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Learn postulates about lines and planes, interpret geometric diagrams, and find angle measurements using special angles. Understand basic geometric principles and logical reasoning in geometry.
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2.4 Use Postulates & Diagrams Objectives: • To illustrate and understand postulates about lines and planes • To accurately interpret geometric diagrams • To use properties of special pairs of angles to find angle measurements
Example 1 What is the length of ?
Example 1 You basically used the Segment Addition Postulate to get the length of the segment, where SA + AM = SM.
Postulates As you build a deductive system like geometry, you demonstrate that certain statements are logical consequences of other previously accepted or proven statements.
Postulates This chain of logical reasoning must begin somewhere, so every deductive system must contain some statements that are never proved. In geometry, these are called postulates.
Postulates and Theorems • Postulates are statements in geometry that are so basic, they are assumed to be true without proof. • Sometimes called axioms. • Theorems are statements that were once conjectures but have since been proven to be true based on postulates, definitions, properties, or previously proven conjectures. Both postulates and theorems are ordinarily written in conditional form. (If-Then statement)
Postulates Are Easy! For example, under the second strip write: Through any two points there exists exactly one line.
Example 1 State the postulate illustrated by the diagram. 2 lines intersect at exactly one point 2 planes intersect at a line
Example 2 How does the diagram shown illustrate one or more postulates? Answer in your notebook
Interpreting Diagrams When you interpret a diagram, you can assume only information about size or measure if it is marked. Don’t make assumptions
Example 3 Sketch and carefully label a diagram with plane A containing noncollinear points R, O, and W, and plane B containing noncollinear points N, W, and R. Do this in your notebook
Perpendicular Figures A line is perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point.
Example 4 Which of the following cannot be assumed from the diagram? • A, B, and F are collinear. • E, B, and D are collinear. • AB plane S 2
Example 4 Which of the following cannot be assumed from the diagram? • CD plane T • AF intersects BC at point B. 4
Example 5a • Identify all linear pairs of angles. • Identify all pairs of vertical angles. Answer in notebook
Example 5b • If m<1 = 40°, find the measures of the other angles in the diagram. Answer in you notebook
Linear Pair Postulate If two angles form a linear pair, then they are supplementary. Do we have to prove this?
Vertical Angle Congruence Theorem Vertical angles are congruent.
Example 6 Find the missing measure of each angle. A D B C <A=65, <B=55, <C=60, <D=55
Example 7 Find the value of x and y. x=70, y=12
Example 8 Find the value(s) of x. x=1/2, 7 ½ does not create congruent angles, therefore the only reasonable answer is 7
Example 9: SAT For the two intersecting lines, which of the following must be true? • a > c • a = 2b • a + 60 = b + c lll
Example 10: SAT In the figure, what is the value of y? 20
