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Regents Review #1
ESSENTIALS OF GEOMETRY:
Point - may be represented by a dot on a piece of paper, and is usually named by a capital
letter. A point has no length, width, or thickness. It only indicates a position.
Line – an infinite set of points extending endlessly in both directions.
Plane – a set of points that form a flat surface extending indefinitely in all directions.
Congruent – Equal in measure (  )
Line Segment – a set of points consisting of two points on a line, called endpoints, and all of
the points on the line between the endpoints. *[name with at least 2 letters]*
Midpoint – the point of that line segment that divides the segment into two congruent
segments.
Bisector of a line segment – any line, or subset of a line, that intersects the segment at its
midpoint.
Ray – a ray consists of a point on a line and all points on one side of that point.
Angle – the union of two rays having the same endpoint. *[name with 3 letters  ]*
Straight Angle – an angle whose degree measure is 180˚
Acute Angle – an angle whose degree measure is greater than 0˚ and less than 90˚
Right Angle – an angle whose degree measure is 90˚
Obtuse Angle – an angle whose degree measure is greater than 90˚ and less than 180˚
Angle Bisector – a ray whose endpoint is the vertex of the angle, and that divides that
angle into two congruent angles.
Perpendicular Lines – two lines that intersect to form right angles. (  )
Parallel Lines – two lines, both in the same plane, that never intersect. ( || )
Adjacent Angles – two angles, in the same plane, that have a common vertex and a common
side.
Complementary Angles – two angles, the sum of whose degree measure is 90˚
Supplementary Angles – two angles, the sum of whose degree measure is 180˚
Linear Pair – two adjacent angles whose sum is 180˚ (a straight angle)
Vertical Angles – angles formed opposite each other when two line intersect.
Transversal - a line that intersects two other coplanar lines in two different points.
Parallel Lines:
If two lines are parallel then
-
alternate interior angles are congruent.
-
corresponding angles are congruent.
-
interior angles on the same side of the transversal are supplementary.
1
3 4
5 6
7 8
2
Alternate Interior Angles:
,
Corresponding Angles:
,
,
,
Interior Angles on the Same Side of the Transversal:
,
Vertical Angles:
,
,
,
Scalene Triangle – has no congruent sides
Acute Triangle – has 3 acute angles
Isosceles Triangle – has 2 congruent sides
Right Triangle – has 1 right angle
Equilateral Triangle – has 3 congruent
sides
Obtuse Triangle – have 1 obtuse angle
Equiangular Triangle – has 3 congruent
angles (all 60˚)
Isosceles Triangle:
A
Base Angles – the 2 congruent angles opposite the legs (B and C)
Vertex Angle – the angle formed by the 2 congruent sides, opposite the
base (A)
Side/Legs – the 2 congruent sides (AB and AC)
Base – the side opposite the vertex angle (BC)
B
C
LOGIC:
Logic is the science of reasoning. The principles of logic allow us to determine if a
statement is true, false, or uncertain on the basis of the truth of related statements.
Closed sentence (statement): a sentence that can be judged to be true or false.
Open sentence: a sentence that contains a variable and has no truth value.
The negation of a statement always has the opposite truth value of the original statement
(~)
*(add “not” to the sentence if it isn’t there and take away the “not” from the
sentence if it is there)*
A conjunction is a compound statement formed by combining two simple statements using
the word AND. ( )
*(Only true when both statements are true. T T = T)*
A disjunction is a compound statement formed by combining two simple statements using
the word OR. ( )
*(Only false when both statements are false. F V F = F)*
A conditional is a compound statement formed by using the words IF… THEN to combine
two simple sentences. ( )
-
The hypothesis comes after the IF part of the statement
-
The conclusion comes after the THEN part of the statement
*(Only false when a true is followed by a false. T F = F)*
The inverse of a conditional statement is formed by negating the
hypothesis and the conclusion.
The converse of a conditional statement is formed by switching
the position of the hypothesis and the conclusion.
The contrapositive is formed by negating and switching the
hypothesis and the conclusion.
The biconditional is a conjunction of a conditional and its converse. ( )
*(Only true when both statements have the same truth value.
T
T=T
F
F = T)*
INTRODUCTION TO PROOFS:
Reasoning:
Inductive reasoning is when a series of particular examples leads to a conclusions.
A proof in geometry is a valid argument that establishes the truth of a statement.
