, repeat with the compass tip at T. Make sure the arcs intersect. Label the intersection of the arcs Z. Draw WZ. Constructions Quick Check 9-7

Constructions PRE-ALGEBRA LESSON 9-7 1–3. Check students’ work. Draw and construct the figures. 1.Draw QR; then construct ST QR. 2.Draw LMN; then construct UVW LMN. 3.Construct the bisectors of QR and LMN that you drew for Questions 1 and

Translations PRE-ALGEBRA LESSON The short sides of a kite measure 54 cm each, and the long sides each measure 78 cm. What is the perimeter of the kite? 264 cm

Translations PRE-ALGEBRA LESSON 9-8 (For help, go to Lesson 1-10.) Graph each point. 1.A(–4, 3)2.B(0, 2)3.C(1, 4) 4.D(4, –2)5.E(–2, –3) Check Skills You’ll Need 9-8

Translations PRE-ALGEBRA LESSON 9-8 Solutions 1–5. 9-8

Graph the image of BCD after a translation 3 units to the left and 4 units down. PRE-ALGEBRA LESSON 9-8 Translations Quick Check 9-8

Use arrow notation to describe the translation of X to X. PRE-ALGEBRA LESSON 9-8 Translations The point moves from X(–2, 3) to X (3, 1), so the translation is X(–2, 3) X (3, 1). Quick Check 9-8

Write a rule to describe the translation of RST to R S T. PRE-ALGEBRA LESSON 9-8 Use R(–2, –3) and its image R (–3, 2) to find the horizontal and vertical translations. Horizontal translation: –3 – (–2) = –1 Vertical translation: 2 – (–3) = 5 The rule is (x, y) (x – 1, y + 5). Translations Quick Check 9-8

Translations PRE-ALGEBRA LESSON 9-8 For Exercises 1–3, use a translation 2 units to the left and 4 units down. 1.Graph EFG with vertices E(1, 1), F(4, 4), and G(4, 1). Also graph its image, E F G. 2.Use arrow notation to describe the translation of E to E. 3.Write a rule to describe the translation of EFG to E F G. E(1, 1) E (–1, –3) (x, y) (x – 2, y – 4) 9-8

Symmetry and Reflections PRE-ALGEBRA LESSON A line graph of Dana’s weight in one month resembles a horizontal line. Describe the situation the graph reflects. Dana’s weight has stayed the same during the one-month period.

Symmetry and Reflections PRE-ALGEBRA LESSON 9-9 (For help, go to Lesson 8-3.) Graph each line. 1.x = 02.y = 03.x = 3 4.y = 25.x = –16.x = y Check Skills You’ll Need 9-9

Symmetry and Reflections PRE-ALGEBRA LESSON 9-9 Solutions

Identify the lines of symmetry. Tell how many there are. PRE-ALGEBRA LESSON 9-9 a. b. 2 lines of symmetry 8 lines of symmetry Symmetry and Reflections Quick Check 9-9

Graph the image of FG after a reflection over the x-axis. PRE-ALGEBRA LESSON 9-9 Reflect the other endpoint. Symmetry and Reflections Since F is 2 units below the x-axis, F is 2 units above the x-axis. Draw F G. Quick Check 9-9

Graph the image of FG after a reflection over y = –1. PRE-ALGEBRA LESSON 9-9 Graph y = –1 (in red). Reflect the other endpoint. Symmetry and Reflections Since F is 1 unit below the red line, F is 1 unit above the red line. Draw F G. Quick Check 9-9

Symmetry and Reflections PRE-ALGEBRA LESSON 9-9 W (3, 3), X (2, 0), Y (0, 2) 1 W (–3, –3), X (–2, 0), Y (0, –2) Use a graph of WXY with vertices W(–3, 3), X(–2, 0), and Y(0, 2). 1.Graph WXY. How many lines of symmetry does WXY have? 2.Give the vertices of W X Y, the image of WXY after a reflection over the x-axis. 3.Give the vertices of W X Y, the image of WXY after a reflection over the y-axis. 9-9

Rotations PRE-ALGEBRA LESSON A rectangular field is 120 yd long and 53 yd 1 ft wide. How much longer is the field than it is wide? 66 yd 2 ft

Rotations PRE-ALGEBRA LESSON 9-10 (For help, go to Lesson 1-10.) Graph each triangle. 1.A(1, 3), B(4, 1), C(3, –2)2.J(–2, 1), K(1, –3), L(1, 4) 3.X(4, 0), Y(0, 2), Z(–2, –3) Check Skills You’ll Need 9-10

Rotations PRE-ALGEBRA LESSON 9-10 Solutions

Find the vertices of the image of RST after a rotation of 90° about the origin. PRE-ALGEBRA LESSON 9-10 Step 1 Use a blank transparency sheet. Trace RST, the x-axis, and the y-axis. Then fix the tracing in place at the origin. Rotations 9-10

(continued) PRE-ALGEBRA LESSON 9-10 Rotations Step 2 Rotate the tracing 90° counterclockwise. Make sure the axes line up. Label the vertices R, S, and T. Connect the vertices of the rotated triangle. The vertices of the image are R (1, 1), S (4, 1), and T (4, 5). Quick Check 9-10

Judging from appearance, tell whether the star has rotational symmetry. If so, what is the angle of rotation? PRE-ALGEBRA LESSON 9-10 The star can match itself in 6 positions. The pattern repeats in 6 equal intervals. 360° ÷ 6 = 60° The figure has rotational symmetry. The angle of rotation is 60°. Rotations Quick Check 9-10

Rotations PRE-ALGEBRA LESSON 9-10 No; you cannot rotate the figure 180° or less so that its image matches the original figure. D (2, –4), F (3, –1), G (1, –2) yes; 45° 1. DFG has vertices D(–2, 4), F(–3, 1), and G(–1, 2). Find the vertices of the image D F G after a rotation of 180° about the origin. Judging from appearance, tell whether each figure has rotational symmetry. If so, what is the angle of rotation? If not, explain