Geometry Review
T/F A point represents a location.
True
T/F Hilbert's axioms changed Euclid's geometry.
True
T/F If p is a true statement and q is false, then p V q is true.
True
T/F If two points of a ray lie in a plane, then all of the ray lies in the plane.
True
How many lines are determined by two points?
1
If line d intersects plane F, then how many points on line d also lie on plane F? 0 1 2 infinite
1
Which is the characteristic of a line? width depth color length
length
The inverse of p -> q is:
~p -> ~q
The contrapositive of p -> q is:
~q -> ~p
The inverse of x -> y is:
~x -> ~y
Given the conditional statement, match the following items. Conditional statement: If two angles are adjacent, then the two angles have the same vertex. 1. if two angles have the same vertex, then they are adjacent 2. if two angles are not adjacent, then the angles do not have the same vertex 3. if two angles do not have the same vertex, then the two angles are not adjacent
1. converse 2. inverse 3. contrapositive
Given the conditional statement, match the following. Conditional statement: If a polygon is a square, then it is rectangle. 1. If a polygon is a rectangle, then it is a square. 2. If a polygon is not a square, then it is not a rectangle. 3. If a polygon is not a rectangle, then it is not a square.
1. converse 2. inverse 3. contrapositive
Given the conditional statement, match the following. Conditional statement: If today is Thursday, then tomorrow is Wednesday. 1. If tomorrow is Wednesday, then today is Thursday. 2. If Today is not Thursday, then tomorrow is not Wednesday. 3. If tomorrow is not Wednesday, then today is not Thursday.
1. converse 2. inverse 3. contrapositive
Given the conditional statement, match the following. Conditional statement: If two lines intersect, then their intersection is one point. 1. If the intersection of two lines is a point, then they intersect. 2. If two lines do not intersect, then their intersection is not one point. 3. If the intersection of two lines is not one point, then the two lines do not intersect.
1. converse 2. inverse 3. contrapositive
Match the following statements and negations. 1. statement formed by interchanging the hypothesis and the conclusion in a conditional statement 2. a new statement formed by negating both the hypothesis and the conclusion 3. formed by exchanging the hypothesis and the conclusion and negating both of them
1. converse of a conditional 2. inverse of a conditional 3. contrapositive of a conditional
Match the reasons with the statement. Given: 12 - x = 20 - 5x To Prove: x = 2 1. 12 - x = 20 - 5x 2. 12 + 4x = 20 3. 4x = 8 4. x = 2
1. given 2. addition property of equality 3. subtraction property of equality 4. division property of equality
Match the following items. 1. 2x + x + 4 = -17 2. 3x + 4 =-17 3. 3x =- 21 4. x = -7
1. given 2. combine like terms 3. subtraction 4. division
Match the reasons with the statements. GIVEN: x 2 + 6x + 2x + 12 = 0 TO PROVE: x = -6 or x = -2 1. x2 + 6x + 2x + 12 = 0 2. x2 + 8x + 12 = 0 3. (x + 6)(x + 2) = 0 4. x + 6 = 0 or x + 2 = 0 5. x = -6 or x = -2
1. given 2. combining like terms 3. distributive postulate 4. zero product postulate 5. subtraction property of equality
Match the reasons with the statements. Given: 2 (x + 3) = 8 To Prove: x = 1 1. 2(x + 3) = 8 2. 2x + 6 = 8 3. 2x = 2 4. x = 1
1. given 2. distributive postulate 3. subtraction property of equality 4. division property of equality
List the six parts in order, for the format of a two-column proof.
1. statement of the theorem 2. figure 3. given information 4. conclusion to prove 5. plan of proof 6. proof
How many parts are there in the format of a two-column proof?
6
Select a counter-example that makes the conclusion false. 3 is prime, 5 is prime, 7 is prime Conclusion: All odd numbers are prime. Counterexample:
9 is not prime
From the statement select the related given statement. In a triangle a segment joining the midpoints of two sides is one-half the length of the third side.
<>ABC with midpoints M and N.
