Warm - Up
Segments line segment segment The ______ _________ or __________ AB consists of the endpoints A and B on the line AB that lie between A and B (inclusive). Notation: AB A B
Rays ray The ______ AB consists of the initial point A and all points on line AB that lie on the same side of A as B lies. Notation: AB A B Note: AB and BA do not refer to the same ray.
Notation Summary AB Line AB Segment AB AB AB Ray AB A B
Point C is between A and B. Note: a point is between two others if all three points are collinear and it is “between” the other two. C A B Point C is between A and B.
A pair of opposite rays form a line. If C is between A and B, then CA and CB are __________ ________. C A B A pair of opposite rays form a line.
Parallel Lines __________ _______ are coplanar lines that do not intersect. Parallel lines l m
Use the figure below. Name all segments that are parallel to AE. Name all segments that are skew to AE. Skew lines are lines that do not lie in the same plane. The four lines in the figure that do not lie in the same plane as AE are BC, CD, FG, and GH. Parallel segments lie in the same plane, and the lines that contain them do not intersect. The three segments in the figure above that are parallel to AE are BF, CG, and DH.
Coplanar coplanar Points that lie in the same plane are ______________. A B C M D A,B,C are coplanar A,B,C,D are not coplanar Are B,C,D coplanar? Yes. Just because the plane is not drawn does not mean it does not exist.
Skew Lines __________ _______ are noncoplanar lines that do not intersect. Skew lines a b M
Parallel Planes Parallel planes are planes that do not intersect. M N
Collinear Points that lie on the same line are ______________. Name as many sets of collinear points as you can.
Axioms and Postulates __________ or __________: a rule that is accepted without _______________ Axiom postulate proof
Postulate 1-1 Through any two points there is exactly one _____. line B Line t is the only line that pass through points A and B
Postulate 1-2 If two lines intersect, then they intersect in exactly one __________. point A E B D C
Postulate 1-3 If two planes intersect, then they intersect in exactly one ___________. line
Use the diagram below. What is the intersection of plane HGC and plane AED? As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED. The back and left faces of the cube intersect at HD. Planes HGC and AED intersect vertically at HD.
Shade the plane that contains X, Y, and Z. Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z.
Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. D, J, and H planes BCGF and CGHD HE 1 Sample: Planes AEHD and BFGC never intersect.
Equality vs. Congruence Equality means the same object Congruence means different objects that are equivalent in some way
Ruler Postulate The distance between 2 points equals the absolute value between the coordinates. The distance between C and D is Notation: CD
Notation Summary AB Line AB Segment AB AB AB Ray AB AB Length of segment AB A B
Congruence congruent Two segments are __________ if they have the same length. 3 A B 4 C D 3 E F
Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC. 5 6 AC = A B C 11
If AC = 60, find the value of x. B C
If AB = 25, find the value of x. Then find AN and NB. Use the Segment Addition Postulate to write an equation. AN + NB = AB Segment Addition Postulate (2x – 6) + (x + 7) = 25 Substitute. 3x + 1 = 25 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3. AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25.
If AC = 20, find the value of x. B C