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Section 2.4

Section 2.4. Use Postulates and Diagrams. Objective: Use postulates involving points, lines and planes. Postulates and Theorems. Postulates are statements in geometry that are so basic, they are assumed to be true without proof. Sometimes called axioms .

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Section 2.4

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  1. Section 2.4 Use Postulates and Diagrams • Objective: • Use postulates involving points, lines and planes

  2. 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.

  3. Postulates

  4. b. a. a. Postulate 7: If two lines intersect, then their intersection is exactly one point. b. Postulate 11: If two planes intersect, then their intersection is a line. EXAMPLE 1 Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. SOLUTION

  5. EXAMPLE 1 Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. P5: Through any two points there exists exactly one line. Postulate 10: If two points lie in a plane, then the line containing them lies in the plane

  6. EXAMPLE 2 Identify postulates from a diagram Use the diagram to write examples of Postulates 9 and 10. Postulate 9: Plane Pcontains at least three noncollinear points, A, B, and C. Postulate 10: Point Aand point Blie in plane P, so line ncontaining Aand Balso lies in plane P.

  7. EXAMPLE 2 Identify postulates from a diagram Use the diagram to write examples of Postulates 6 and 8 P8:Line l contains at least two points R and S Postulate 6: Through noncollinear points R, S, and W, there exists exactly one plane M

  8. 1. Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane Pand plane Qis a line? ANSWER Postulate 11If two planes intersect, then their intersection is a line. for Example 1 and 2 GUIDED PRACTICE

  9. 2. Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. ANSWER Postulate 5 : Linenpasses through pointsAandB LinencontainsAandB Postulate 6 : Postulate 7 : Linemand n intersect at pointA for Example 1 and 2 GUIDED PRACTICE

  10. Section 2.4 Use Postulates and Diagrams Interpreting a Diagram You can’t assume: You can Assume:

  11. Sketch a diagram showing TVintersecting PQat point W, so that TWWV. Draw TVand label points Tand V. Draw point Wat the midpoint of TVMark the congruent segments. Draw PQthrough W. EXAMPLE 3 Use given information to sketch a diagram SOLUTION STEP 1 STEP 2 STEP 3

  12. Section 2.4 Perpendicular Figures Perpendicular Figures: A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is a perpendicular to every line in the plane that intersects it at that point.

  13. Which of the following statements cannot be assumed from the diagram? ABplane S CDplane T AFintersects BCat point B. No drawn line connects E, B, and D, so you cannot assume they are collinear. With no right angle marked, you cannot assume CDplane T. EXAMPLE 4 Interpret a diagram in three dimensions A, B, and Fare collinear. E, B, and Dare collinear. SOLUTION

  14. 3. If the given information stated PWand QWare congruent, how would you indicate that in the diagram? ANSWER for Examples 3 and 4 GUIDED PRACTICE In Questions 3 and 4, refer back to Example 3.

  15. 4. Name a pair of supplementary angles in the diagram. Explain. ANSWER In the diagram the angle TWP and VWP form a linear pair So, angle TWP, VWP are a pair of supplementary angles. for Examples 3 and 4 GUIDED PRACTICE

  16. 5. In the diagram for Example 4, can you assume plane Sintersects plane Tat BC? ANSWER Yes for Examples 3 and 4 GUIDED PRACTICE

  17. 6. Explain how you know that ABBCin Example 4. ANSWER It is given that line AB is perpendicular to plane S, therefore line AB is perpendicular to every line in the plane that intersect it at point B. So, AB BC for Examples 3 and 4 GUIDED PRACTICE

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