2.3 ostulates and iagrams ssential uestion In a diagram, what can be assumed and what needs to be labeled? Looking at a iagram Work with a partner. On a piece of paper, draw two perpendicular lines. Label them and. Look at the diagram from different angles. o the lines appear perpendicular regardless of the angle at which you look at them? escribe all the angles at which you can look at the lines and have them appear perpendicular. NING O RISION o be proficient in math, you need to state the meanings of the symbols you choose. view from above Interpreting a iagram Work with a partner. When you draw a diagram, you are communicating with others. It is important that you include sufficient information in the diagram. Use the diagram to determine which of the following statements you can assume to be true. xplain your reasoning. a. ll the points shown are coplanar. b. oints, G, and I are collinear. c. oints,, and H are collinear. d. G and H are perpendicular. e. and are a linear pair. f. and are perpendicular. h. and are coplanar. j. and intersect. view from upper right g. G and are parallel. i. G and do not intersect. k. G and are perpendicular. l. and are vertical angles. m. and H are the same line. ommunicate Your nswer 3. In a diagram, what can be assumed and what needs to be labeled? 4. Use the diagram in xploration 2 to write two statements you can assume to be true and two statements you cannot assume to be true. Your statements should be different from those given in xploration 2. xplain your reasoning. G H I Section 2.3 ostulates and iagrams 83
2.3 Lesson ore Vocabulary line perpendicular to a plane, p. 86 revious postulate point line plane What You Will Learn Identify postulates using diagrams. Sketch and interpret diagrams. Identifying ostulates Here are seven more postulates involving points, lines, and planes. ostulates oint, Line, and lane ostulates ostulate 2.1 wo oint ostulate hrough any two points, there exists exactly one line. 2.2 Line-oint ostulate line contains at least two points. xample hrough points and, there is exactly one line. Line contains at least two points. 2.3 Line Intersection ostulate If two lines intersect, then their intersection is exactly one point. m n he intersection of line m and line n is point. 2.4 hree oint ostulate hrough any three noncollinear points, there exists exactly one plane. 2.5 lane-oint ostulate plane contains at least three noncollinear points. R hrough points,, and, there is exactly one plane, plane R. lane R contains at least three noncollinear points. 2.6 lane-line ostulate If two points lie in a plane, then the line containing them lies in the plane. R oints and lie in plane R, so lies in plane R. 2.7 lane Intersection ostulate If two planes intersect, then their intersection is a line. S he intersection of plane S and plane is line. 84 hapter 2 Reasoning and roofs
Identifying a ostulate Using a iagram State the postulate illustrated by the diagram. a. If then b. If then SOLUION a. Line Intersection ostulate If two lines intersect, then their intersection is exactly one point. b. lane Intersection ostulate If two planes intersect, then their intersection is a line. Identifying ostulates from a iagram Use the diagram to write examples of the lane-oint ostulate and the lane-line ostulate. m n SOLUION lane-oint ostulate lane contains at least three noncollinear points,,, and. lane-line ostulate oint and point lie in plane. So, line n containing points and also lies in plane. Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. Use the diagram in xample 2. Which postulate allows you to say that the intersection of plane and plane is a line? 2. Use the diagram in xample 2 to write an example of the postulate. a. wo oint ostulate b. Line-oint ostulate c. Line Intersection ostulate Section 2.3 ostulates and iagrams 85
Sketching and Interpreting iagrams Sketching a iagram Sketch a diagram showing V intersecting at point W, so that W WV. NOHR WY In xample 3, there are many ways you can sketch the diagram. nother way is shown below. W V SOLUION Step 1 raw V and label points and V. Step 2 raw point W at the midpoint of V. Mark the congruent segments. Step 3 raw through W. t q p line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. In a diagram, a line perpendicular to a plane must be marked with a right angle symbol, as shown. W V Interpreting a iagram Which of the following statements cannot be assumed from the diagram? oints,, and are collinear. oints,, and are collinear. plane S plane S intersects at point. SOLUION No drawn line connects points,, and. So, you cannot assume they are collinear. With no right angle marked, you cannot assume plane. Monitoring rogress Refer back to xample 3. Help in nglish and Spanish at igideasmath.com 3. If the given information states that W and W are congruent, how can you indicate that in the diagram? 4. Name a pair of supplementary angles in the diagram. xplain. Use the diagram in xample 4. 5. an you assume that plane S intersects plane at? 6. xplain how you know that. 86 hapter 2 Reasoning and roofs