From the statement select the related given statement. If two angles of a triangle are equal, the sides opposite those angles are equal.
<>ABC with opposite angles for which <1 = <2.
What is indicated by arrowheads on a line? A line extends indefinitely in both directions. A line has no dimension or thickness. A line is only as long as the arrowheads.
A line extends indefinitely in both directions.
Which of the following statements about a plane is not true? A plane can be thought of as flat. The surface of a plane is made up of points. A plane can be seen. A plane extends infinitely in all directions.
A plane can be seen.
Prove the following theorem indirectly. We will give you a start. Prove that a triangle cannot have two right angles. A triangle cannot have two right angles. Suppose a triangle had two right angles.
A triangle is 180 degrees. If a triangle were to have 2 right angles it would add up to be 180 degrees without even adding in the last angle
From the figure and statement provided, select the proper TO PROVE statement. If B is between A and C, then AB + BC = AC
AB + BC = AC
From the figure and statement provided, select the proper TO PROVE statement. If two angles of a triangle are equal, the sides opposite those angles are equal.
AB = BC
State a conclusion that seems reasonable. 6 + 0 = 6, 8 + 0 = 8, 9 + 0 = 9, 100 + 0 = 100. Conclusion:
Any number added by 0 remains the same
State a true conclusion. 1) If it's Saturday, then Cathy's dad goes fishing. 2) Today is Saturday. Conclusion:
Cathy's dad is going fishing
Select inductive reasoning, deductive reasoning, or neither. If 5x + 7 = 12, then x = 1. Inductive reasoning Deductive reasoning neither
Deductive reasoning
State a conclusion that seems reasonable. Donald is older than Jeanette; Donald is older than Ethel. Donald is older than Allen. Conclusion:
Donald is the oldest
A line has measurable length.
False
T/F A plane can have only two points.
False
T/F A postulate is a statement requiring proof.
False
T/F A postulate is a statement that must be proved.
False
T/F A segment has exactly one end point.
False
T/F A theorem is a statement that is proved by inductive reasoning.
False
T/F An undefined term cannot be used to state a postulate.
False
T/F In the six essential parts of a two-column proof, the "Proof" comes before the "Plan of the Proof."
False
T/F Inductive reasoning is the process of making a specific conclusion based on general sampling.
False
T/F Two intersecting lines lie in two planes.
False
T/F Two planes can intersect in exactly one point.
False
T/F One method to develop a plan for a proof is to work "backwards" by starting with the conclusion.
True
Point O is between points G and D on line m. Which of the following statements is not necessarily true? O, G and D are collinear. O, G and D are coplanar. GO + OD = GD GO = OD
GO = OD
Which of the following statements has the same truth value as the statement, "If it is Friday, then Bruce has beans for supper."? If it is not Friday, then Bruce does not have beans for supper. If Bruce has beans for supper, then it is Friday. If Bruce does not have beans for supper, then it is not Friday.
If Bruce does not have beans for supper, then it is not Friday.
Write the following statement in if - then form. David will sing provided Eric plays the piano.
If Eric plays the piano, then David will sing
Which of the following statements is the contrapositive of "If I have money, then I spend it"?
If I don't spend money, then I don't have it.
How should the following statement be written in if - then form? The sum of the measures of the angles of a triangle is 180 degrees.
If a triangle exists, then the sum of the measures of the angles is 180 degrees.
Write the following statement in if - then form. A woman that lives in Houston lives in Texas.
If a women lives in Houston, then she lives in Texas.
Select a counter-example that makes the conclusion false. 7 - 3 = 4, 8 - 5 = 3, 9 - 8 = 1 Conclusion: the difference of two positive numbers is always positive. Counterexample:
If you subtract a number by a much larger number, you will end up with a difference that is negative
Select inductive reasoning, deductive reasoning, or neither. You watch an ant taste a liquid and die. Several more ants sample the liquid and they die. You conclude the liquid is an ant poison.
Inductive reasoning
From the statement select the related given statement. Through a point outside a line one line can be drawn parallel to the line.
Line l; point P not on l.
