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Download Chapter 1 Honors Notes 1.2 Points, Lines, and Planes Objective: I
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Transcript
Chapter 1 Honors Notes 1.2 Points, Lines, and Planes Objective: I can understand basic terms and postulates of geometry. Term Description How to Name It A ______________ indicates a location and has no size. You can represent a point by a dot and name it by a capital letter such as A. A ______ is represented by a straight path that extends in two directions without end. A ____________________ is represented by a flat surface that extends without end. Diagram You can name a line by any two points on the line or by a single lower case letter that is written at the end of the line. You can name a plane by a capital letter or at least three points in the plane that are not all on the same line. ___________________________________________________ - points that lie on the same line __________________________________________ - points and lines that lie on the same plane Naming Points, Lines, And Planes 1.) What are two other ways to name line AD? 2.) What are two other ways to name Plane P? 3.) Name three collinear points. 4.) Name four coplanar points. Definition A ______________________ is part of a line that consists of two endpoints and all the points between. How To Name It Diagram You can name a segment by its two endpoints. A _________________ is part of a line that consists of one endpoint and all the points on one side of the endpoint. You can name a ray by its endpoint and another point on the ray. The order of points indicates the ray’s direction. (ORDER MATTERS!) _____________ are two rays You can name opposite rays that share the same endpoint by their common endpoint extend in opposite and any other point on each directions. ray. ___________________________________________ - a rule that is accepted without proving it Postulate 1-1: Through any two points there is exactly one distinct _______________________. Postulate 1-2: If two lines intersect, then they intersect in exactly one ____________________. Postulate 1-3: If two planes intersect, then they intersect in exactly ______________________. Postulate 1-4: Through any three noncollinear points there is exactly _____________________. Use the Diagram to Answer the Questions Below 5.) Name three segments. 6.) Name three rays. 7.) Name a pair of opposite rays. 8.) Name the intersection of Plane ABC and Plane BCD. 1.3 Measuring Segments/1.4 Measuring Angles Objective: I can find and compare length of segments and measures of angles. Postulate 1-5 Ruler Postulate: Every point on a line can be paired with a real number known as a coordinate. Postulate 1-6 Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then __________________________________________________________. 1.) If PR = 25, PQ = 2x + 1, and QR = 3x + 4, then find QR. 2.) Find LN if M is between L and N, LN = 8x + 2, LM = 2x + 4, and MN = 3x + 19. ___________________________________________________ - segments with the same length ________________________________ - point dividing a segment into two congruent segments ______________ - a segment, ray, line, or plane that passes through the midpoint of a segment 3.) AB bisects CD at E. Find the value of x. Naming and Classifying Angles _______________ _______________ ________________ __________________ ___________________________________________________ - angles with the same measure Postulate 1-8 Angle Addition Postulate: If B is in the interior of Angle AOC, then mAOB + mBOC = mAOC. Diagram: 4.) If the measure of angle ABD = 53, then find the value of x. 5.) AB and AC are opposite rays. If D is not on either line and the mBAD = 2x + 5 and mDAC = 8x, find mDAC. 1.5 Exploring Angle Pairs Objective: I can identify special angle pairs and use their relationships to find angle measures. Definition Example/Diagram _______________________ are two coplanar angles that share a side and a vertex. They also have no common interior points. __________________________________ are two angles whose sides form opposite rays. _____________________________________ are two angles whose sum is 90 degrees. _____________________________________ are two angles whose sum is 180 degrees. _____________________________________ are a pair of adjacent angles whose noncommon sides form opposite rays. ___________________________ is a ray that divides an angle into two congruent angles. Postulate 1-9 Linear Pair Postulate: If two angles form a linear pair, then _____________________________________________________________________________. Diagram: Use the Diagram Below. 1.) Name two pairs of vertical angles. 2.) Name two pairs of adjacent angles. 3.) Name a linear pair. 4.) If Angle ABD and angle CBD are complementary, then the measure of angle CBD = ___? 5.) QS bisects angle PQR. Find the value of x. 6.) Angle PQS and SQR form a linear pair. Find the measure of angle PQS. 1.7 Midpoint and Distance Formula Objective: I can find the midpoint of a segment and find the distance between two points in the coordinate plane. Description On the Number Line The coordinate of the midpoint is the average of the two coordinates of the endpoints. In the Coordinate Plane The coordinates of the midpoint are the average of the x-coordinates and the ycoordinates of the endpoints. Formula Diagram The coordinates of the 𝑎+𝑏 midpoint M of AB is 2 . Given AB where A (x1, y1) and B(x2, y2), the coordinates of the midpoint of AB are x x y y2 M 1 2 , 1 2 2 1.) Find the midpoint of A(2,5) and B(3,-15). 2.) The midpoint of AB is M(-2,1). One endpoint is B(-5,7). What are the coordinates of the other endpoint A? Distance Formula: The distance between two points A (x1, y1) and B(x2, y2) is d = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2. 3.) Find the distance between P(-6,3) and Q(5,7). Partitioning a Line Segment 4.) On a number line, A is at -2 and B is at 4. What is the location of C between A and B, 2 such that AC is 3 the length of AB? (draw the situation!) 5.) Points A(-2,-3) and B(8,2) are the endpoints of AB. What are the coordinates of point C 2 on AB such that AC is 5 the length of AB?