2.3 xercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. OML H SNN hrough any noncollinear points, there exists exactly one plane. 2. WRIING xplain why you need at least three noncollinear points to determine a plane. Monitoring rogress and Modeling with Mathematics In xercises 3 and 4, state the postulate illustrated by the diagram. (See xample 1.) In xercises 13 20, use the diagram to determine whether you can assume the statement. (See xample 4.) 3. 4. If then If then W R N J K L M X In xercises 5 8, use the diagram to write an example of the postulate. (See xample 2.) J p K H G 5. Line-oint ostulate (ostulate 2.2) 6. Line Intersection ostulate (ostulate 2.3) 7. hree oint ostulate (ostulate 2.4) q L M 13. lanes W and X intersect at KL. 14. oints K, L, M, and N are coplanar. 15. oints, J, and M are collinear. 16. MN and R intersect. 17. JK lies in plane X. 19. NKL and JKM are vertical angles. 18. LK is a right angle. 20. NKJ and JKM are supplementary angles. 8. lane-line ostulate (ostulate 2.6) In xercises 9 12, sketch a diagram of the description. (See xample 3.) 9. plane and line m intersecting plane at a 90 angle 10. XY in plane, XY bisected by point, and point not on XY 11. XY intersecting WV at point, so that X = V 12.,, and are all in plane, and point X is the midpoint of all three segments. RROR NLYSIS In xercises 21 and 22, describe and correct the error in the statement made about the diagram. 21. M is the midpoint of and. 22. intersects at a 90 angle, so. M Section 2.3 ostulates and iagrams 87
23. NING O RISION Select all the statements about the diagram that you cannot conclude. S H,, and are coplanar. lane intersects plane S in. intersects. H,, and are coplanar. lane plane S. oint bisects H. G H H and H are a linear pair.. 24. HOW O YOU S I? Use the diagram of line m and point. Make a conjecture about how many planes can be drawn so that line m and point lie in the same plane. Use postulates to justify your conjecture. m 25. MHMIL ONNIONS One way to graph a linear equation is to plot two points whose coordinates satisfy the equation and then connect them with a line. Which postulate guarantees this process works for any linear equation? 26. MHMIL ONNIONS way to solve a system of two linear equations that intersect is to graph the lines and find the coordinates of their intersection. Which postulate guarantees this process works for any two linear equations? In xercises 27 and 28, (a) rewrite the postulate in if-then form. hen (b) write the converse, inverse, and contrapositive and state which ones are true. 27. wo oint ostulate (ostulate 2.1) 28. lane-oint ostulate (ostulate 2.5) 29. RSONING hoose the correct symbol to go between the statements. number of points to number of points to determine a line determine a plane < = > 30. RIIL HINKING If two lines intersect, then they intersect in exactly one point by the Line Intersection ostulate (ostulate 2.3). o the two lines have to be in the same plane? raw a picture to support your answer. hen explain your reasoning. 31. MKING N RGUMN Your friend claims that even though two planes intersect in a line, it is possible for three planes to intersect in a point. Is your friend correct? xplain your reasoning. 32. MKING N RGUMN Your friend claims that by the lane Intersection ostulate (ost. 2.7), any two planes intersect in a line. Is your friend s interpretation of the lane Intersection ostulate (ost. 2.7) correct? xplain your reasoning. 33. SR RSONING oints,, and G all lie in plane and in plane. What must be true about points,, and G so that planes and are different planes? What must be true about points,, and G to force planes and to be the same plane? Make sketches to support your answers. 34. HOUGH ROVOKING he postulates in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. plane is the surface of the sphere. ind a postulate on page 84 that is not true in spherical geometry. xplain your reasoning. Maintaining Mathematical roficiency Solve the equation. ell which algebraic property of equality you used. (Skills Review Handbook) 35. t 6 = 4 36. 3x = 21 37. 9 + x = 13 38. Reviewing what you learned in previous grades and lessons x 7 = 5 88 hapter 2 Reasoning and roofs