Glencoe Geometry Interactive Chalkboard
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Lesson 2-1 Inductive Reasoning and Conjecture
Lesson 2-2 Logic
Lesson 2-3 Conditional Statements
Lesson 2-4 Deductive Reasoning
Lesson 2-5 Postulates and Paragraph Proofs
Lesson 2-6 Algebraic Proof
Lesson 2-7 Proving Segment Relationships
Lesson 2-8 Proving Angle Relationships
Example 1 Patterns and Conjecture
Example 2 Geometric Conjecture
Example 3 Find a Counterexample
Make a conjecture about the next number based on
the pattern.
2, 4, 12, 48, 240
Find a pattern:
2
4
×2
12
×3
48
×4
240
×5
The numbers are multiplied by 2, 3, 4, and 5.
Conjecture: The next number will be multiplied by 6.
So, it will be
or 1440.
Answer: 1440
Make a conjecture about the next number based on
the pattern.
Answer: The next number will be
For points L, M, and N,
and
make a conjecture and draw a figure to illustrate
your conjecture.
Given: points L, M, and N;
Examine the measures of the segments. Since
the points can be collinear with
point N between points L and M.
Answer:
Conjecture: L, M, and N are collinear.
,
ACE is a right triangle with
Make a
conjecture and draw a figure to illustrate your
conjecture.
Answer:
Conjecture: In ACE, C is a right angle and
hypotenuse.
is the
County
Civilian Labor Force
Rate
Shawnee
90,254
3.1%
Jefferson
9,937
3.0%
Jackson
8,915
2.8%
Douglas
55,730
3.2%
Osage
10,182
4.0%
3,575
3.0%
11,025
2.1%
Wabaunsee
Pottawatomie
Source: Labor Market Information Services–
Kansas Department of Human Resources
UNEMPLOYMENT Based on the table showing
unemployment rates for various cities in Kansas,
find a counterexample for the following statement.
The unemployment rate is highest in the cities with
the most people.
Examine the data in the table. Find two cities such that
the population of the first is greater than the population
of the second while the unemployment rate of the first is
less than the unemployment rate of the second.
Shawnee has a greater population than Osage while
Shawnee has a lower unemployment rate than Osage.
Answer: Osage has only 10,182 people on its civilian
labor force, and it has a higher rate of
unemployment than Shawnee, which has
90,254 people on its civilian labor force.
DRIVING The table on the next screen shows
selected states, the 2000 population of each state,
and the number of people per 1000 residents who
are licensed drivers in each state. Based on the
table, find a counterexample for the following
statement.
The greater the population of a state, the lower the
number of drivers per 1000 residents.
Population
Licensed Drivers
per 1000
Alabama
4,447,100
792
California
33,871,648
627
Texas
20,851,820
646
608,827
831
West Virginia
1,808,344
745
Wisconsin
5,363,675
703
Vermont
Source: The World Almanac and Book of Facts 2003
State
Answer: Alabama has a greater population than West
Virginia, and it has more drivers per 1000 than
West Virginia.
Example 1 Truth Values of Conjunctions
Example 2 Truth Values of Disjunctions
Example 3 Use Venn Diagrams
Example 4 Construct Truth Tables
Use the following statements to write a compound
statement for the conjunction p and q. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: One foot is 14 inches, and September has
30 days. p and q is false, because p is false
and q is true.
Use the following statements to write a compound
statement for the conjunction
. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: A plane is defined by three noncollinear
points, and one foot is 14 inches.
is false,
because r is true and p is false.
Use the following statements to write a compound
statement for the conjunction
. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: September does not have 30 days, and a
plane is defined by three noncollinear points.
is false because
is false and r is true.
Use the following statements to write a compound
statement for the conjunction p  r. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: A foot is not 14 inches, and a plane is defined
by three noncollinear points. ~p  r is true,
because ~p is true and r is true.
Use the following statements to write a compound
statement for each conjunction.
Then find its truth value.
p: June is the sixth month of the year.
q: A square has five sides.
r: A turtle is a bird.
a. p and r
Answer: June is the sixth month of the year, and a
turtle is a bird; false.
b.
Answer: A square does not have five sides, and a turtle
is not a bird; true.
Use the following statements to write a compound
statement for each conjunction. Then find its truth
value.
p: June is the sixth month of the year.
q: A square has five sides.
r: A turtle is a bird.
c.
Answer: A square does not have five sides, and June is
the sixth month of the year; true.
d.
