Transcript 2-5 / 2-6 Class Notes

Five-Minute Check (over Lesson 2–4)
CCSS
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 a Paragraph Proof
Theorem 2.1: Midpoint Theorem
Bellringer 9/24/12
Define the following terms:
• 1. Postulate
• 2. Proof
• 3. Theorem
• 4. Algebraic Proof
• 5. Two Column Proof
• postulate
• proof
• theorem
• algebraic proof
• two column 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 the points 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 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.
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 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
Justify Each Step When Solving an Equation
Solve -5(x+4) = 70.
Algebraic Steps
-5(x+4) = 70
Properties
Original equation
-5x + (-5)4 = 70
Distributive Property
-5x - 20 = 70
Substitution Property
-5x - 20 + 20 = 70 + 20
Addition Property
Justify Each Step When Solving an Equation
–
5x = 90
Substitution Property
-5x = 90
-5
-5
Division Property
x = -18
Answer: a =–18
Substitution Property
Justify Each Step When Solving an Equation
Solve 2(5 – 3a) – 4(a + 7) = 92.
Algebraic Steps
2(5–3a)–4(a + 7) = 92
Properties
Original equation
10–6a–4a–28 = 92
Distributive Property
–
18–10a = 92
Substitution Property
–
18–10a + 18 = 92 + 18
Addition Property
Justify Each Step When Solving an Equation
–
10a = 110
Substitution Property
Division Property
a=–
11
Answer: a =–11
Substitution Property
Solve –3(a + 3) + 5(3 – a) = –50.
A. a = 12
B. a = –37
C. a = –7
D. a = 7
Write an Algebraic Proof
Begin by stating what is given and what you are to prove.
Write an Algebraic Proof
Proof:
Statements
Reasons
1. d = 20t + 5
1. Given
2. d–5 = 20t
2. Addition Property of Equality
3.
4.
=t
3. Division Property of Equality
4. Symmetric Property of
Equality
Which of the following statements would complete
the proof of this conjecture?
If the formula for the area of a trapezoid is
, then the height h of the trapezoid is
given by
.
Proof:
Statements
Reasons
1.
1. Given
?
2. _____________
2. Multiplication Property of
Equality
3.
3. Division Property of Equality
4.
4. Symmetric Property of
Equality
A. 2A = (b1 + b2)h
B.
C.
D.
Write a Geometric Proof
If A B, mB = 2mC, and mC = 45, then
mA = 90. Write a two-column proof to verify this
conjecture.
Write a Geometric Proof
Proof:
Statements
Reasons
1. A B;
mB = 2mC;
mC = 45
1. Given
2. mA = mB
2. Definition of
3. mA = 2mC
3. Transitive Property of
Equality
4. mA = 2(45)
4. Substitution
5. mA = 90
5. Substitution
angles
Proof:
Statements
1.
2.
Reasons
1. Given
?
2. _______________
3. AB = RS
3. Definition of congruent
segments
4. AB = 12
4. Given
5. RS = 12
5. Substitution
A. Reflexive Property of Equality
B. Symmetric Property of Equality
C. Transitive Property of Equality
D. Substitution Property of Equality