From the figure and statement provided, select the proper TO PROVE statement. If a segment joins the midpoints of two sides of a triangle, its length is one-half the length of the third side.
MN =1/2AB
Type a counter-example that would have to exist in order for the conclusion to be false. 5 0, 6 0, 12 0, 16 0, 20 0, 100 0. Conclusion: All numbers are greater than 0. Counterexample:
Negative numbers are not greater than 0.
State a true conclusion. 1) If it is June, then it is summer. 2) It is summer. Conclusion:
No conclusion can be drawn. It could be summer but that does not mean it is necessarily June.
State a true conclusion. 1) If the mill is on strike, then no one works. 2) The mill is on strike. Conclusion:
No one is working
Point O is between points M and N on line segment MN. O is the midpoint of line segment MN if _____. OM = NM ON = NM ON = OM ON + OM = MN
ON + OM = MN
From the statement select the related given statement. If two parallel planes are cut by a third plane, the lines of intersection are parallel.
Plane R is parallel to plane S; Plane T cuts planes R and S.
From the statement select the related given statement. If B is between A and C, then AB + BC = AC
Point B is between points A and C
Select the postulate that proves this fact. If G and H are different points in plane R, then a third point exists in R not on .
Postulate 1a: A plane contains at least three points not all on one line.
Select the postulate that specifies the minimum number of points in space.
Postulate 1b: Space contains at least four points not all on one plane.
Select the postulate that states a line is determined by two points.
Postulate 2: Through any two different points, exactly one line exists.
Select the postulate that states points A and B lie in only one line.
Postulate 2: Through any two different points, exactly one line exists.
A table with four legs will sometimes wobble if one leg is shorter than the other three, but a table with three legs will not wobble. Select the postulate that substantiates this fact.
Postulate 3: Through any three points that are not one line, exactly one plane exists.
State the postulate that verifies [RAY] AB is in plane Q when points A and B are in Q.
Postulate 4: If two points lie in a plane, the line containing them lies in that plane.
Select the postulate about two planes.
Postulate 5: If two planes intersect, then their intersection is a line.
State a true conclusion. 1) If B is between A and C, then AB + BC = AC. 2) Q is between R and S. Conclusion:
RQ+QS=RS
Which of the following in NOT an essential part of a two-column proof? Statement of the Theorem Figure Given information Plan of proof Staircase logic
Staircase logic
State a true conclusion. 1) If a figure had four equal sides, then it is a square or a rhombus. 2) This figure has four equal sides. Conclusion:
The figure is either a square or a rhombus
State a true conclusion. 1) If two planes intersect, their intersection is a line. 2) Plane R and plane S intersect. Conclusion:
The intersection of plane R and S are a line
State a conclusion that seems reasonable. You find a nest with 12 eggs in it. The first 5 hatch out to be snakes. Conclusion:
The rest of the eggs will also hatch out to be snakes.
T/F A line and a point not on the line lie in exactly one plane.
True
T/F A line is imaginary.
True
T/F A point has no physical characteristics.
True
State a conclusion that seems reasonable. A teacher gave all her classes a true-false test on seven consecutive Mondays. Today is Monday. Conclusion:
There are true-false tests every Monday
State a true conclusion. 1) If a triangle has a right angle, then the triangle is a right triangle. 2) Triangle ABC is not a right triangle. Conclusion:
Triangle ABC does not have a right angle
T/F Select true or false to tell whether the following conditional p q is true or false. Use the truth table if needed. If A and B are the names of two points, then is the name of a line through A and B.
True
T/F The "given" information should be considered as true.
True
T/F The diameter is the longest chord that can be drawn in a circle.
True
T/F The three undefined terms in geometry are point, line, and plane.
True
T/F There are an infinite number of points that lie on a period at the end of a sentence.
True
T/F To prove a statement in geometry means to demonstrate that the statement follows logically from other accepted statements.
True
T/F Two lines intersect in one point
True
T/F Using the two given statements, is the conclusion true or false ? 1.) Three noncollinear points determine a plane. 2.) Points S, O, N are noncollinear. Conclusion: The points S, O, N form a plane.