Answer: A turtle is not a bird, and a square has five
sides; false.
Use the following statements to write a compound
statement for the disjunction p or q. Then find its
truth value.
p:
is proper notation for “line AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
Answer:
is proper notation for “line AB,” or centimeters
are metric units. p or q is true because q is true. It
does not matter that p is false.
Use the following statements to write a compound
statement for the disjunction
. Then find its truth
value.
p:
is proper notation for “line AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
Answer: Centimeters are metric units, or 9 is a prime
number.
is true because q is true. It does
not matter that r is false.
Use the following statements to write a compound
statement for each disjunction. Then find its truth value.
p: 6 is an even number.
q: A cow has 12 legs
r: A triangle has 3 sides.
a. p or r
Answer: 6 is an even number, or a triangle as 3 sides;
true.
b.
Answer: A cow does not have 12 legs, or a triangle
does not have 3 sides; true.
DANCING The Venn diagram shows the number of
students enrolled in Monique’s Dance School for
tap, jazz, and ballet classes.
How many students are enrolled in all three
classes?
The students that are enrolled in
all three classes are represented
by the intersection of all three
sets.
Answer: There are 9 students enrolled in all three
classes
How many students are enrolled in tap or ballet?
The students that are enrolled
in tap or ballet are represented
by the union of these two sets.
Answer: There are 28 + 13 + 9 + 17 + 25 + 29 or
121 students enrolled in tap or ballet.
How many students are enrolled in jazz and ballet
and not tap?
The students that are enrolled
in jazz and ballet and not tap
are represented by the
intersection of jazz and ballet
minus any students enrolled
in tap.
Answer: There are 25 + 9 – 9 or 25 students enrolled in
jazz and ballet and not tap.
PETS The Venn diagram shows the number of
students at Manhattan School that have dogs, cats,
and birds as household pets.
a. How many students in
Manhattan School have one
of three types of pets?
Answer: 311
b. How many students have
dogs or cats?
Answer: 280
c. How many students have
dogs, cats, and birds as pets?
Answer: 10
Construct a truth table for
.
Step 1 Make columns with the headings
p, q, ~p, and ~p
p
q
~p
~p
Construct a truth table for
.
Step 2 List the possible combinations of truth values for
p and q.
p
T
T
F
F
q
T
F
T
F
~p
~p
Construct a truth table for
.
Step 3 Use the truth values of p to determine the truth
values of ~p.
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~p
Construct a truth table for
.
Step 4 Use the truth values for ~p and q to write the
truth values for ~p  q.
Answer:
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~p
T
F
T
T
Construct a truth table for
.
Step 1 Make columns with the headings
p, q, r, ~q, ~q  r, and p  (~q  r).
p
q
r
~q
~q  r
p  (~q  r)
Construct a truth table for
.
Step 2 List the possible combinations of truth values for
p, q, and r.
p
q
r
T
T
T
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
F
F
F
F
~q
~q  r
p  (~q  r)
Construct a truth table for
.
Step 3 Use the truth values of q to determine the truth
values of ~q.
p
q
r
~q
T
T
T
F
T
F
T
T
T
T
F
F
T
F
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
F
F
T
~q  r
p  (~q  r)
Construct a truth table for
.
Step 4 Use the truth values for ~q and r to write the
truth values for ~q  r.
p
q
r
~q
~q  r
T
T
T
F
F
T
F
T
T
T
T
T
F
F
F
T
F
F
T
F
F
T
T
F
F
F
F
T
T
T
F
T
F
F
F
F
F
F
T
F
p  (~q  r)
Construct a truth table for
.
Step 5 Use the truth values for p and ~q  r to write the
truth values for p  (~q  r).
Answer:
p
q
r
~q
~q  r
p  (~q  r)
T
T
T
F
F
T
T
F
T
T
T
T
T
T
F
F
F
T
T
F
F
T
F
T
F
T
T
F
F
F
F
F
T
T
T
T
F
T
F
F
F
F
F
F
F
T
F
F
Construct a truth table for (p  q)  ~r.
