chapter 2-4 & 2-5 reasoning and proofs. lesson 5 mi/vocab postulate axiom theorem proof...
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Chapter 2-4 & 2-5Reasoning and Proofs
• postulate
• axiom
• theorem
• proof
• paragraph proof
• informal proof
• Identify and use basic postulates about points, lines, and planes.
• Write paragraph proofs.
Standard 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. (Key)
Standard 2.0 Students write geometric proofs, including proofs by contradiction. (Key)
Standard 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. (Key)
Determine Valid Conclusions
A. The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.
Answer: Since the conditional is true and the hypothesis is true, the conclusion is valid.
Given:
Conclusion:
The hypothesis states that
Determine Valid Conclusions
B. The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.
Answer: According to the hypothesis for the conditional, you must have two pairs of congruent segments. The given only has one pair of congruent segments. Therefore, the conclusion is not valid.
Given:
Conclusion:
The hypothesis states that
1. A.
2. B.
3. C.
0% 0%0%
A. The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a polygon is a convex quadrilateral, then the sum of the interior angles is 360.
Given: mX + mN + mO = 360Conclusion: If you connect X, N, and O with segments, the figure will be a convex quadrilateral.
A. valid
B. not valid
C. cannot be determined
1. A.
2. B.
3. C.
0% 0%0%
B. The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a polygon is a convex quadrilateral, then the sum of the interior angles is 360.
Given: ABCD is a convex quadrilateral.Conclusion: The sum of the interior angles of ABCD is 360.
A. valid
B. not valid
C. cannot be determined
Law of Syllogism (the name is not important)
•If you do your homework, then you will do well on the tests.
•If you do well on the tests, then you will pass the class.
Conclusion:
If you do your homework, then you will pass the class.
A. PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.
(1) If Salline attends the prom, she will go with Mark.(2) If Salline goes with Mark, Donna will go with Albert.
Answer: If Salline attends the prom, Donna will go with Albert.
Determine Valid Conclusions From Two Conditionals
B. PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.
(1) If Mel and his date eat at the Peddler Steakhouse before going to the prom, they will miss the
senior march. (2) The Peddler Steakhouse stays open until 10 P.M.
Answer: There is no valid conclusion. While both statements may be true, the conclusion of each statement is not used as the hypothesis of the other.
Determine Valid Conclusions From Two Conditionals
1. A.
2. B.
3. C.
0% 0%0%
A. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.
(1) If you ride a bus, then you attend school.(2) If you ride a bus, then you go to work.
A. valid
B. not valid
C. cannot be determined
1. A.
2. B.
3. C.
0% 0%0%
B. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.
(1) If your alarm clock goes off in the morning, then you will get out of bed. (2) If you ride a bus, then you go to work.
A. valid
B. not valid
C. cannot be determined
Attention:
You do not have to write down or memorize these postulates.
You will need to refer to them on tonight’s assignment.
They are on page 105 in the book.
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.
Points and Lines
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.
Points and Lines
A. A
B. B
C. C
D. D0% 0%0%0%
A. 20
B. 70
C. 35
D. 45
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?
Attention:
You do not have to write down or memorize these postulates.
You will need to refer to them on tonight’s assignment.
They are on page 106 in the book.
Use Postulates
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.
A. Determine whether the following statement is always, sometimes, or never true. Explain.
If plane T contains contains point G, then plane T contains point G.
Use Postulates
B. 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.
Use Postulates
C. Determine whether the following statement is always, sometimes, or never true. Explain.
Answer: Never; noncollinear points do not lie on the same line by definition.
1. A.
2. B.
3. C.
0% 0%0%
A. Determine whether each statement is always, sometimes, or never true. Explain.
Plane A and plane B intersect in exactly one point.
A. always
B. sometimes
C. never
1. A.
2. B.
3. C.
0% 0%0%
B. Determine whether each statement is always, sometimes, or never true. Explain.
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.
A. always
B. sometimes
C. never
1. A.
2. B.
3. C.
0% 0%0%
C. Determine whether each statement is always, sometimes, or never true. Explain.
Two planes will always intersect to form a line.
A. always
B. sometimes
C. never
Homework
Pg 102
9 – 20 and Pg 108
8 – 20, 32 – 35
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