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Lesson 2-1 Inductive Reasoning and Conjecture
Lesson 2-2 Logic
Lesson 2-3 Conditional Statements
Lesson 2-4 Deductive Reasoning
Lesson 2-5 Postulates and Paragraph Proofs
Lesson 2-6 Algebraic Proof
Lesson 2-7 Proving Segment Relationships
Lesson 2-8 Proving Angle Relationships
Example 1 Points and Lines
Example 2 Use Postulates
OBJECTIVE: To identify and use basic postulates about
points, line, and planes (2.8.8E) (M8.D.1.1)
KEY CONCEPT: Five essential parts of a good proof:
1. State the theorem or conjecture to be proven.
2. List the given information
3. If possible, draw a diagram to illustrate given info.
4. State what is to be proved.
5. Develop a system of deductive reasoning
SNOW CRYSTALS Some snow crystals are shaped
like regular hexagons. How many lines must be
drawn to interconnect all vertices of a hexagonal
snow crystal?
Explore The snow crystal has six vertices since a regular
hexagon has six vertices.
Plan
Draw a diagram of a hexagon to illustrate the
solution.
Solve
Label the vertices of the hexagon A, B, C, D,
E, and F. Connect each point with every other
point. Then, count the number of segments.
Between every two points there is exactly one
segment. Be sure to include the sides of the
hexagon. For the six points, fifteen segments
can be drawn.
Examine In the figure,
are all segments
that connect the vertices of the snow crystal.
Answer: 15
ART Jodi is making a string art design. She has
positioned ten nails, similar to the vertices of a
decagon, onto a board. How many strings will she
need to interconnect all vertices of the design?
Answer: 45
Determine whether the following statement is
always, sometimes, or never true. Explain.
If plane T contains
plane T contains point G.
contains point G, then
Answer: Always; Postulate 2.5 states that if two points
lie in a plane, then the entire line containing
those points lies in the plane.
Determine whether the following statement is
always, sometimes, or never true. Explain.
For
, if X lies in plane Q and Y lies in plane R,
then plane Q intersects plane R.
Answer: Sometimes; planes Q and R can be parallel,
and
can intersect both planes.
Determine whether the following statement is
always, sometimes, or never true. Explain.
contains three noncollinear points.
Answer: Never; noncollinear points do not lie on the
same line by definition.
Determine whether each statement is always,
sometimes, or never true. Explain.
a. Plane A and plane B intersect in one point.
Answer: Never; Postulate 2.7 states that if two planes
intersect, then their intersection is a line.
b. Point N lies in plane X and point R lies in plane Z.
You can draw only one line that contains both points
N and R.
Answer: Always; Postulate 2.1 states that through any
two points, there is exactly one line.
Determine whether each statement is always,
sometimes, or never true. Explain.
c. Two planes will always intersect a line.
Answer: Sometimes; Postulate 2.7 states that if the two
planes intersect, then their intersection is a
line. It does not say what to expect if the
planes do not intersect.
Review
1. Determine the number of segments that
can be drawn connecting 4 noncollinear
points.
6
2. Determine the number of segments that
can be drawn connecting 6 noncollinear
points.
15
Line BD and Ray BR are in plane P, and W
is on Line BD. State the postulate or
definition that can be used to show each
statement is true
3. B, D, and W are collinear.
Definition of collinear.
4. E, B, and R are coplanar.
Through any 3 points not on the same line,
there is exactly one plane.
5. R and W are collinear.
Through any two points, there is exactly one
line.
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