Step 1 Make columns with the headings
p, q, r, ~r, p  q, and (p  q)  ~r.
p
q
r
~r
pq
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 2 List the possible combinations of truth values for
p, q, and r.
p
q
r
T
T
T
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
F
F
F
F
~r
pq
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 3 Use the truth values of r to determine the truth
values of ~r.
p
q
r
~r
T
T
T
F
T
F
T
F
T
T
F
T
T
F
F
T
F
T
T
F
F
F
T
F
F
T
F
T
F
F
F
T
pq
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 4 Use the truth values for p and q to write the truth
values for p  q.
p
q
r
~r
pq
T
T
T
F
T
T
F
T
F
T
T
T
F
T
T
T
F
F
T
T
F
T
T
F
T
F
F
T
F
F
F
T
F
T
T
F
F
F
T
F
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 5 Use the truth values for p  q and ~r to write the
truth values for (p  q)  ~r.
Answer:
p
q
r
~r
pq
(p  q)  ~r
T
T
T
F
T
T
T
F
T
F
T
F
T
T
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
T
F
F
T
F
F
F
F
T
F
T
T
F
F
F
F
T
F
F
Construct a truth table for the following compound
statement.
a.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
F
F
F
F
F
F
T
T
F
T
T
F
F
T
F
F
F
F
T
F
F
F
F
F
F
F
F
F
F
Construct a truth table for the following compound
statement.
b.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
T
T
T
T
T
F
T
T
T
T
F
F
T
F
F
F
T
T
T
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
F
F
F
F
F
Construct a truth table for the following compound
statement.
c.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
T
F
T
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
F
F
T
F
T
F
T
F
F
F
F
F
F
Example 1 Identify Hypothesis and Conclusion
Example 2 Write a Conditional in If-Then Form
Example 3 Truth Values of Conditionals
Example 4 Related Conditionals
Identify the hypothesis and conclusion of the
following statement.
If a polygon has 6 sides, then it is a hexagon.
If a polygon has 6 sides, then it is a hexagon.
hypothesis
conclusion
Answer: Hypothesis: a polygon has 6 sides
Conclusion: it is a hexagon
Identify the hypothesis and conclusion of the
following statement.
Tamika will advance to the next level of play if she
completes the maze in her computer game.
Answer: Hypothesis: Tamika completes the maze in her
computer game
Conclusion: she will advance to the next level
of play
Identify the hypothesis and conclusion of each
statement.
a. If you are a baby, then you will cry.
Answer: Hypothesis: you are a baby
Conclusion: you will cry
b. To find the distance between two points, you can use
the Distance Formula.
Answer: Hypothesis: you want to find the distance
between two points
Conclusion: you can use the Distance Formula
Identify the hypothesis and conclusion of the
following statement. Then write the statement
in the if-then form.
Distance is positive.
Sometimes you must add information to a statement.
Here you know that distance is measured or determined.
Answer: Hypothesis: a distance is determined
Conclusion: it is positive
If a distance is determined, then it is positive.
Identify the hypothesis and conclusion of the
following statement. Then write the statement
in the if-then form.
A five-sided polygon is a pentagon.
Answer: Hypothesis: a polygon has five sides
Conclusion: it is a pentagon
If a polygon has five sides, then it is
a pentagon.
Identify the hypothesis and conclusion of each
statement. Then write each statement in if-then form.
a. A polygon with 8 sides is an octagon.
Answer: Hypothesis: a polygon has 8 sides
Conclusion: it is an octagon
If a polygon has 8 sides, then it is an octagon.
b. An angle that measures 45º is an acute angle.
Answer: Hypothesis: an angle measures 45º
Conclusion: it is an acute angle
If an angle measures 45º, then it is an acute
angle.
Determine the truth value of the following statement
for each set of conditions. If Yukon rests for 10 days,
his ankle will heal.
Yukon rests for 10 days, and he still has a hurt ankle.
The hypothesis is true, but the conclusion is false.
Answer: Since the result is not what was expected, the
conditional statement is false.
Determine the truth value of the following statement
for each set of conditions. If Yukon rests for 10 days,
his ankle will heal.
Yukon rests for 3 days, and he still has a hurt ankle.
The hypothesis is false, and the conclusion is false. The
statement does not say what happens if Yukon only rests
for 3 days. His ankle could possibly still heal.
Answer: In this case, we cannot say that the statement
is false. Thus, the statement is true.
Determine the truth value of the following statement
for each set of conditions. If Yukon rests for 10 days,
his ankle will heal.
Yukon rests for 10 days, and he does not have a hurt
ankle anymore.
The hypothesis is true since Yukon rested for 10 days,
and the conclusion is true because he does not have a
hurt ankle.
Answer: Since what was stated is true, the conditional
statement is true.
