2.4 Use Postulates & Diagrams
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2.4 Use Postulates & Diagrams Objectives: 1. To illustrate and understand postulates about lines and planes 2. To use properties of special pairs of angles to find angle measurements Assignment: • P. 99-102: 1-23 odd, 29, 45, 46, 47 • P. 128-131: 8-16 even, 28, 49, 50 • Challenge Problems Objective 1 You will be able to illustrate and understand postulates about lines and planes Exercise 1 What is the length of SM ? S A 6 cm 20 cm M Exercise 1 You basically used the Segment Addition Postulate to get the length of the segment, where ππ΄ + π΄π = ππ. S A 26 cm M 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. Eight Window Foldable 1. Fold blank piece of paper in half length-wise. 2. While it’s still folded, fold in half in the opposite direction. Eight Window Foldable 3. Now in the same direction, fold the paper in half two more times. 4. Unfold the paper and cut along the fold lines on the right side of the paper to create eight windows. Postulates Are Easy! On the first strip, write “Postulate.” Under that strip write the definition of postulate. Postulates Are Easy! The other seven windows are for specific postulates. The outside strip should have a picture that illustrates the postulate that appears under the strip. Postulates Are Easy! For example, under the second strip write: Through any two points there exists exactly one line. On the front of the second strip, draw an illustration: B A B A Repeat Times Seven! Exercise 2 State the postulate illustrated by the diagram. Exercise 3 How does the diagram shown illustrate one or more postulates? Interpreting Diagrams When you interpret a diagram, you can assume only information about size or measure if it is marked. Interpreting Diagrams Interpreting Diagrams Exercise 4 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. 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. Exercise 5 Which of the following cannot be assumed from the diagram? 1. A, B, and F are collinear. 2. E, B, and D are collinear. 3. π΄π΅ ⊥ plane π Exercise 5 Which of the following cannot be assumed from the diagram? 4. πΆπ· ⊥ plane π 5. π΄πΉ intersects π΅πΆ at point π΅. Objective 2 You will be able to use properties of special pairs of angles to find angle measurements Exercise 6a 1. Identify all linear pairs of angles. 2. Identify all pairs of vertical angles. 2 3 1 4 Exercise 6b 3. If π∠1 = 40°, find the measures of the other angles in the diagram. 2 3 1 4 Click me! 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. Exercise 7 Find the missing measure of each angle. 60ο° 65ο° Exercise 8 Find the value of x and y. ο¨3y - 1ο©ο° ο¨2x +5ο©ο° 35ο° Exercise 9 Find the value(s) of x. Exercise 10: SAT For the two intersecting lines, which of the following must be true? I. a > c II. a = 2b III. a + 60 = b + c aο° 60ο° bο° cο° Exercise 11: SAT In the figure, what is the value of y? xο° 3xο° yο° 2xο° 2.4 Use Postulates & Diagrams Objectives: 1. To illustrate and understand postulates about lines and planes 2. To use properties of special pairs of angles to find angle measurements Assignment: • P. 99-102: 1-23 odd, 29, 45, 46, 47 • P. 128-131: 8-16 even, 28, 49, 50 • Challenge Problems
