Transcript Glencoe Geometry - Dolfanescobar's Weblog
Five-Minute Check (over Lesson 2 –4)
Postulates: Points, Lines, and Planes
Key Concept: Intersections of Lines and Planes
Example 1: Real-World Example: Identifying Postulates
Example 2: Analyze Statements Using Postulates
Key Concept: The Proof Process
Example 3: Write a Paragraph Proof
Over Lesson 2 –4 Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.
Given:
A and
B are supplementary.
Conclusion: m
A + m
B = 180 A.
valid
A. A B. B
B.
invalid A 0% 0% B
Over Lesson 2 –4 Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.
Given: Polygon RSTU has 4 sides.
Conclusion: Polygon RSTU is a square.
A.
valid B.
invalid
A. A B. B
A 0% 0% B
Over Lesson 2 –4 Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.
Given:
A and
B are congruent.
Conclusion: ΔABC exists.
A.
valid
A. A B. B
B.
invalid A 0% 0% B
Over Lesson 2 –4 Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.
Given:
A and
B are congruent.
Conclusion:
A and
B are vertical angles.
A.
valid B.
invalid
A. A B. B
A 0% 0% B
Over Lesson 2 –4 Determine whether the stated conclusion is valid based on the given information. If not, choose invalid.
Given: m
Y in ΔWXY = 90.
Conclusion: ΔWXY is a right triangle.
A.
valid
A. A B. B
B.
invalid A 0% 0% B
Over Lesson 2 –4 How many points define a plane?
A.
1 B.
2 C.
3 D.
4
A. A B. B C. C
A 0% 0% B C D
MA.912.D.6.4
Use methods of direct and indirect proof and determine whether a short proof is logically valid.
MA.912.G.8.1
Analyze the structure of Euclidean geometry as an axiomatic system. Distinguish between undefined terms, definitions, postulates and theorems.
Also addresses MA.912.G.8.5.
You used deductive reasoning by applying the Law of Detachment and the Law of Syllogism. (Lesson 2 –4) • Identify and use basic postulates about points, lines, and planes.
• Write paragraph proofs.
• postulate • axiom • proof • theorem • deductive argument • paragraph proof • informal proof
Identifying Postulates ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.
A.
Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q.
Answer:
Points
F
and
G
lie on line
m
, and the line lies in plane
Q
. Postulate 2.5, which states that if two points lie in a plane, the entire line containing that point lies in that plane, shows that this is true.
Identifying Postulates ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.
B.
Points A and C determine a line.
Answer:
Points
A
and
C
lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true.
ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true.
A.
Plane P contains points A, B, and D.
A.
Through any two points there is exactly one line.
B.
C.
D.
A line contains at least two points.
A plane contains at least two noncollinear points.
A plane contains at least three noncollinear points.
A. A B. B C. C
A 0%
D. D
0% 0% 0% B C D
ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true.
B.
Line AB and line BC intersect at point B.
A.
B.
C.
D.
Through any two points there is exactly one line.
A line contains at least two points.
If two lines intersect, then their intersection is exactly one point.
If two planes intersect, then their intersection is a line.
A. A B. B C. C
A 0%
D. D
0% 0% 0% B C D
Analyze Statements Using Postulates 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.
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.
Analyze Statements Using Postulates B.
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.
A. Determine whether each statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point.
A.
always B.
sometimes C.
never A 0%
A. A B. B
0%
C. C
0% B C
B. Determine whether each statement is always, sometimes, or never true. 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.
B.
always sometimes 0%
A. A B. B
0%
C. C
0% C.
never A B C
Write a Paragraph Proof Given: Prove:
ACD
is a plane.
Proof:
and must intersect at
C
because if two lines intersect, then their intersection is exactly one point. Point
A
is on and point
D
is on . Points
A, C,
and
D
are not collinear. Therefore,
ACD
is a plane as it contains three points not on the same line.
Proof:
?
A.
Definition of midpoint B.
Segment Addition Postulate C.
Definition of congruent segments D.
Substitution
A. A B. B C. C
A 0% 0% B C D