Determine the truth value of the following statement
for each set of conditions. If Yukon rests for 10 days,
his ankle will heal.
Yukon rests for 7 days, and he does not have a hurt
ankle anymore.
The hypothesis is false, and the conclusion is true. The
statement does not say what happens if Yukon only rests
for 7 days.
Answer: In this case, we cannot say that the statement
is false. Thus, the statement is true.
Determine the truth value of the following statements
for each set of conditions. If it rains today, then
Michael will not go skiing.
a. It does not rain today; Michael does not go skiing.
Answer: true
b. It rains today; Michael does not go skiing.
Answer: true
c. It snows today; Michael does not go skiing.
Answer: true
d. It rains today; Michael goes skiing.
Answer: false
Write the converse, inverse, and contrapositive of the
statement All squares are rectangles. Determine
whether each statement is true or false. If a statement
is false, give a counterexample.
First, write the conditional in if-then form.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.
Write the converse by switching the hypothesis and
conclusion of the conditional.
Converse: If a shape is a rectangle, then it is a square.
The converse is false. A rectangle with = 2
and w = 4 is not a square.
Inverse:
If a shape is not a square, then it is not a
rectangle. The inverse is false. A 4-sided
polygon with side lengths 2, 2, 4, and 4 is
not a square, but it is a rectangle.
The contrapositive is the negation of the hypothesis and
conclusion of the converse.
Contrapositive: If a shape is not a rectangle, then it is
not a square. The contrapositive is true.
Write the converse, inverse, and contrapositive of the
statement The sum of the measures of two
complementary angles is 90. Determine whether each
statement is true or false. If a statement is false, give
a counterexample.
Answer: Conditional: If two angles are complementary,
then the sum of their measures is 90; true.
Converse: If the sum of the measures of two
angles is 90, then they are complementary;
true.
Inverse: If two angles are not complementary,
then the sum of their measures is not 90; true.
Contrapositive: If the sum of the measures of
two angles is not 90, then they are not
complementary; true.
Example 1 Determine Valid Conclusions
Example 2 Determine Valid Conclusions From Two
Conditionals
Example 3 Analyze Conclusions
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.
Given:
Conclusion:
The hypothesis states that
Answer: Since the conditional is true and the
hypothesis is true, the conclusion is valid.
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.
Given:
Conclusion:
The hypothesis states that
is a segment and
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.
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.
a. Given:
Conclusion: If you connect X, N, and O with segments,
the figure will be a convex quadrilateral.
Answer: not valid
b. Given: ABCD is a convex quadrilateral.
Conclusion: The sum of the interior angles of ABCD
is 360.
Answer: valid
PROM Use the Law of Syllogism to determine whether
a valid conclusion can be reached from the following
set of statements.
(1) If Salline attends the prom, she will go with Mark.
(2) Mark is a 17-year-old student.
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.
PROM Use the Law of Syllogism to determine whether
a valid conclusion can be reached from the following
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.
Use the Law of Syllogism to determine whether a
valid conclusion can be reached from each set of
statements.
a. (1) If you ride a bus, then you attend school.
(2) If you ride a bus, then you go to work.
Answer: invalid
b. (1) If your alarm clock goes off in the morning, then you
will get out of bed.
(2) You will eat breakfast, if you get out of bed.
Answer: valid
Determine whether statement (3) follows from
statements (1) and (2) by the Law of Detachment or
the Law of Syllogism. If it does, state which law was
used. If it does not, write invalid.
(1) If the sum of the squares of two sides of a triangle is
equal to the square of the third side, then the triangle
is a right triangle.
(2) For XYZ, (XY)2 + (YZ)2 = (ZX)2.
(3) XYZ is a right triangle.
p: the sum of the squares of the two sides of a triangle
is equal to the square of the third side
q: the triangle is a right triangle
By the Law of Detachment, if
then q is also true.
is true and p is true,
Answer: Statement (3) is a valid conclusion by the Law
of Detachment
Determine whether statement (3) follows from
statements (1) and (2) by the Law of Detachment or
the Law of Syllogism. If it does, state which law was
used. If it does not, write invalid.
(1) If Ling wants to participate in the wrestling competition,
he will have to meet an extra three times a week to
practice.
(2) If Ling adds anything extra to his weekly schedule, he
cannot take karate lessons.
