Transcript QA - Cloudfront.net

QA Review # 1
Which of the following is an example of
inductive reasoning?
a) You must be 21 years old to watch a R movie. Jack is 17 years
old. Therefore, Jack can’t watch the movie.
b) Jacky is taller than Joe. Joe is taller than Amie. Therefore, Jacky
is taller than Amie.
c) Movie tickets are on sale for groups of 7 or more people. There
are 6 people in Juan’s family. They can buy tickets on sale.
d) Maria’s palm tree was 10 inches tall last year. This year, it grew
to be 11 inches tall. Next year, Maria’s palm tree will be 12
inches tall.
QA Review # 2
• I. A number is even if it is divisible by 2. The
number 10 is divisible by 2. Therefore, 10 is even.
• II. Mr. Julian ate 20 eggs on January, 25 eggs on
February, and 30 eggs on March. Therefore, Mr.
Julian will eat 35 eggs on April.
• Which statement(s) use(s) deductive reasoning?
Inductive reasoning? Why?
QA Review # 3
1. Give me an example a real life example of
inductive reasoning?
2. Find 1 counterexample to show that each
conjecture is false.
a. The sum of two integer numbers is always positive.
b. The quotient of any two number is less than 1.
c. The product of two fraction numbers is always equal to a
fraction.
d. The difference of any two numbers is always negative.
QA Review # 4
1. “A triangle with sides of three different lengths
must be a right triangle.”
Give me 1 counterexamples.
2. “If rectangle A is parallel to rectangle B, then all
the lines in rectangle A are parallel to all of the
lines in rectangle B.”
Give me 1 counterexamples.
3. Mr. Julian ran 1 mile on Monday, 4 miles on
Tuesday, 9 miles on Wednesday. Therefore, Mr.
Julian will run 16 miles on Thursday.
Is it Inductive reasoning? Why or why not?
QA Review # 5
1. “A triangle with sides of three different lengths
must be an obtuse triangle.”
Give me 1 counterexamples.
2. “If Plane A is parallel to Plane B, then all the lines
in Plane A are parallel to all of the lines in Plane
B.”
Give me 1 counterexamples.
3. For a regular polygon with n sides, the measure of each
interior angle is given by the formula:
If there is a Polygon with 3 sides
what is its interior angle? Hint: A triangle has 3 sides.
QA Review # 6
Two parallel lines are cut by a transversal as
shown below. Of the two angles shown, what
is the measure of the larger angle?
3x +40
2x - 30
QA Review # 7
Look at the figure below. If m< 1 ≠ m< 5, then lines j
and k are not parallel to one another. What can
you assume about the properties of this figure?
1) m<1 __ m<4?
3) m<5, m<3
2) m<3__ m<6?
Supplementary angle?
1 2
3 4
≠
≠
4) m<5 and m<6
Complementary angles?
5 6
7 8
QA Review # 8
Look at the figure below. B is an obtuse angle.
Complementary Angles : 2 angles that have their sum equal to 90.
Supplementary Angles: 2 angles that have their sum equal to 180.
Assuming that m< A = m< B leads to a contradiction with the given
statement, tell me statements that leads to the contradiction?
1) m<A and m< B are supplementary
2) Since m<A = m<B , then m<A = 90* and
m<B = 90* because their sum has to
equal 180*.
3) This is impossible since m<B is an
Obtuse angle.
a b
c d
QA Review # 9
• Two parallel lines are cut by a transversal as
shown below. If m< A = 75º, what is m< F?
f
e
d
b
c
a
g
h
QA Review # 10
Lines J and K are cut by a transversal as shown below.
4
1
J
5
2
K
3
Given that lines J and K are parallel,
What theorem can be used to prove <1≈ <2 ?
What theorem can be used to prove <4 ≈<2?
What can we say about <5 and <2= ?
QA Review # 11
What is m< ABC ? m<YXB? m<XBY?
A
123
X
B
27
Y
C
QA Review # 12
1) What can you say about m<BAC = ? ; m<BCA = ?
2) What else can you say about m< b and m<A?
B
50
D
C
A
E
QA Review # 13
1) Prove that two lines intersected by a transversal are parallel. Just
write down the Theorem you would use.
1)Converse of the Alternate Interior Angles Theorem: If two
lines and a transversal form alternate interior angles that are
congruent, then the two lines are parallel.
2)Converse of the Same-Side interior angles that are
supplementary, then the two lines are parallel.
3) Converse Corresponding Angles Postulate: If a transversal
intersects two parallel lines, then corresponding angles are
congruent.
<1
<4
<2
<3
2) What is the measure of an exterior angle of a
regular pentagon?
3) What is the measure of an exterior angle of a
regular hexagon?
4) What is the measure of an exterior angle of a
regular octagon?
