Basics of Geometry Chapter 1

1.2 Points, Lines, and Planes Three undefined terms in Geometry: Point: No size, no shape, only LOCATION.  Named by a single Capital letter Line: No thickness, only DIRECTION.  Named by two points on the line. Plane: No thickness, flat surface, infinate in ALL DIRECTIONS.  Named by any three or more points on the plane.

Vocabulary Collinear points – Three or more points that lie on the same line. Coplanar points – Four or more points that lie on the same plane. Segment – Part of a line that consists of two endpoints and all points in between. Name by the two endpoints in any order.

More Vocabulary Ray – Part of a line that consists of one endpoint and all points in one direction. Named by the endpoint and a point on the ray (in that order) Opposite rays – two rays that share an endpoint and form a straight line.

Examples Describe what each of these symbols means: PQ QP

Examples True or False? Point A lies on line l. Point B lies on line l. A, B, and C are collinear. A, B, and C are coplanar. D B A E C l m

1.3 Segments and Their Measures The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the length of AB. AB AB AB = l x 2 – x 1 l

Segments and Their Lengths Another way to find the length of a segment is by using the distance formula:

Examples Given A(3, 2) and B(-2, -1), find AB.

Terminology Definitions: statements that are known facts (do not have to be proven true) Postulates (Axioms): Statements that are accepted as true (do not have to be proven) Theorems: Statements that MUST be proven true.

Segment Addition Postulate If B is between A and C, then AB + BC = AC. Likewise, if AB + BC = AC, then B is between A and C. ABC AC BCAB

Examples If AB = 5, AC = 13, and BD = 15, what is AD?

Examples Suppose M is between L and N, find x and find the lengths of the segments LM and MN. LM = 3x + 8 MN = 2x + 5 LN = 23

Examples Use the distance formula to decide whether J (3, -5) K (-1, 2) L (-5, -5)

1.4 Angles and Their Measures Angle – Consists of two different rays that have the same endpoint. Sides of the angle are the rays Vertex of the angle is the common endpoint. An angle that has sides AB and AC is denoted by BAC, CAB, or A, if point A is the vertex. C A B

Angle Addition Postulate If P is in the interior of RST, then

Angle Measures MEASURES are EQUAL ANGLES are CONGRUENT

Classifying Angles Acute: Right: Obtuse: Straight:

Examples What is the measure of ABC?

1.5 Segment and Angle Bisectors Midpoint of a segment is the point that divides, or bisects, the segment into 2 congruent segments. (A midpoint divides it in half.) Segment bisector is a segment, line, ray, or plane that intersects a segment at its midpoint. AB M C D

Midpoint Formula If A(x 1, y 1 ) and B(x 2, y 2 ) are points in a coordinate plane, then the midpoint of AB has coordinates:

Example If AB has endpoints at (-2, 3) and (5, -2), what is the midpoint?

Example If the midpoint of RP is M(2, 4), and one endpoint is R(-1, 7), what is the coordinate for P?

Angle Bisector An angle bisector is a ray that divides an angle into 2 congruent, adjacent angles. (The angle bisector divides the angle in half.) Adjacent angles: share a common vertex and side, but have no common interior points. C B D A

Example The ray RQ bisects PRS. The measures of the two congruent angles are (x + 40) and (3x – 20). Solve for x. R S Q P

1.6 Angle Pairs Two angles are vertical angles if their sides form two pairs of opposite rays. 1 & 3 2 & 4 Two adjacent angles are a linear pair if their noncommon sides are opposite rays. 5 &

Example Solve for x and y. Then find the angle measures.

Vocabulary Complementary Angles are two angles that the sum of their measures is 90. Supplementary Angles are two angles that the sum of their measures is 180.

Examples Are 5 and 6 a linear pair? Are 5 and 9 a linear pair? Are 5 and 8 a linear pair? Are 5 and 8 vertical angles? Are 5 and 7 vertical angles? Are 9 and 6 vertical angles?

Always, Sometimes, or Never If m 1 = 40, then m 2 = 140. If m 4 = 130, then m 2 = and 4 are congruent. m 2 + m 3 = m 1 + m 4 2 = 1 m 2 = 90 - m 3. ~