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## Introduction to Geometry: Points, Lines, and Planes

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**Introduction to Geometry: Points, Lines, and Planes**PRE-ALGEBRA LESSON 9-1 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 9-1**<**< – – > > – – 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 3 2.a 0 3.a 5 4. 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.**Name a point with a capital letter. H, I, J, and K HO Name a segment by its endpoints. , HJ , KI , and OI , OK , and OI Horizontal line KI has several names. IK , KO , IO HO The first letter names the endpoint. , OJ , KI , OK , and JH Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 Quick Check a. four points b. four different segments c. five other names for KI d. five different rays 9-1**MP, OP, QT, ST**MQ, NR, OS MN, NO, QR, RS Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 You are looking directly down into a wooden crate. Name each of the following. a. four segments that intersectPT b. three segments parallel to PT c. four segments skew to PT Quick Check 9-1**First draw two lines that intersect.**Then draw a segment that is parallel to one of the lines. Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 Draw two intersecting lines. Then draw a segment that is parallel to one of the intersecting lines. Use the lines on notebook or graph paper. Quick Check 9-1**EQ**AQ, AE, GR EF, DC, AB AB, AD, EF, EH AB, AD, EF, EH 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 9-1**Angle Relationships and Parallel Lines**PRE-ALGEBRA LESSON 9-2 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 9-2**Angle Relationships and Parallel Lines**PRE-ALGEBRA LESSON 9-2 (For help, go to Lesson 7-5.) Solve. 1.n + 45 = 180 2. 75 + x = 90 3. 3y = 2y + 90 4. 2a + 15 = a + 45 Check Skills You’ll Need 9-2**Angle Relationships and Parallel Lines**PRE-ALGEBRA LESSON 9-2 Solutions 1.n + 45 = 180 2. 75 + x = 90 n + 45 – 45 = 180 – 45 75 – 75 + x = 90 – 75 n = 135 x = 15 3. 3y = 2y + 90 4. 2a + 15 = a + 45 3y – 2y = 2y – 2y + 90 2a – a + 15 = a – a + 45 y = 90 a + 15 = 45 a + 15 – 15 = 45 – 15 a = 30 9-2**m 3 + m 4 = 180°**3 and 4 are supplementary. m 3 + 110° = 180° Replace m 4 with 110°. m 3 + 110° – 110° = 180° – 110° Solve for m 3. m 3 = 70° Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 Find the measure of 3 if m 4 = 110°. Quick Check 9-2**1 3, 2 4, 5 7, 6 8**2 7, 6 3 Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 In the diagram, p || q. Identify each of the following. a. congruent corresponding angles b. congruent alternate interior angles Quick Check 9-2**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, 3 7 4 7, 2 5 Angle Relationships and Parallel Lines PRE-ALGEBRA LESSON 9-2 145° 9-2**Classifying Polygons**PRE-ALGEBRA LESSON 9-3 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. 9-3**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. acute 2. obtuse 3. obtuse 4. right 5. acute 6. obtuse 9-3**Classifying Polygons**PRE-ALGEBRA LESSON 9-3 Classify the triangle by its sides and angles. The triangle has no congruent sides and one obtuse angle. The triangle is a scalene obtuse triangle. Quick Check 9-3**Classifying Polygons**PRE-ALGEBRA LESSON 9-3 Name the types of quadrilaterals that have at least one pair of parallel sides. All parallelograms and trapezoids have at least one pair of parallel sides. Parallelograms include rectangles, rhombuses, and squares. Quick Check 9-3**Classifying Polygons**PRE-ALGEBRA LESSON 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. 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. = 36 Simplify. Quick Check For a side length of 3 ft, the perimeter is 36 ft. 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. isosceles right triangle parallelogram, rhombus P = 7x; 84 in. 9-3**Sample answer**They are all parallelograms. Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 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. 9-4**Problem Solving Strategy: Draw a Diagram**PRE-ALGEBRA LESSON 9-4 (For help, go to Lesson 9-3.) Sketch each figure. 1. equilateral triangle 2. rectangle 3. pentagon 4. hexagon 5. octagon Check Skills You’ll Need 9-4**Solutions**1.2.3. 4.5. Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 9-4**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. AH, AG, AF, AE, AD, and AC are some of the diagonals. Problem Solving Strategy: Draw a Diagram PRE-ALGEBRA LESSON 9-4 How many diagonals does a nonagon have? 9-4**Problem Solving Strategy: Draw a Diagram**PRE-ALGEBRA LESSON 9-4 (continued) 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. Vertex Number of Diagonals 6 A 6 B 5 C 4 D E 3 F 2 G 1 H 0 I 0 Total 27 A nonagon has 27 diagonals. 