QA Review # 14
1) What is the value of x? 2) What is the value of x?
30
30
3x + 20
3x
x + 10
x
x + 30
2x + 30
2x
x
x + 30
QA Review # 15
Find the value for x?
45
70
5x +25
QA Review # 16
1) If the measure of each interior angle of a regular
polygon is 140º, how many sides does this
polygon have?
2) What is the length AB ?
B
(b, e)
A
(a,0)
C
(b, 0)
QA Review # 17
What is the measure of x, y, z ?
x
120
x
z
105 y
QA Review # 18
1) If the measure of each interior angle of a regular
polygon is 150º, how many sides does this
polygon have?
2) What is the length BC ?
B
(b, e)
A
(0,0)
C
(c, 0)
QA Review # 19
What is the value of x? y? z?
2x
x + 70
x – 15
z
y
x
y
x
x+5
x
x
x
x
x
x
QA
Given : AB
Construct : XY so that XY | AB
at the midpoint M of AB.
Review # 20
A
B
Take out your compass and construct a perpendicular bisector.
1. Put the compass point on point A and draw a long arc as
shown. Be sure the opening is greater than ½ AB.
2. With the same compass setting, put the compass point on
the point B and draw another long arc. Label the points
where the two arcs intersect as X and Y.
3. Draw XY. The point of intersection of AB and XY is M, the
midpoint of AB.
XY | AB at the midpoint of AB, so XY is the perpendicular
bisector of AB.
QA Review # 21
Given : Line l and point N not on l .
Construct : Line m through
N
N with m || l.
l
Take out your compass and construct the line parallel to a
given point that is not on the line.
1. Label two points H and J on l . Draw HN.
2. Construct <1 with vertex at N so that <1 ≈ < NHJ and the
two angles are corresponding angles. Label the line you
just constructed
3.m|| l .
m.
Bell Work
1) What is the value of x? 2) What is the value of x?
50
30
x + 20
3x
x + 20
x
110
2x + 60
2x
x
x + 30
Bell Work
1) If the measure of each interior angle of a regular
polygon is 180º, how many sides does this
polygon have?
2) What is the length BC ?
B
(b, e)
A
(0,0)
C
(c, 0)
Bell Work
1) If the measure of each interior angle of a regular
polygon is 135º, how many sides does this
polygon have?
2) What is the length AB ?
B
(b, e)
A
(x,y)
C
(c, d)
Bell Work
1) A) If the measure of each interior angle of a
regular polygon is 120º, how many sides does this
polygon have?
B) 144º? C)140º? D) 156º?
1) What is the length AB ?BC?AC?
B
A
(x,y)
(w, z )
C
(u, v)
Bell Work
Two parallel lines are cut by a transversal as
shown below. Of the two angles shown, what
is the measure of the larger angle?
3x +25
2x - 15
Bell Work
Two parallel lines are cut by a transversal as
shown below. Of the two angles shown, what
is the measure of the larger angle?
2x +45
2x - 15
Look at the figure below. If m< 1 ≠ m< 5, then lines j
and k are not parallel to one another. What can
you assume about the properties of this figure?
1) m<1 __ m<4?
3) m<5, m<3
2) m<3__ m<6?
Supplementary angle?
1 2
3 4
≠
≠
4) m<5 and m<6
Complementary angles?
5 6
7 8
1) Prove that two lines intersected by a transversal are
parallel?
1)Converse of the Alternate Interior Angles Theorem: If two
lines and a transversal form alternate interior angles that are
congruent, then the two lines are parallel.
2)Converse of the Same-Side interior angles that are
supplementary, then the two lines are parallel.
3) Corresponding Angles Postulate: If a transversal
intersects two parallel lines, then corresponding angles are
congruent.
<1
<4
<2
2) What is the measure of an exterior angle of a regular
pentagon?
3) What is the measure of an exterior angle of a regular
hexagon?
4) What is the measure of an exterior angle of a regular
octagon?
Bell Work
Look at the figure below. If m< 1 ≠ m< 5, then lines j
and k are not parallel to one another. What can
you assume about the properties of this figure?
1 2
3 4
5 6
7 8
≠
≠
Warm Up # 9
Indirect Proof:
Assume what you need to prove is false, and then show that something contradictory (absurd) happens.
•
Assume that the opposite of what you are trying to prove is true.
•
From this assumption, see what conclusions can be drawn. These conclusions must be based upon the assumption and the
use of valid statements.
•
Search for a conclusion that you know is false because it contradicts given or known information. Oftentimes you will be
contradicting a piece of GIVEN information.
•
Since your assumption leads to a false conclusion, the assumption must be false.
•
If the assumption (which is the opposite of what you are trying to prove) is false, then you will know that what you are
trying to prove must be true.