Download Lesson 15

Algebra III
Lesson 15
Assumptions - Proofs
Assumptions
Postulates and Axioms
- An assumption accepted without proof
Postulate
- An assumption of geometry
Axiom
- An assumption for all of mathematics
The First Postulates
By Euclid about 300 B.C.
Postulate 1) Two points determine a unique straight line.
2) A straight line extends indefinitely far in either direction.
3) A circle may be drawn with any given center and any given
radius.
4) All right angles are equal.
5) Given a line n and a point P not on that line, there exists in the
plane of P and n and through P one and only one line m, which
does not meet the given line n.
P
P
n
m
n
Postulate 6) Things equal to the same thing are equal to each other.
7) If equals be added to equals, the sums are equal.
8) If equals be subtracted from equals, the remainders are
equal.
9) Figures which can be made to coincide are equal.
10) The whole is greater than any part.
Use these postulates with the following postulates to create proofs:
SSS, SAS, ASA, HL, Parallel line relationships, Inscribed angle, …
Proofs
- The method of showing a conclusion to be true.
- The first line of the proof is the givens. The last line is
the thing to be proven.
- Make notes of the major steps that must be done to get
the job done.
- If unsure of the path from beginning to end, try figuring
out what has to true for the conclusion to be true. Work
backwards.
- We will be using the Two Column Proof format.
The Two Column Proof Format
Givens: ---------------
Drawings
Prove: ----------
Statements
1)
-------------------
Reasons
1) Given
2)
2)
3)
3)
....
....
n) Thing to be proven
n) Reason
Example 15.1
A median of a triangle is a segment that connects a vertex with the midpoint
of the side opposite the vertex. Prove that the median drawn from the
vertex formed by the equal sides of an isosceles triangle bisects the angle at
this vertex.
B
Givens: ∆ABC is isosceles.
BD is a median.
Prove:
BD bisects ∠ABC .
Statements
1)
? ?
∆ABC is isosceles.
BD is a median.
Reasons
1) Given
A
2) AB ≅ CB
2) Def. of Isosceles ∆
3) AD ≅ DC
3) Def. of median
4) BD ≅ BD
4) Reflexive
∆ABD ≅ ∆CBD
6) ∠ABD ≅ ∠CBD
5) SSS
7) BD bisects ∠ABC
7) Def. of Bisect
5)
6) CPCTC
D
Notes
Mark triangle
∆ABD ~ ∆CBD
CPCTC
C
Example 15.2
Prove that if two tangents to a circle intersect at a point outside the circle
then the lengths of the tangent segments are equal.
A
Givens: PA & PB are tangent to
circle O
Statements
Reasons
1) PA & PB are tangent
to circle O
1) Given
2) Draw OA , OB , & PO
2) Euclid’s 1st Postulate
3) OA ⊥ PA; OB ⊥ PB
3) Tangent-Radius Properties
4) OA ≅ OB
4) Radii if a circle are congruent
5)
PO ≅ PO
5) Reflexive
6) Def. of perpendicular lines.
7) ∆OAP & ∆OBP are
right triangles
7) Def. of right triangles
∆OAP ≅ ∆OBP
B
Notes
6) ∠OAP & ∠OBP are right angles.
8)
P
O
Prove: PA = PB
8) HL Postulate
9) PA ≅ PB
9) CPCTC
10) PA = PB
10) Def. of Congruent
Make triangles
∆PAO ≅ ∆PBO
CPCTC
Example 15.3
Prove that the sum of the measures of the angles of a triangle is 180°.
Givens: ∆ABC
1
Prove: Sum of angles in
∆ABC = 180°.
B
2
3
C
A
Statements
Reasons
1) ∆ABC
1) Given
2) Draw line parallel to AC
through B
2) Euclid’s 5th Postulate
3) m∠1 + m∠2 + m∠3 = 180°
3) Def. of straight line
4) ∠2 = ∠B
4) Two names for same angle (Reflexive)
5)
∠1 = ∠A; ∠3 = ∠C
6) m∠A + m∠B + m∠C = 180°
5) Alternate Interior Angles of Parallel Lines
6) Substitution
Notes
A + B + C = 180
Straight line = 180
Parallel lines
Substitute
Example 15.4
Use the sixth and eighth postulates of Euclid to prove that vertical angles
are equal in measure.
Givens:
∠A & ∠C 
 are vertical angles.
∠B & ∠D
B
Prove: m∠A = m∠C
Statements
1) ∠A & ∠C are
vertical angles.
2)
m∠A + m∠B = 180°
m∠C + m∠B = 180°
3) m∠A + m∠B = m∠C + m∠B
4)
m∠A = m∠C
C
A
D
Reasons
1) Given
2) Def. of straight line
3) Euclid’s 6th Postulate (Substitution)
4) Euclid’s 8th Postulate; subtract
from both sides
Notes
Straight lines
Example 15.5
B
Givens: BD is the angle bisector of angle B.
AB ≅ CB
Prove: AD ≅ CD
Statements
1) BD is the angle
bisector of angle B.
Reasons
1) Given
∠ABD ≅ ∠CBD
3) BD ≅ BD
4)
∆ABD ≅ ∆CBD
5) AD ≅ DC
D
C
Notes
AB ≅ CB
2)
A
2) Def. of bisector
3) Reflexive
4) SAS
5) CPCTC
Mark triangle
∆ABD ~ ∆CBD
CPCTC
Example 15.6
Q
Givens: ∠Q ≅ ∠S
R
PQ ║ SR
Prove: ∆PQR ≅ ∆RSP
P
Statements
1) ∠Q ≅ ∠S
Reasons
1) Given
PQ ║ SR
2) ∠QPR ≅ ∠SRP
∠QRP ≅ ∠SPR
3) PR ≅ PR
4)
∆PQR ≅ ∆RSP
S
Notes
2) Alternate Interior Property
of Parallel Lines
3) Reflexive
4) ASA
Alt. Int. angles of parallel
ASA
Practice
D
A
a)
B
Givens: AC ≅ CB
∠ACD ≅ ∠BCD
Prove: ∆ACD ≅ ∆BCD
C
Statements
1) AC ≅ CB
Reasons
1) Given
∠ACD ≅ ∠BCD
2) CD ≅ CD
3)
∆ACD ≅ ∆BCD
2) Reflexive
3) SAS
Notes
CD = CD
SAS
A
b)
Givens:
AB ≅ AD
BC ≅ DC
Prove: ∆ABC ≅ ∆ADC
D
B
C
Statements
1) AB ≅ AD
Reasons
1) Given
BC ≅ DC
2) AC ≅ AC
3)
∆ABC ≅ ∆ADC
2) Reflexive
3) SSS
Notes
AC = AC
SSS
c)
a
(b)
2 z −3 4
4
4 z +8
a 4 z + 2b 2 z −3
Do a’s first
b’s second
a 2 z −3 a − ( 4 z + 2 )
b 4 z +8b − ( 2 z − 3 )
a 2 z − 3− ( 4 z + 2 )
b 4 z +8 − ( 2 z −3 )
a 2 z − 3− 4 z − 2
b 4 z − 2 z +8+ 3
a −2 z −5
b 2 z +11
a −2 z −5 b 2 z +11