(3) If Ling wants to participate in the wrestling competition,
he cannot take karate lessons.
p: Ling wants to participate in the wrestling competition
q: he will have to meet an extra three times a week to
practice
r: he cannot take karate lessons
By the Law of Syllogism, if
Then
is also true.
and
are true.
Answer: Statement (3) is a valid conclusion by the
Law of Syllogism.
Determine whether statement (3) follows from
statements (1) and (2) by the Law of Detachment of
the Law of Syllogism. If it does, state which law was
used. If it does not, write invalid.
a. (1) If a children’s movie is playing on Saturday, Janine
will take her little sister Jill to the movie.
(2) Janine always buys Jill popcorn at the movies.
(3) If a children’s movie is playing on Saturday, Jill will
get popcorn.
Answer: Law of Syllogism
b. (1) If a polygon is a triangle, then the sum of the interior
angles is 180.
(2) Polygon GHI is a triangle.
(3) The sum of the interior angles of polygon GHI is
180.
Answer: Law of Detachment
Example 1 Points and Lines
Example 2 Use Postulates
Example 3 Write a Paragraph Proof
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.
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.
Examine In the figure,
are all segments
that connect the vertices of the snow crystal.
Answer: 15
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?
Answer: 45
Determine whether the following statement is
always, sometimes, or never true. Explain.
If plane T contains
plane T contains point G.
contains point G, then
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.
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.
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.
Determine whether each statement is always,
sometimes, or never true. Explain.
a. Plane A and plane B intersect in one point.
Answer: Never; Postulate 2.7 states that if two planes
intersect, then their intersection is a line.
b. 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.
Answer: Always; Postulate 2.1 states that through any
two points, there is exactly one line.
Determine whether each statement is always,
sometimes, or never true. Explain.
c. Two planes will always intersect a line.
Answer: Sometimes; Postulate 2.7 states that if the two
planes intersect, then their intersection is a
line. It does not say what to expect if the
planes do not intersect.
Given
intersecting
, write a paragraph proof
to show that A, C, and D determine a plane.
Given:
intersects
Prove: ACD is a plane.
Proof:
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
Therefore, points A and D are not collinear.
Therefore, ACD is a plane as it contains three
points not on the same line.
Given
midpoint of
is the midpoint of
and X is the
write a paragraph proof to show that
Proof: We are given that S is the midpoint of
X is the midpoint of
and
By the definition of midpoint,
Using the definition of congruent
segments,
Also using the given
statement
and the definition of congruent
segments,
If
then
Since S and X are midpoints,
By substitution,
congruence,
and by definition of
Example 1 Verify Algebraic Relationships
Example 2 Write a Two-Column Proof
Example 3 Justify Geometric Relationships
Example 4 Geometric Proof
Solve
Algebraic Steps
Properties
Original equation
Distributive Property
Substitution Property
Addition Property
Substitution Property
Division Property
Substitution Property
Answer:
Solve
Algebraic Steps
Properties
Original equation
Distributive Property
Substitution Property
Subtraction Property
Substitution Property
Division Property
Substitution Property
Answer:
Write a two-column proof. If
Proof:
then
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
3. Substitution
4.
4. Subtraction Property
5.
5. Substitution
6.
6. Division Property
7.
7. Substitution
Write a two-column proof. If
Proof:
Statements
Reasons
then
1.
1. Given
2.
2. Multiplication Property
3.
3. Distributive Property
4.
4. Subtraction Property
5.
5. Substitution
6.
6. Addition Property
Write a two-column proof. If
Proof:
Statements
Reasons
then
7.
7. Substitution
8.
8. Division Property
9.
9. Substitution
Write a two-column proof for the following.
a.
Proof:
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
4.
5.
3. Substitution
4. Subtraction Property
5. Substitution
6.
6. Division Property
7.
7. Substitution
Write a two-column proof for the following.
b. Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Multiplication Property
3.
4.
3. Distributive Property
4. Subtraction Property
5.
5. Substitution
6.
6. Subtraction Property
7.
7. Substitution
MULTIPLE- CHOICE TEST ITEM
If
and
then which of the
following is a valid conclusion?
I
II
III
A I only
B I and II
C I and III
D I, II, and III
Read the Test Item
Determine whether the statements are true based
on the given information.
Solve the Test Item
Statement I:
Examine the given information, GH JK ST and
From the definition of congruence of segments, if
then ST RP. You can substitute RP for ST in
GH JK ST to get GH JK RP. Thus, Statement I
is true.