True
T/F We think of a line as a collection of points.
True
T/F When developing a plan for a proof, it is important to look at the "given" information.
True
What type of proof is used extensively in geometry?
Two-column
S, U, N are collinear. Select the point that satisfies the definition of betweenness if NU + US = NS. N S U
U
State a conclusion that seems reasonable. You eat a new kind of fruit and suffer from hives for the first time in your life. A week later, you try the fruit and the hives recur. A month later, you have a similar experience. Conclusion:
You are allergic to the new kind of fruit
State a true conclusion. 1) All rabbits like lettuce. 2) My pet is a rabbit. Conclusion:
Your pet likes lettuce
State a true conclusion. 1) If people live in Zook, then they live in Zee. 2) Zela lives in Zook. Conclusion:
Zela lives in Zee
From the figure and statement provided, select the proper TO PROVE statement. If two parallel planes are cut by a third plane, the lines of intersection are parallel.
a is parallel to b
A point represents: line segment a location intersection of a plane none of the above
a location
A line can be named by using: a number a capital letter a lowercase letter none of the above
a lowercase letter
Which of the following would represent a line in the real world? an airplane a sheet of paper a pencil the Internet
a pencil
The top of a desk would be a representation of: a point a plane a line none of the above
a plane
Which of the following would represent a plane in the real world? an airplane a sheet of paper a pencil the Internet none of the above
a sheet of paper
Consider this equation: 2x + 2 = 11 - x Now consider the equation written as: 3x + 2 = 11 Which of the following is the correct property of equality that justifies rewriting the equation? addition subtraction division
addition
A point has: no shape no color no size no physical characteristics all of the above
all of the above
The expression "an established fact" refers to: given information postulates definitions a theorem that has already been proven all of the above
all of the above
Which of the following can be used as "reasons" in a two-column proof? given information postulates definitions previously proven theorems all of the above
all of the above
Which of the following is a characteristic of a plane? does not have thickness is not visible does not have any physical properties all of the above
all of the above
If C is between A and B then AC + CB = AB
always
In a proof the figure should fit the hypothesis.
always
The conclusion of a statement is the then part.
always
The hypothesis of a statement is the if part.
always
Three points are coplanar. always sometimes never
always
Two planes intersect in a line. always sometimes never
always
Where p and q are statements, "p and q" is false if p is false.
always
In Geometry, "line" is _____.
an undefined term
If p∨q is true, then what must be true about the truth values of p and q?
at least one is true
Planes A and B intersect in line s. If point V is a point on line s, then it lies on: plane A only plane B only both plane A and B neither plane A or B
both plane A and B
You ________ always prove a conclusion by inductive reasoning. can cannot must should none of the above
cannot
Points are usually labeled as: numbers capital letters lower case letters Roman numerals none of the above
capital letters
When two statements are connected with the word and, the new statement is called a:
conjunction
Where p and q are statements, p q is called the _________ of p and q.
conjunction
Select inductive reasoning, deductive reasoning, or neither. The ceiling and wall of a room meet in a line segment.
deductive reasoning
Select all that apply. What types of statements can be used to support conclusions made in proving statements by deductive reasoning? definitions axioms postulates previously proved theorems logic
definitions postulates previously proved theorems
When two statements are connected with the word or, the new statement is called a:
disjunction
Where m and n are statements m V n is called the _____ of m and n.
disjunction
A point is represented by a: line dot square none of the above
dot
Given the following statements determine whether the conditional statement is true or false p: 10 > 7 q: 10 > 5 q -> ~p
false
Given the following statements determine whether the conditional statement is true or false. p: 10 > 7 q: 10 > 5 p -> ~q
false
If a statement is true, then its negation is _______.
false
T/F An undefined term cannot be used in a theorem
false
When drawing a figure, it is best to draw the most __________ figure, unless given other information.
general
The "if" in the "if-then" statement is the:
given information
Let U be the set of students in a high school. The school has 800 students with 20 students on the gymnastic team and 10 students on the chess team. Select the Venn diagram if three students are on both teams.
