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Transcript
1.2 Points, Lines, & Planes 1 • Point – indicates a location and has no size. –Name it with a capital letter, such as A. • Line –a straight path that extends in opposite directions without end and has no thickness. –Contains infinitely many points. –Name a line by any two points on the line, or by a single lowercase letter. • Plane –a flat surface that extends without end and has no thickness. –Contains infinitely many lines. –Name a plane by capital letter or by at least three points in the plane that aren’t on same line. •Collinear points – points that lie on the same line. •Coplanar points – points that lie in the same plane. –All the points of a line are coplanar. Naming Points, Lines, and Planes 1) What are two other ways to name 𝑨𝑪? 2) What are two ways to name Plane S? 3) What are the names of three collinear points? 4) What are the names of four coplanar points? • Segment – part of line that consists of two endpoints and all points between them. –Name a segment by its two endpoints. • Ray – segment extended infinitely in one direction from single endpoint. –Name a ray by its endpoint and another point on the ray. • Opposite rays – two rays that share the same endpoint and form a line. –Can name opposite rays by their shared endpoint and any other point on each ray. Name Segments and Rays 5) What are the names of the segments in the figure? 6) What are the names of the rays in the figure? 7) Which of the rays in the previous question are opposite rays? 8) What is the intersection of plane RST and plane STW? • Postulate 1-4 • Through any three noncollinear points, there is exactly one plane. 9) The intersection of two lines is a . 10) The intersection of two planes is a . 11) Any three points can define a . 12) Any two points can define a . Warm Up Problems 1. Name the intersection of plane AEH and plane GHE. 2. What plane contains points B, F, and C? 3. What plane contains points E, F, and D? 1.3 Measuring Segments 11 • Postulate 1.5 – Ruler Postulate –Every point on a line can be paired with a real number. –The real number that corresponds to a point is called the coordinate of the point. –Allows you to measure lengths of segments and will allow you to find the distances between points. Measuring Segment Lengths 1) What is 𝐴𝐵? 2) What is 𝐴𝐶? 3) What is 𝐷𝐸? 4) What is 𝐵𝐷? Postulate 1.6 Segment Addition Postulate • If three points A, B, and C are collinear and B is between A and C, then 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶. Using the Segment Addition Postulate 5) If 𝐴𝐶 ,38 = what are 𝐴𝐵 and 𝐵𝐶? 6) Find 𝑅𝑇 and 𝑅𝑆. Congruent Segments If two segments have the same length, then the segments are congruent segments. Comparing Segment Lengths 7) 8) • The midpoint of a segment is a point that divides the segment into two congruent segments. • A point, line, ray, or other segment that intersects a segment at its midpoint is said to bisect the segment. –The point, line, ray, or segment is called a segment bisector. Using the Midpoint 9) In the figure, B is the midpoint of 𝑨𝑪. Find the length of 𝑨𝑪, 𝑨𝑩, & 𝑩𝑪. 1.7 Midpoint and Distance in the Coordinate Plane 20 Formulas for Midpoint Number Line Coordinate Plane Find the midpoint with the given endpoints. 1)-9 and 6 2) -7 and -14 Find the midpoint of 𝑨𝑩. 3) A (-1, 3) and B (-9, 2) 4) A (-4, -5) and B (-1, 1) 5) Given the midpoint of 𝑿𝒀 is (-3, 4) and point Y is located at (-12, 0), find point X. Distance Formula The distance between two points A(x1 , y1) and B(x1 , y1) is Find the distance between each pair of points. If necessary, round to the nearest hundredth. 6) G (3, 0) and Y (0, 12) 7) M (-3, 5) and S (2, -9) 8) T (4, -3) and W (11, -3) 1.4 Measuring Angles 27 An angle is formed by two rays with the same endpoint. The rays are the sides of the angle and the endpoint is the vertex of the angle. Ways to name an angle: • The vertex • A point on each ray & the vertex (the vertex must go in the middle) • A number 1) Name angle 1 and angle 2 in two other ways. The interior of an angle is the region containing all of the points between the two sides of the angle. The exterior of an angle is the region containing all of the points outside of an angle. Postulate 1.7 Protractor Postulate • allows you to find the measure of an angle. • The measure of the angle is the absolute value of the difference of the real numbers paired with the sides of the angle. Types of Angles 2) Given the figure, find the measure of angle APB, angle BPC, and angle APC. Congruent Angles Angles are congruent if they have the same angle measure. iff reads “if and only if” 3) Name the congruent angles in the figure. 4) If the measure of angle QPW is 55, what is the measure of angle PWV? Angle Addition Postulate 5) Assume angle ABC is a straight line, find the measure of angle ABD & DBC. 6) Given that angle ABD = 120, find the measure of angle ABC and angle CBD. 1.5 Exploring Angle Pairs 38 Adjacent angles – two coplanar angles with a common side, a common vertex, and no common interior points. Vertical angles – two angles whose sides are opposite rays; vertical angles are congruent to one another. Complementary angles – two angles whose measures have a sum of 90°; each angle is called the complement of the other. Supplementary angles – two angles whose measures have a sum of 180°; each angle is called the supplement of the other. 2) Find the value of x, angle CEA, and angle DEA. 1) Name a pair of adjacent, vertical, complementary, and supplementary angles in the given figure. Linear pair- a pair of adjacent angles whose noncommon sides are opposite rays; angles of a linear pair form a straight angle. Postulate 1.9 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. Angle bisector- a ray that divides an angle into two congruent angles; its endpoint is at the angle vertex. Within the ray, a segment with the same endpoint is also an angle bisector; the ray or segment bisects the angle. 3) Given the figure, find q and then find the measure of angle ABD, angle CBD. 4) BD bisects angle CBA. Find the measure of angle CBA