1. 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

2. 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.

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

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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

10. 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

11. 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.

12. 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.

13. 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

14. 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.)

15. 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

16. 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