https://cciu24.owschools.com/media/g_geo_2015/1/m80129b.gif
The given information is the _____ of the if-then statement.
if
The transitive property of equality states that:
if a = b and b = c, then a = c
Write the following statement in if - then form. Two opposite rays form a straight line. Which of the following is the hypothesis? if two rays are opposite if a straight line is formed then rays are opposite
if two rays are opposite
The Statement of Theorem should be in what form?
if-then
A point is _____.
imaginary
Reaching a conclusion by looking at several examples is called _______ reasoning.
inductive
Select inductive reasoning, deductive reasoning, or neither. If Mary is older than Bill and Bill is older than Frank, then Mary is older than Frank. Inductive reasoning Deductive reasoning neither
inductive reasoning
How many points are contained on a line? 20,000 none 1 infinitely many
infinitely many
Which of the following is a characteristic of a line? its length can be measured it can be seen its width can be measured it is infinite in length
it is infinite in length
Which of the following does NOT describe a line? it goes on forever it is imaginary it is typically named by a Roman numeral it has no physical characteristics
it is typically named by a Roman numeral
Which of the following is NOT a characteristic of a plane? it cannot be seen it is imaginary its width can be measured it is infinite none of the above
its width can be measured
Opposite rays form a: point ray line plane
line
Select all that apply. Which of the following are the building blocks of geometry, according to Hilbert? ray line angle point plane
line point plane
From the figure and statement provided, select the proper TO PROVE statement. Through a point outside a line one line can be drawn parallel to the line.
m is parallel to l
Select inductive reasoning, deductive reasoning, or neither. A courtroom spectator merely looks at the defendant and says, "He's guilty, I tell you." Inductive reasoning Deductive reasoning neither
neither
Points A and B lie in plane R. Line AB does not lie in plane R. always sometimes never
never
Select whether the following statement is always true, sometimes true, or never true. A proof should have more steps in the reason column than steps in the statement column.
never
Two planes intersect in exactly one point.
never
State a true conclusion. 1) If you accept the hypothesis, then you accept the conclusion. 2) You accept the conclusion. Conclusion:
no conclusion can be made
Points have the following physical properties: depth height thickness none
none. points have no physical properties
If points S, O, and N are collinear, how many lines do they determine? one two three four
one
p -> q is false when
p is true and q is false
All of the following are defined terms except: ray angle plane segment
plane
A plane is a set of ____________ on a flat surface that extends forever. lines points other planes none of the above
points
A statement we accept as true without proof is a:
postulate
The converse of p -> q is:
q -> p
What conclusion can be based on the given statement? q is a whole number between 42.3 and 43.1 q = 42.1 q = 43.0 q = 43.4
q = 43.0
Which of the following postulates states that a quantity must be equal to itself? symmetric identity reflexive closure
reflexive
If C is between A and B, then AC = CB. always sometimes never
sometimes
If points A, B, and C lie on a line, then AB + BC = AC. always sometimes never
sometimes
Select whether the following statement is always true, sometimes true, or never true. In the plan of a proof, you should use the plan that was used on previous theorems.
sometimes
Three points are collinear.
sometimes
Unless otherwise indicated, a line in geometry is understood to be _____. straight curved either
straight
Which of the following is the conclusion of the statement "An angle has exactly one bisector"?
the it has exactly one bisector
An indirect proof assumes:
the negative of the conclusion is true
The "to prove" part of the information is the _____ of the "if-then" statement.
then
Select a counter-example that makes the conclusion false. You select three marbles from a bag and each of them is black. Conclusion: All the marbles in the bag are black. Counterexample:
there is a non-black marble in the bag
Given the following statements determine whether the conditional statement is true or false. p: 10 > 7 q: 10 > 5 ~p -> q
true
Given the following statements determine whether the conditional statement is true or false. p: 10 > 7 q: 10 > 5 ~p -> ~q
true
Given the following statements determine whether the conditional statement is true or false. p: 10 > 7 q: 10 > 5 ~q -> ~p
true
T/F Deductive reasoning is the process of making a conclusion by fitting a specific example into a general statement.
