Download Date: Geometry Unit 3 Day 4 Introduction to Proofs Wha

Name: ____________________________________ Date: __________________________
Geometry Unit 3 Day 4
Introduction to Proofs
What is a proof?
A proof is a convincing argument that something is true. In mathematics, a proof starts with things
that are agreed upon, called postulates or axioms, and then uses logic to reach a conclusion.
Conclusions are often reached in geometry by observing data and looking for patterns. This type of
reasoning is called inductive reasoning. The conclusions reached by inductive reasoning is called a
conjecture.
A proof in geometry consists of a sequence of statements, each supported by a reason, that starts
with a given se of premises and leads to a valid conclusion. This type of reasoning is called
deductive reasoning. Each statement in a proof follows from one or more of the previous
statements. A reason for a statement can come from the set of given premises or from one of the four
types of other premises: definitions; postulates; properties of algebra, equality, or congruence; or
previously proven theorems. Once a conjecture is proved, it is called a theorem. As a theorem, it
becomes a premise for geometric arguments you can use to prove other conjectures.
The four common methods of geometric proofs are:
1) Paragraph proofs – a proof in which the statements and their justifications are written
together in logical order in complete sentences. There is always a diagram and a statement of
the given and prove sections before the paragraph.
2) Two-column proofs - a proof in which the statements are listed in logical order in one column,
and the reason each statement is true is in another column. The last statement is always what
is proven.
3) Flow chart proofs - a concept map that shows the statements and reasons needed for a proof
in a structure that helps indicate the logical order. Statements, written in the logical order, are
placed in the boxes. The reason for each statement is placed under that box.
4) Coordinate proofs - a proof involving midpoints, slopes, and distances on the coordinate
plane
Think of proofs like a game. The object of the proof game is to have all the statements in your chain
linked so that one fact leads to another until you reach the prove statement. However, before you
start playing the proof game, you should survey the playing field (your figure), look over the given and
the prove parts, and develop a plan on how to win the game.
Here are some important theorems we will prove before using them in formal proofs.

Vertical Angle Theorem: If two angles are vertical angles, then they have equal measures.

Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.

Complements of the Same Angele Theorem: If two angles are complements of the same
angle, then they have equal measures.
Angles 1 and 3 are both complements to
angle 2, therefore, they have the same
measure.

Supplements of the Same Angle Theorem: If two angles are supplements of the same
angle, then they have equal measures.
Angles 1 and 3 are both supplements to
angle 2, therefore, they have the same
measure.

Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the
alternate interior angles
have
equal measures.
Converse of Alternate Interior Angles Theorem: If
two coplanar lines are cut by a transversal so that two
alternate interior angles have the same measure, then
the lines are parallel.

Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then the
corresponding angles have equal measures.
Converse of Corresponding Angles Theorem: If two
coplanar lines are cut by a transversal so that two
corresponding angles have the same measure, then
the lines are parallel.
What does the word converse mean in the above context?
Vertical Angle Theorem
If two angles are vertical angles, then they have equal measures.
Given: Lines AB and CD intersect at E
Prove:  AEC   BED &  AED   BEC
For this proof, we will use transformations,
specifically, a rotation. Using patty paper, trace the
intersecting lines and points. Then rotate  AEC to
map onto  BED. On the diagram, mark and label
C’ and A’. Then do the same for  AED.
Paragraph Proof
A rotation of ______about point ______ will map point A onto EB such that A will lie on EB since we are
dealing with straight segments. It will also map point C onto _______such that C will lie on ________. The
rotation will create  A’EC’, which will be congruent to  ________ since they are the same angles with the
same sides (rays) and the same vertex. Since  A’EC’ is a 180o rotation of  _______ about point _______,
 A’EC’   AEC since rotations are rigid transformations which preserve ____________________________
and congruent angles are equal in measure.  AEC   BED by the __________________ property of
congruence (or substitution). The same argument will apply to proving  AED   BEC.
Two Column Proof
Statements
1) Lines AB and CD intersect at E
Reasons
1)
2)  1,  2 &  3,  4 &  1,  4 form linear
pairs
2) A linear pair is
3)  1,  2 &  3,  4 &  1,  4 are
supplementary
3)
4) m  1 + m  2 = 180
m  3 + m  4 = 180
m  1 + m  4 = 180
5) m  1 + m  2 = m  1 + m  4
m1 + m4 = m 3 + m 4
4) Supplementary angles are
6) m  1 = m  1; m  4 = m  4
6)
7) m  2 = m  4; m  1 = m  3
7)
8)  2   4;  1   3 or
8)
5)
 AEC   BED &  AED   BEC
Complements of the Same Angele Theorem
If two angles are complements of the same angle, then they have equal measures.
Given:  1 and  2 are complementary
 2 and  3 are complementary
Prove:  1   3
Flow Chart Proof
1 and 2 are
complementary
2 and 3 are
complementary
m
m
1+m
2+m
2 =90
3 = 90
m
1+m
2=
m
2+m
3
m
2=m
2
m
1=m
3
1
3
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary
Given:  1 and  2 form a linear pair
Prove:  1 and  2 are supplementary
Two Column Proof
Statements
1)  1 and  2 form a linear pair
2)
BA and BC are opposite rays
Reasons
1)
2)
3)  ABC is a straight angle
3)
4) m  ABC = 180o
4)
5) m  1 + m  2 = m  ABC
5)
6) m  1 + m  2 = 180o
6)
7)  1 and  2 are supplementary
7)
Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the corresponding angles have equal measures
Given:
AD ║ BC
Prove:  ABC   EAD
Paragraph Proof
After a translation of  ABC by vector
map onto
BA , B’ will map onto A, A’ will map onto AE , and C’ will
AD .  A’BC’   ABC since a translation is a rigid transformation that preserves
________________________________ and angles are congruent if they have equal measure.
 A’BC’   EAD since they are the same angles with the same sides (rays) and the same vertex.
 ABC   EAD by the _____________________ property of congruence (or substitution).
Teacher’s Copy
Vertical Angle Theorem
If two angles are vertical angles, then they have equal measures.
Given: Lines AB and CD intersect at E
Prove:  AEC   BED &  AED   BEC
For this proof, we will use transformations,
specifically, a rotation. Using patty paper, trace the
intersecting lines and points. Then rotate  AEC to
map onto  BED. On the diagram, mark and label
C’ and A’. Then do the same for  AED.
Paragraph Proof
A rotation of 180o about point E will map point A onto EB such that A will lie on EB since we are
dealing with straight segments. It will also map point C onto ED such that C will lie on ED . The
rotation will create  A’EC’, which will be congruent to  BED since they are the same angles with the
same sides (rays) and the same vertex. Since  A’EC’ is a 180o rotation of  AEC about point E
 A’EC’   AEC since rotations are rigid transformations which preserve angle measure
and congruent angles are equal in measure.  AEC   BED by the transitive property of
congruence (or substitution). The same argument will apply to proving  AED   BEC.
Two Column Proof
Statements
1) Lines AB and CD intersect at E
1) Given
Reasons
2)  1,  2 &  3,  4 &  1,  4 form linear
pairs
2) A linear pair is a pair of adjacent angles that
form a straight line.
3)  1,  2 &  3,  4 &  1,  4 are
supplementary
3) Angles that form a linear pair are
supplementary
4) m  1 + m  2 = 180
m  3 + m  4 = 180
m  1 + m  4 = 180
5) m  1 + m  2 = m  1 + m  4
m1 + m4 = m 3 + m 4
4) Supplementary angles are two angles the
sum of whose measures is 180o
6) m  1 = m  1; m  4 = m  4
5) The transitivity of congruence. (Quantities
equal to the same quantity are equal to each
other)
6) Reflexive Property (Quantity is = to itself)
7) m  2 = m  4; m  1 = m  3
7) Subtraction Property of Equality
8)  2   4;  1   3 or
 AEC   BED &  AED   BEC
8) Congruent angles are angles equal in
measure
Complements of the Same Angele Theorem
If two angles are complements of the same angle, then they have equal measures.
Given:  1 and  2 are complementary
 2 and  3 are complementary
Prove:  1   3
Flow Chart Proof
1 and 2 are
complementary
GIVEN
2 and 3 are
complementary
GIVEN
m
1+m
2 =90
Complementary
angles are two
adjacent angles
whose sum is 90o
m
2+m
3 = 90
Complementary
angles are two
adjacent angles
whose sum is 90o
m
1+m
2=
m
2+m
3
Transitive
Property of
Equality
m
2=m
Reflexive
Property
2
m
1=m
Subraction
Property of
Equality
3
1
3
Congruent angles
have equal
measures
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary
Given:  1 and  2 form a linear pair
Prove:  1 and  2 are supplementary
Two Column Proof
Statements
1)  1 and  2 form a linear pair
2)
BA and BC are opposite rays
Reasons
1) Given
2) Linear pairs are formed by two adjacent
angles whose noncommon sides are
opposite rays
3)  ABC is a straight angle
3) Opposite rays form a straight angle
4) m  ABC = 180o
4) A straight angle measure 180o
5) m  1 + m  2 = m  ABC
5) Angle Addition Postulate
6) m  1 + m  2 = 180o
6) Transitive Property of Equality
(Substitution)
7)  1 and  2 are supplementary
7) Supplementary angles are two angles the
sum of whose measures is 180o
Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the corresponding angles have equal measures
Given:
AD ║ BC
Prove:  ABC   EAD
Paragraph Proof
After a translation of  ABC by vector
map onto
BA , B’ will map onto A, A’ will map onto AE , and C’ will
AD .  A’BC’   ABC since a translation is a rigid transformation that preserves
angle measure and angles are congruent if they have equal measure.
 A’BC’   EAD since they are the same angles with the same sides (rays) and the same vertex.
 ABC   EAD by the transitive property of congruence (or substitution).