Deductive reasoning uses the laws of logic to link together true statements to arrive at a
true conclusion.
Direct Proof – Uses the laws of inference to link together true premises and statements
that lead DIRECTLY to a true conclusion.
Indirect Proof – The laws of inference are used to prove that a statement is false, leading
INDIRECTLY to the conclusion that the negation of the statement must be true.
Postulate – a statement whose truth is accepted without proof.
Theorem – a statement that has been proven.
Properties:
Reflexive Property: a quantity is equal to itself.
Symmetric Property: an equality may be written in either order.
-
If
, then
.
Transitive Property: if quantities are equal to the same quantity, they are equal to each
other.
-
If
and
, then
.
Postulates:
Substitution Postulate: a quantity may be substituted for its equal in any expression.
Addition Postulate: if equal quantities are added to equal quantities, the sums are equal.
-
If
and
, then
.
Subtraction Postulate: if equal quantities are subtracted from equal quantities, the
differences are equal.
-
If
and
, then
.
Multiplication Postulate: if equal quantities are multiplied by equal quantities, the products
are equal.
-
“doubles of equals are equal”
-
If
and
, then
.
Division Postulate: if equal quantities are divided by equal quantities, the quotients are
equal.
-
“halves of equals are equal”
-
If
and
, then
.
Rules for Isosceles Triangles:
1) If 2 sides of a Δ are , then base angles are .
2) If 2 sides of a Δ are , then the Δ is isosceles.
3) If 2 angles of a Δ are , then the opposite sides are .
4) If 2 angles of a Δ are , then the Δ is isosceles.
5) If a Δ is isosceles, then the base angles are .
6) If a Δ is isosceles, then the opposite sides are .
Angle Theorems:
If two angles are right angles then they are congruent. “All right angles are congruent”
If two angles are straight angles then they are congruent. “All straight angles are
congruent”
If two angles form a linear pair, then they are supplementary.
If two angles are vertical angles, then they are congruent. “Vertical angles are congruent”
If two angles are complements of the same angle, then they are congruent.
If two angles are supplements of the same angle, then they are congruent.
If two lines intersect to form congruent adjacent angles, then they are perpendicular.
If two angles are congruent, then their supplements are congruent. (HONORS)
If two angles are congruent, then their complements are congruent. (HONORS)
Tips for proving ∆’s congruent
1) Mark any given information on your diagram.
2) Look to see if the pieces you need are "parts" of the triangles that can be proven
congruent.
3) If not given all needed pieces to prove the triangles congruent, look to see what else
you might know about the diagram.
4) Know your definitions! If the given information contains definitions, consider these as
"hints" to the solution and be sure to use them.
5) Stay open-minded. There may be more than one way to solve a problem.
6) Look to see if your triangles "share" parts. These common parts are automatically one
set of congruent parts.
Segments of a Triangle:
A median of a triangle is a line segment that joins any vertex of a triangle to the midpoint
of the opposite side.
An altitude of a triangle is a line segment drawn from any vertex of the triangle,
perpendicular to and ending in the line that contains the opposite side.
Angle Bisector of a triangle is a line segment that bisects any angle of the triangle and
terminates in the side opposite that angle. An angle bisector of a triangle divides the
angle into two congruent angles.
A perpendicular bisector is any line or subset of a line that is perpendicular to the line
segment at its midpoint. **The perpendicular bisector does not have to terminate in the
side of the triangle.
CONGRUENT TRIANGLES:
SAS: Two triangles are congruent if two sides and the included angle of one triangle are
congruent to two sides and the included angle of another triangle. (side-angle-side)
ASA: Two triangles are congruent if two angles and the included side of one triangle are
congruent to two angles and the included side of another triangle. (angle-side-angle)
SSS: If three sides of one triangle are congruent to three sides of another triangle, the
triangles are congruent. (side-side-side)
AAS: Two triangles are congruent if two angles and the consecutive side of one triangle
are congruent to two angles and the consecutive side of another triangle. (angle-angleside)
HL: Two triangles are congruent if the hypotenuse and leg of one right triangle are
congruent to the hypotenuse and leg of another right triangle. (hypotenuse-leg) [need to
state that the triangles are right triangles first in a proof.]
*****ASS and AAA are NOT ways to prove two triangles are congruent*****
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Part I:
_____1) Which statement is logically equivalent to the statement “If it’s cold, then we go
skiing.”