true
T/F Order is important in a two-column proof.
true
True or False? The following is an example of inductive reasoning: In your study of geometry, you notice that every rectangle you have seen is also a square. You conclude that this is true in all cases.
true
True or False? The following is an example of inductive reasoning: In your study of geometry, you notice that every square you have seen is also a rectangle. You conclude that this is true in all cases.
true
True or False? The following is an example of inductive reasoning: Out your window you notice that seven elderly people have walked by on the sidewalk. You conclude that only elderly people live in your neighborhood.
true
True or False? The following is an example of inductive reasoning: You observe that it has rained on the past three Tuesdays and you conclude that it rains on every Tuesday.
true
t/f An indirect proof has been successful if you reach a contradiction of a known fact.
true
t/f Indirect proofs are often used to prove theorems.
true
What type of proof is used extensively in Geometry?
two-column proof
What conclusion can be based on the given statement? x is a number such that 7x = 7y x = y x = 1 y = 1
x = y
State a true conclusion. 1) If a · b = 0, then a = 0 or b = 0 2) 3x = 0 Conclusion:
x=0
What conclusion can be based on the given statement? y is a number such that 6y = 12 y = 1/2 y = 2 y = 72
y = 2
Form the intersection for the following sets. E = {1, 2, 3} F = {101, 102, 103, 104} E ∩ F = { } {1, 2, 3} {101, 102, 102, 103, 104} {1, 2, 3, 101, 102, 103, 104}
{ }
Form the intersection for the following sets. M = { } N = {6, 7, 8, 9, 10} M ∩ N = {0, 6, 7, 8, 9, 10} {Ø, 6, 7, 8, 9, 10} {6, 7, 8, 9, 10} { }
{ }
Form the union for the following sets. X = {0, 10, 100, 1000} Y = {100, 1000} X ∪ Y = { } {0, 10} {100, 1000} {0, 10, 100, 1000}
{0, 10, 100, 1000}
Form the union for the following sets. E = {1, 2, 3} F = {101, 102, 103, 104} E ∪ F = {1, 2, 3, 101, 102, 103, 104} {1, 2, 3} {101} { }
{1, 2, 3, 101, 102, 103, 104}
Form the union for the following sets. R = {10, 15, 20} S = {20, 25} R ∪ S = { } {20} {10, 15, 25} {10, 15, 20, 25}
{10, 15, 20, 25}
Form the intersection for the following sets. X = {0, 10, 100, 1000} Y = {100, 1000} X ∩ Y = {0, 10, 100, 1000} {100, 1000} {0, 10} { }
{100, 1000}
Given the following Venn diagram, choose the correct set for the following: https://cciu24.owschools.com/media/g_geo_2015/1/m80130a.gif Universal Set = ____. {3, 4, 6, 7, 9} {2, 3, 4, 5, 6, 7, 8, 9} {2, 3, 5, 6, 8, 9}
{2, 3, 4, 5, 6, 7, 8, 9}
Given the following Venn diagram, choose the correct set for M ∪ N. https://cciu24.owschools.com/media/g_geo_2015/1/m80130a.gif {2, 5, 8} {2, 4, 6, 8} {3, 4, 6, 7}
{2, 5, 8}
Form the intersection for the following sets. R = {10, 15, 20} S = {20, 25} R ∩ S = { } {10, 15, 25} {20} {10, 15, 20, 25}
{20}
Given the following Venn diagram, choose the correct set for the following: https://cciu24.owschools.com/media/g_geo_2015/1/m80130a.gif M ∪ N = ____. {3, 4, 6, 7, 9} {3, 4, 5, 6, 8, 9} {2, 4, 6, 8}
{3, 4, 6, 7, 9}
Given the following Venn diagram, choose the correct set for the following: https://cciu24.owschools.com/media/g_geo_2015/1/m80130a.gif M ∩ N = ____. {3, 9} {2, 5, 8} {4, 7}
{4, 7}
