Download Prove It! - Harding Charter Preparatory High School

Developing Flow Proofs
A Geometric Activity
High School Geometry
By Mrs. C. Henry
COMPARING TYPES OF PROOFS
TYPES OF PROOFS
Three types of proofs:
3.2
1)
TWO-COLUMN PROOF is the most formal type. Lists numbered
statements in the left column and the reason for each statement
in the right column.
2)
PARAGRAPH PROOF describes the logical argument with
sentences. It is more conversational than a two-column proof.
3)
FLOW PROOF uses the same statements and reasons as a twocolumn proof, but the logical flow connecting the statements is
indicated by arrows.
Proof and Perpendicular Lines
Flow Proof
GIVEN
PROVE
5 and 6 are a linear pair.
6 and 7 are a linear pair.
5
7
5
6
7
5 and 6 are a
linear air.
Linear Pair Postulate
Given
5  7
Congruent
6 and 7 are a
linear pair.
Given
3.2
Supplements
Theorem
Linear Pair Postulate
Proof and Perpendicular Lines
IDENTIFYING ANGLES
FLOW PROOF CARDS
Neatly write the following terms on index cards. Use these terms and
others to create your Flow Proof.
Reflexive Property
Substitution
Δ Sum Theorem
Bisector Theorem
 Bisector Theorem
Corresponding angles
SSS
Alternate Exterior angles
SAS
Alternate Interior angles
ASA
Consecutive Interior angles
AAS
Vertical angles
HL
Supplementary angles
Linear pair
3.1
Prove It!
GIVEN: AC is  bisector of BD.
PROVE: ΔABC  ΔADC
A
B
C
D
Prove It!
GIVEN: AC is  bisector of BD.
PROVE: ΔADM  ΔABM
A
M
B
C
D
Prove It!
GIVEN: AC is  bisector of BD.
PROVE: ΔHGF  ΔHEF
F
E
G
H
Prove It!
PROVE: ΔFGE  ΔJGH
F
G
E
Prove It!
PROVE: ΔLMN  ΔNPL
L
M
P
N
Prove It!
GIVEN: Point L bisects KM.
 KLJ   MLN.
J
N
PROVE: ΔJKL  ΔNML
K
L
M
Prove It!
GIVEN: GH ‖ FE.
PROVE: ΔGHJ  ΔEFJ
Prove It!
GIVEN: AD & CD are  bisectors.
m ABC = 100
PROVE: m ADC = 140.
B
D
A
C
Prove It!
PROVE: ΔAEB  ΔDEC
A
B
E
C
D
Proving Two Triangles are Congruent
Prove that
AEB 
DEC .
A
B
E
SOLUTION
D
C
Statements
Reasons
AB || DC , AB  DC
Given
 EAB   EDC,
 ABE   DCE
Alternate Interior Angles Theorem
 AEB   DEC
Vertical Angles Theorem
E is the midpoint of AD,
E is the midpoint of BC
Given
AE  DE , BE  CE
Definition of midpoint
AEB 
DEC
Definition of congruent triangles
Example
Using the Third Angles Theorem
Find the value of x.
SOLUTION
In the diagram,  N   R and  L   S.
From the Third Angles Theorem, you know that  M   T. So, m M = m T.
From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚.
m M = m T
60˚ = (2 x + 30)˚
Third Angles Theorem
Substitute.
30 = 2 x
Subtract 30 from each side.
15 = x
Divide each side by 2.
Example
Proving Triangles are Congruent
Decide whether the triangles are congruent. Justify your reasoning.
SOLUTION
Paragraph Proof
From the diagram, you are given that all three corresponding sides are congruent.
RP  MN, PQ  NQ , and QR  QM
Because  P and  N have the same measures,  P   N.
By the Vertical Angles Theorem, you know that  PQR   NQM.
By the Third Angles Theorem,  R   M.
So, all three pairs of corresponding sides and all three pairs of
corresponding angles are congruent. By the definition of congruent
triangles, PQR  NQM .
Goal 2
Flow Proof
GIVEN
PROVE
5 and 6 are a linear pair.
6 and 7 are a linear pair.
5
5 and 6 are a
linear pair.
6
7
5 and 6 are
supplementary.
Linear Pair Postulate
Given
7
5
5  7
Congruent
6 and 7 are a
linear pair.
Given
3.2
6 and 7 are
supplementary.
Linear Pair Postulate
Proof and Perpendicular Lines
Supplements
Theorem