(1) If we go skiing, then it is cold.”
(2) If we did not go skiing, then it is cold.
(3) If it is not cold, then we do not go skiing.
(4) If we do not go skiing, then it is not cold.
_____2) ∆ABC  ∆DEF with A  D and B  E. If BC = 8, which side of ∆DEF has
the same measure?
(1) ED
(2) EF
(3) DF
(4) None of the above
_____3) What is the converse of the statement “If x is even, then x + 1 is odd?”
(1) If x + 1 is odd, then x is even.
(2) If x is odd, then x + 1 is even.
(3) If x is not even then x + 1 is odd.
(4) If x is even, then x + 1 is odd.
_____4) If AB  AD it can be proved that BC  CD if it is also known that
(1) BAC  CAD
(2) B  D
(3) BAC  ACD
(4) A  A
_____5) State which method(s) can be used to prove the triangles congruent. If no
method applies, choose none.
(1) AAS
(2) ASA
(3) ASA and AAS
(4) none
_____6) What is the inverse of “If a quadrilateral is a square, then it has four right
angles”?
(1) If a quadrilateral is not a square, then it does not have four right angles.
(2) If a quadrilateral has four right angles, then it is a square.
(3) If a quadrilateral is not a square, then it has four right angles.
(4) If a quadrilateral does not have four right angles, then it is not a square.
B
_____7) In the figure given, if 1 2, then:
(1) BAC  BCA
(2) AE is a median
(3) AE is an altitude
(4) AE bisects <BAC
D
E
F
A
1
2
C
_____8) What is the value of x if line m is parallel to line k?
(1) 29
(2) 25
(3) 5
(4) 26.4
_____9) Two supplementary angles have measures in the ratio 2:7. What is the measure
of the smaller angle?
(1) 40˚
(2) 20˚
(3) 70˚
(4) 90˚
_____10) Given ∆KEY with KEY a right and angle and K Y, which of the following is
not a true statement?
(1) EK  EY
(2) KE  EY
(3) ∆KEY is an isosceles ∆
(4) ∆KEY  ∆YES
_____11) ∆GHI  ∆LMN. Which is not a congruence statement for these triangles?
(1) ∆HGI  ∆MLN
(2) ∆IHG  ∆MNL
(3) ∆HIG  ∆MNL
(4) ∆GIH  ∆LNM
In exercises 12-14: Refer to pairs I to IV of triangles.
II
I
III
IV
_____12) Which pair of triangles are congruent by SAS?
(1) I
(2) II
(3) III
(4) IV
_____13) Which pair of triangles are congruent by ASA?
(1) I
(2) II
(3) III
(4) IV
_____14) Which pair of triangles can not be shown to be congruent?
(1) I
(2) II
(3) III
(4) IV
Part II:
15) Given: Tenth graders are sophomores and a hamburger is a soft drink.
Determine the truth value of the compound sentence. Justify how you arrived at your
answer.
16) If BE bisects the mABD, mABE = (y – 8) and m<ABD = (5y – 100), find the value of y.
17) In ΔABC, AB  BC. If mA = (4x + 50), mB = (2x + 60), and mC = (14x + 30). Find
the value of x.
18) Given: Twelve is a prime number or 36 is a perfect square number.
Determine the truth value of the compound sentence. Justify how you arrived at your
answer.
Part III:
19) For the statement “If I don’t study, I will get poor grades.”
a) Write the converse.
b) Write the inverse.
c) Write the contrapositive.
d) Identify which statement is logically equivalent to the original statement.
20)
Given: AB  ED, C is midpoint BD, AB  BD; ED  BD
Prove: ∆ABC  ∆EDC
STATEMENT
21)
Given: C is the midpoint of AE, 1  2, 3  BCD, 4  BCD
Prove: AB  DE
REASON
STATEMENT
REASON
STATEMENT
REASON
B
A
1
D
3
C
4
2
E
Part IV:
22) Given: DF  BE, DC  AD, AB  AD, 1  2
Prove: ∆ABF  ∆CDE
A
B
2
F
E
1
D
C
23)
If CD  DE, 1  2, and DA bisects BDF, prove that AE  AC.
STATEMENT
A
F
E
B
REASON
C
2
1
D
24)
Given: EF  GF, DF  HF
Prove: 1  2
STATEMENT
H
D
2
1
C
G
E
F
REASON