State which postulate justifies each statement. 1. If D is in the interior of ABC, then m ABD + m DBC = m ABC. ANSWER Angle Addition Postulate 2. If M is between X and Y, then XM + MY = XY. ANSWER Segment Addition Postulate

3. Find the length of MN if N is between M and P, MP = 6x – 2, MN = 4x, and MP = 16. ANSWER 12 4. Determine whether the following conditional is true or false. If false, provide a counterexample. If true, identify the property of logic used. “If a shape is not a rectangle, then it is not a square.” Contrapositive: “If a shape is a square, then it is a rectangle.” True, logically equivalent.

Understand Geometric Relationships in Diagrams Target Understand Geometric Relationships in Diagrams You will… Use postulates involving points, lines, and planes.

Postulates Postulate 1 – Ruler Postulate Postulate 2 – Segment Addition Postulate Postulate 3 – Protractor Postulate Postulate 4 – Angle Addition Postulate About lines: Postulate 5 – Through any two points there exists exactly one line. Postulate 6 – A line contains at least two points. Postulate 7 – If two lines intersect, then their intersection is exactly one point.

Postulates About planes: Postulate 8 – Through any three noncollinear points there exists exactly one plane. Postulate 9 – A plane contains at least three noncollinear points. Postulate 10 – If two points lie in a plane, then the line containing them lies in the plane. Postulate 11 – If two planes intersect, then their intersection is a line.

Vocabulary line perpendicular to a plane – a line is perpendicular to a plane if and only if the line intersects the plane in a point and the line is perpendicular to every line in the plane that intersects it at that point.

EXAMPLE 1 Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. a. b. SOLUTION 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.

Identify postulates from a diagram EXAMPLE 2 Identify postulates from a diagram Use the diagram to write examples of Postulates 9 and 10. Postulate 9 – A plane contains at least three noncollinear points. Postulate 10 – If two points lie in a plane, then the line containing them lies in the plane. Postulate 9: Plane P contains at least three noncollinear points, A, B, and C. Postulate 10: Point A and point B lie in plane P, so line n containing A and B also lies in plane P.

GUIDED PRACTICE for Example 1 and 2 Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane P and plane Q is a line? 1. ANSWER Postulate 11

Line n passes through points A and B GUIDED PRACTICE for Example 1 and 2 Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. 2. Postulate 5 – Through any two points there exists exactly one line. Postulate 6 – A line contains at least two points. Postulate 7 – If two lines intersect, then their intersection is exactly one point. ANSWER Postulate 5 : Line n passes through points A and B Postulate 6 : Line n contains A and B Postulate 7 : Line m and n intersect at point A

EXAMPLE 3 Use given information to sketch a diagram Sketch a diagram showing TV intersecting PQ at point W, so that TW WV . SOLUTION STEP 1 Draw TV and label points T and V. STEP 2 Draw point W at the midpoint of TV Mark the congruent segments. STEP 3 Draw PQ through W.

EXAMPLE 4 Interpret a diagram in three dimensions Which of the following statements cannot be assumed from the diagram? A, B, and F are collinear. E, B, and D are collinear. AB plane S CD plane T AF intersects BC at point B. SOLUTION No drawn line connects E, B, and D, so you cannot assume they are collinear. With no right angle marked, you cannot assume CD plane T.

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

GUIDED PRACTICE for Examples 3 and 4 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. (Sample answer; there are four possible pairs.)

GUIDED PRACTICE for Examples 3 and 4 5. In the diagram for Example 4, can you assume plane S intersects plane T at BC ? ANSWER Yes. Postulate 11

GUIDED PRACTICE for Examples 3 and 4 6. Explain how you know that AB BC in Example 4. 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 ANSWER