# Points, Lines and Planes

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Points, Lines and Planes

Prerequisite Skills VOCABULARY CHECK Copy and complete the statement. 1. The distance around a rectangle is called its ? , and the distance around a circle is called its ? . ANSWER perimeter, circumference 2. The number of square units covered by a figure is called its ? . ANSWER area

Prerequisite Skills SKILLS AND ALGEBRA CHECK Evaluate the expression. 3. |4 – 6| ANSWER 2 4. |3 – 11| ANSWER 8 5. |–4 + 5| ANSWER 1 6. |–8 – 10| ANSWER 18

Prerequisite Skills SKILLS AND ALGEBRA CHECK Evaluate the expression when x = 2. 7. 5x ANSWER 10 8. 20 – 8x ANSWER 4 9. –18 + 3x –12 ANSWER 10. –5x – 4 + 2x –10 ANSWER

Prerequisite Skills SKILLS AND ALGEBRA CHECK Solve the equation. 11. 274 = –2z –137 ANSWER 12. 8x + 12 = 60 ANSWER 6 13. 2y – 5 + 7y = –32 ANSWER –3

Prerequisite Skills SKILLS AND ALGEBRA CHECK Solve the equation. 14. 6p p = –7 –2 ANSWER 15. 8m – 5 = 25 – 2m ANSWER 3 16. –2n + 18 = 5n – 24 ANSWER 6

EXAMPLE 1 Name points, lines, and planes a. Give two other names for PQ and for plane R. b. Name three points that are collinear. Name four points that are coplanar. SOLUTION Other names for PQ are QP and line n Other names for plane R are plane SVT and plane PTV.

EXAMPLE 1 Name points, lines, and planes b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T,and V lie in the same plane, so they are coplanar.

GUIDED PRACTICE for Example 1 Use the diagram in Example 1. Give two other names for ST . Name a point that is not coplanar with points Q, S, and T. ANSWER Other names for ST are TS and PT. Point V is not coplanar with points Q,S and T

EXAMPLE 2 Name segments, rays, and opposite rays a. Give another name for GH . b. Name all rays with endpoint J Which of these rays are opposite rays? SOLUTION a. Another name for GH is HG . b. The rays with endpoint J are JE , JG , JF , and JH The pairs of opposite rays with endpoint J are JE and JF , and JG and JH .

GUIDED PRACTICE for Example 2 2. Give another name for EF ANSWER Another name for EF is FE 3. Are HJ and JH the same ray ? Are HJ and HG the same ray? Explain. ANSWER No; HJ and JH have different endpoints, so they are not the same ray Yes; points J and G lie on the same side of H, so they are the same ray

EXAMPLE 3 Sketch intersections of lines and planes a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point. SOLUTION a. b. c.

EXAMPLE 4 Sketch intersections of planes Sketch two planes that intersect in a line. Draw: a vertical plane. Shade the plane. STEP 1 SOLUTION Draw: a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. STEP 2 Draw: the line of intersection. STEP 3

GUIDED PRACTICE for Examples 3 and 4 Sketch two different lines that intersect a plane at the same point. ANSWER Use the diagram at the right. 5. Name the intersection of PQ and line k. ANSWER Point M

GUIDED PRACTICE for Examples 3 and 4 6. Name the intersection of plane A and plane B. ANSWER Line K Name the intersection of line k and plane A. ANSWER Line K