Lesson Menu
Five-Minute Check (over Lesson 2–3)
Mathematical Practices
Then/Now
New Vocabulary
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 an Algebraic Flow Proof
Theorem 2.1: Midpoint Theorem
Example 4: Write a Geometric Flow Proof
Key Concept: How to Write a Paragraph Proof
Example 5: Write a Paragraph Proof
Over Lesson 2–3
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
B. invalid
Over Lesson 2–3
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
Over Lesson 2–3
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
B. invalid
Over Lesson 2–3
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
Over Lesson 2–3
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
B. invalid
Over Lesson 2–3
How many noncollinear points define a plane?
A. 1
B. 2
C. 3
D. 4
Mathematical Practices
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique
reasoning of others.
Content Standards
G.CO.9 Prove theorems about lines and
angles.
G.MG.3 Apply geometric methods to solve
problems.
You used deductive reasoning to prove
statements.
• Analyze figures to identify and use
postulates about points, lines, and planes.
• Analyze and construct viable arguments in
several proof formats.
• postulate
• axiom
• proof
• deductive argument
• flow proof
• paragraph 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 determine a line.
Postulate 2.1, which says through any two points there is
exactly one line.
Identifying Postulates
Points F and G lie along an edge, the line that they
determine. Postulate 2.1 shows that this is true.
Answer: Points F and G lie along an edge, the line they
determine. Postulate 2.1 states that through
any two points, there is exactly one line.
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 D lie in plane BEC
and on line n. Line n lies entirely in
plane BEC.
Postulate 2.5, states that if two points lie in a plane, the
entire line containing the points lies in that plane.
Identifying Postulates
Points A and D lie on line n, and the line lies in plane
BEC. Postulate 2.5 shows that this is true.
Answer: Points A and D lie on line n, and the line lies in
plane BEC. Postulate 2.5 states that if two
points lie in a plane, then the entire line
containing the points lies in the plane.
ARCHITECTURE Refer to the
picture. State the postulate that
can be used to show the
statement is true.
A. Plane P contains points E, B,
and G.
A. Through any two points there
is exactly one line.
B. A line contains at least two
points.
C. A plane contains at least three
noncollinear points.
D. A plane contains at least two
noncollinear points.
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. Through any two points there is
exactly one line.
B. A line contains at least two points.
C. If two lines intersect, then their
intersection is exactly one point.
D. If two planes intersect, then their
intersection is a line.
Analyze Statements Using Postulates
A. Determine whether the following statement is
always, sometimes, or never true. Explain your
reasoning.
The intersection of three planes is a line
Postulate 2.7, states that if two planes intersect then their
intersection is exactly one line.
Visualize adding a third plane. What will the intersection
look like?
Analyze Statements Using Postulates
The three plane could intersect in a line or the three
planes could intersect at a point.
So, sometimes the intersection of three planes will be
a line.
Answer: Sometimes; if three planes intersect, then their
intersection could be a line or a point.
Analyze Statements Using Postulates
B. Determine whether the following statement is
always, sometimes, or never true. Explain your
reasoning.
Through points H and K, there is exactly one line.
Postulate 2.1, states through any two points there is
exactly one line.
So, always two points will be a line.
Answer: Always; Postulate 2.1 states that through any
two points, there is exactly one line.
A. Determine whether the statement is always,
sometimes, or never true.
Plane A and plane B intersect in exactly one point.
A. always
B. sometimes
C. never
B. Determine whether the 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. always
B. sometimes
C. never
Write an Algebraic Flow Proof
Prove that if 2(x + 7) = 6, then x = −4.
Write a flow proof.
Given: 2(x + 7) = 6
Prove: x = −4
Answer:
Proof:
2  x  7  6
Given
2  x  7
2

Division
Property
6
2
x 7  3
Simplify
x 77  37
Subtraction
Property
x  4
Simplify
Write a Geometric Flow Proof
Given that
x = 7.
Given:
Prove: x = 7.
Answer:
Proof:
, write a flow proof to show that
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. Midpoint Theorem
B. Segment Addition Postulate
C. Definition of congruent
segments
D. Substitution