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? 2 diagonals 3 triangles 8 triangles 9-4**?**? ? ? ? ? ? Congruence PRE-ALGEBRA LESSON 9-5 Replace the question marks with the correct digits. a. 8 9 + 6. = 15.96 b. 13. 0 – . 4 2 = 4.122 9, 9, 7 6, 4, 9, 8 9-5**ABC ~ XYZ. For the given part of ABC, find the**corresponding part of XYZ. 1.A2.C 3.AB4. CA Congruence PRE-ALGEBRA LESSON 9-5 (For help, go to Lesson 6-3.) Check Skills You’ll Need 9-5**Solutions**1.X2.Z 3. XY4.ZX Congruence PRE-ALGEBRA LESSON 9-5 9-5**X**, T W , TUV WUX V WX , TU WU , VU XU TV c. Find the length of WX. WX, TV, and TV = 300 m, WX = 300 m. Since Congruence PRE-ALGEBRA LESSON 9-5 In the figure, TUV WUX. a. Name the corresponding congruent angles. b. Name the corresponding congruent sides. Quick Check 9-5**ACB**ECD Angle AC EC Side CAB CED Angle by ASA. ACB ECD Congruence PRE-ALGEBRA LESSON 9-5 List the congruent corresponding parts of each pair of triangles. Write a congruence statement for the triangles. a. 9-5**MK**LJ Side MKJ LJK Angle JK JK Side MKJ LJK by SAS. Congruence PRE-ALGEBRA LESSON 9-5 (continued) b. Quick Check 9-5**Given that JKLMNO, 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? O MN MO Congruence PRE-ALGEBRA LESSON 9-5 SAS 9-5**35**n = 21 45 n27 Circles PRE-ALGEBRA LESSON 9-6 Solve the proportion: = 9-6**0.8**5.3 1.6 5.3 10 100 x 360 75 100 x 360 x 360 x 360 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. = Check Skills You’ll Need 9-6**Circles**PRE-ALGEBRA LESSON 9-6 Solutions 1. 36 2. 270 3. 54 4. 109 9-6**C =**d Write the formula. C (3.14)(2)6 Replace with 3.14 and d with (2)6. = 37.68 Simplify. Circles PRE-ALGEBRA LESSON 9-6 Find the circumference of the circle. The circumference of the circle is about 37.68 in. Quick Check 9-6**Use proportions to find the measures**of the central angles. Jackie’s Weekly Budget Entertainment (e) 20% e 360 f 360 20 100 20 100 = = Food (f) 20% e = 72° f = 72° Transportation (t) 10% s 360 50 100 = = t 360 10 100 Savings (s) 50% t = 36° s = 180° Circles PRE-ALGEBRA LESSON 9-6 Make a circle graph for Jackie’s weekly budget. 9-6**Use a compass to draw a circle.**Draw the central angles with a protractor. Label each section. Add a title. Jackie’s Weekly Budget Entertainment Savings Food Transportation Circles PRE-ALGEBRA LESSON 9-6 (continued) Quick Check 9-6**First add to find the total number of students.**120 + 82 + 137 + 101 = 440 Spring Dance Attendance Use proportions to find the measures of the central angles. Freshmen (f) 120 Sophomores (p) 82 120 440 f 360 82 440 p 360 = = Juniors (j) 137 f 98° p 67° Seniors (s) 101 137 440 j 360 101 440 s 360 = = j 112° s 83° Circles PRE-ALGEBRA LESSON 9-6 Draw a circle graph of the data. 9-6**Draw the central angles with a protractor.**Label each section. Add a title. Spring Dance Attendance Seniors Freshmen Sophomores Juniors Circles PRE-ALGEBRA LESSON 9-6 (continued) Use a compass to draw a circle. Quick Check 9-6**After-School Number of Activities**Students(for one class) Band 5 Basketball 8 Baby-sitting 10 Library 7 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° 9-6**Constructions**PRE-ALGEBRA LESSON 9-7 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 9-7**State the meaning of each symbol.**1.B2.AB3.AB 4.AB Constructions PRE-ALGEBRA LESSON 9-7 (For help, go to Lesson 9-1.) 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**Step 1 Draw a ray with endpoint G.**Step 2 Open the compass to the length of WX. Constructions PRE-ALGEBRA LESSON 9-7 Construct a segment congruent to WX. 9-7**Constructions**PRE-ALGEBRA LESSON 9-7 (continued) Step 3 With the same compass setting, put the compass tip on G. Draw an arc that intersects the ray. Label the intersection H. GH WX Quick Check 9-7**Step 1 Draw a ray with**endpoint A. Step 2 With the compass point at W, draw an arc that intersects the sides of W. Label the intersection points M and N. Constructions PRE-ALGEBRA LESSON 9-7 Construct an angle congruent to W. 9-7**Step 3 With 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 Constructions PRE-ALGEBRA LESSON 9-7 (continued) Quick Check 9-7**Step 1 Open 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. Constructions PRE-ALGEBRA LESSON 9-7 Construct the perpendicular bisector of WY. 9-7