Introduction to Points, Lines, and Planes in Geometry
Exploring the fundamental concepts of points, lines, and planes in geometry, including definitions, examples, and postulates. Learn about collinear points, coplanar points, segments, rays, and key postulates in geometry.
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1-2: Points, Lines and Planes
Undefined Terms Term Description Name Diagram Point Indicates location One capital letter A Has no size Line Straight path Extends in opposite directions Has no thickness Infinitely many points Any two points: B AB BA or A One lowercase letter: One uppercase letter:P Three points not on same line: ABC Flat surface Extends without end P Plane A B Has no thickness C Infinitely many lines
Types of Points Collinear: Two or more points that lie on the same line A B C Coplanar: Three or more points that lie on the same plane Lines on the same plane are coplanar A C D B Space: Set of all points in three dimensions.
Example 1: B C E A D Q F AB a. What are two other ways name ? line or BA b. What are two ways to name plane Q? plane AEC plane ADC c. What are the names of three collinear points? points A, E, D points A, F, B d. What are the names of four coplanar points? points A, E, C points A, E, B a. RQ b. plane RVS c. N, Q, T Complete Got It? #1 p.12
Defined Terms Term Description Name Diagram Segment Part of a line Two endpoints: A B Two endpoints and all points between AB BA or Ray Part of a line One endpoint and all points on the line on one side of endpoint Endpoint and any point on the ray: AB A B C Shared endpoint and any other point on each ray: CA A Opposite Ray Two rays that share the same endpoint C Form a line B CB
Example 2: P N M a. What are the names of the segments in the figure? , MN NP MP , b. What are the names of the rays in the figure? , , MP NP MN PN NM PM , , , c. Which of the rays in part(b) are opposite rays? and NP NM Complete Got It? #2 p.13
Postulates (statements assumed to be true) Postulate 1-1 Through any two points there is exactly one line. A B Postulate 1-2 If two distinct lines intersect, then they intersect in exactly one point. D A G B C Postulate 1-3 If two distinct planes intersect, then they intersect in exactly one line.
Example 3: D C A B H G E F AG CG a. What is the intersection of and ? G AG b. What is the intersection of plane ABCD and ? A c. What is the intersection of plane ABGH and plane DCHG? HG a. plane BFE b. Two planes intersect in one line, so you need two common points to name the common line. Complete Got It? #3 p.14
Postulate Postulate 1-4 Through any three noncollinear points there is exactly one plane. A C D B
Example 4: Each surface of the box represents part of a plane D C A B H G E F a. Which plane contains points A, B and C? plane ABCD b. Which plane contains points E, H and C? plane EHCB
1-3: Measuring Segments
Ruler Postulate Every point on a line can be paired with a real number. This makes a ____________ ___________________ between the points on the line and the real numbers. The real number that corresponds to a point is called the ____________ of the point. one-to-one correspondence correspondence coordinate What s the distance between A and B? b = a = 3 1 = 3 1 2 A B
Distance Formula (on a number line) Take the absolute value of the difference of the coordinates of two points: AB = AB means the distance from A to B a b Example 1: What is CD? C D B E 2 10 = c d CD CD = 2 10 CD = 10 2 CD = CD = = = 8 8 8 8
Postulate 1-6: If three points A, B and C are collinear and B is between A and C, then: AB Segment Addition Postulate + = B C A C AC A B C BC AB Example 2: If LN=32, what are LM and MN? Complete Got It? #2 p.21 L M x+ x+ 3 8 N 2 4 8 2 + + + 12 5 = = LM = + = 3 4 32 32 20 x x JK = 3(4) 8 20 42 x+ 5 MN = + = KL = 2(4) 4 12 78 x = x = 4
Vocabulary Congruent Segments Two or more segments that have the same length Symbol Midpoint Point that divides a segment into two congruent segments. Segment Bisector Point, line, ray or other segment that intersects a segment at its midpoint. Divides segment into two equal length segments.
Example 3: is a segment bisector of . What are , and RS ST RT AC RT ? C What do you know? = RS ST x+ x R S 3 1 7 3 T How do you use what you know? = + 7 3 3 1 x x A = = 4 7(1) 3 RS = 4 + 7 3 x = x = 3 4 1 1 x x Substitute Solve = = + 4 3(1) 1 ST = 4 4 = + 8 RT
Homework: p. 24 #s 10, 12, 14, 16, 19, 20, 27-29, 39, 43, 48-52