Statement II:
Since the order you name the endpoints of a segment is
not important,
and TS = PR. Thus, Statement II
is true.
.
,
Statement III
If GH JK ST, then
not true.
. Statement III is
Because Statements I and II only are true, choice B is
correct.
Answer: B
MULTIPLE- CHOICE TEST ITEM
If
and
then
which of the following is a valid conclusion?
I.
II.
III.
A I only
B I and II
Answer: C
C I and III
D II and III
SEA LIFE A starfish has five legs. If the length of leg 1
is 22 centimeters, and leg 1 is congruent to leg 2, and
leg 2 is congruent to leg 3, prove that leg 3 has length
22 centimeters.
Given:
m leg 1
22 cm
Prove: m leg 3
22 cm
Proof:
Statements
Reasons
1.
1. Given
2.
2. Transitive Property
3. m leg 1
m leg 3
3. Definition of congruence
4. m leg 1
22 cm
4. Given
5. m leg 3
22 cm
5. Transitive Property
DRIVING A stop sign as shown below is a regular
octagon. If the measure of angle A is 135 and angle
A is congruent to angle G, prove that the measure
of angle G is 135.
Proof:
Statements
1.
Reasons
1. Given
2.
2. Given
3.
3. Definition of congruent angles
4.
4. Transitive Property
Example 1 Proof With Segment Addition
Example 2 Proof With Segment Congruence
Prove the following.
Given: PR = QS
Prove: PQ = RS
Proof:
Statements
1. PR = QS
1. Given
2. PR – QR = QS – QR
2. Subtraction Property
3. PR – QR = PQ;
QS – QR = RS
4. PQ = RS
3. Segment Addition
Postulate
4. Substitution
Reasons
Prove the following.
Given:
Prove:
Proof:
Statements
Reasons
1. AC = AB, AB = BX
1. Given
2. AC = BX
2. Transitive Property
3. CY = XD
3. Given
4. AC + CY = BX + XD
4. Addition Property
5. AC + CY = AY;
BX + XD = BD
6. AY = BD
5. Segment Addition
Property
6. Substitution
Prove the following.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
3.
2. Definition of congruent segments
4.
4. Transitive Property
5.
5. Transitive Property
3. Given
Prove the following.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Transitive Property
3.
3. Given
4.
4. Transitive Property
5.
5. Symmetric Property
Example 1 Angle Addition
Example 2 Supplementary Angles
Example 3 Use Supplementary Angles
Example 4 Vertical Angles
TIME At 4 o’clock, the angle between the
hour and minute hands of a clock is 120º.
If the second hand stops where it bisects
the angle between the hour and minute
hands, what are the measures of the angles
between the minute and second hands and
between the second and hour hands?
If the second hand stops where the angle is bisected,
then the angle between the minute and second hands is
one-half the measure of the angle formed by the hour
and minute hands, or
.
By the Angle Addition Postulate, the sum of the two
angles is 120, so the angle between the second and hour
hands is also 60º.
Answer: They are both 60º by the definition of angle
bisector and the Angle Addition Postulate.
QUILTING The diagram below shows one square for
a particular quilt pattern. If
and
is a right angle, find
Answer: 50
If
and
form a linear pair and
find
Supplement Theorem
Subtraction Property
Answer: 14
If
and
find
.
Answer: 28
are complementary angles and
In the figure,
Given:
Prove:
and
form a linear pair, and
and
Prove that
are congruent.
form a linear pair.
Proof:
Statements
Reasons
1.
1. Given
2.
2. Linear pairs are
supplementary.
3. Definition of
supplementary angles
3.
4.
4. Subtraction Property
5.
5. Substitution
6.
6. Definition of congruent
angles
In the figure, NYR and RYA form a linear pair,
AXY and AXZ form a linear pair, and RYA and
AXZ are congruent. Prove that RYN and AXY
are congruent.
Proof:
Statements
Reasons
1.
1. Given
linear pairs.
2.
2. If two s form a
linear pair, then
they are suppl. s.
3.
3. Given
4.
4.
If 1 and 2 are vertical angles and m1
m2
find m1 and m2.
1
m1
2
Vertical Angles Theorem
m2
Definition of congruent angles
Substitution
Add 2d to each side.
Add 32 to each side.
Divide each side by 3.
and
Answer: m1 = 37 and m2 = 37
If
and
are vertical angles and
find
and
Answer: mA = 52; mZ = 52
and
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