Download 5.1 Indirect Proof

10/15/2014
Learning Scale
4
3
2
1
I can apply what I know about the relationships and properties of quadrilaterals consistently and could easily prove them in real world applications and feel confident I could even use them in proving broader relationships in other polygons.
I can apply what I know about the relationships and properties of quadrilaterals consistently and can consistently create correct proofs for those relationships and properties.
I can sometimes create correct proofs for those relationships and properties of quadrilaterals, but still need help to have a complete proof(missing 1 or 2 steps) I use the properties to set up the problem right but don’t always get the right answer yet.
I am beginning to see the relationships and properties of quadrilaterals, but I get stuck just past the given information in my proofs and I am usually missing more than 2 steps in my proofs on my homework and/or I need help setting up the problems applying properties.
5.1 Indirect Proof
From the given information we can prove the
triangles congruent by ASA, which makes
AC ≅ DF
But wait! This is not possible, since the given
information states that
AC ≅ DF
Therefore, our assumption was false
and we can now say ∠B ≅ ∠E ,
because this is the only other possibility.
Indirect Proof Procedure…
1. List the possibilities for the conclusion.
2. Assume that the negation of the desired
conclusion is correct.
3. Write a chain of reasons until you reach an
impossibility.
This will be a contradiction of either:
a. Given information or
b. A theorem, definition, or other known
fact.
4. State the remaining possibility as the desired
conclusion.
Objective:
After studying this section, you will be able to write indirect proofs.
An indirect proof is useful when a direct proof
is difficult to apply.
Remember to start by looking at the
conclusion!
P
Let’s take a
S
A
B
D
C
E
Given: ∠A ≅ ∠D, AB ≅ DE , AC ≅ DF
Prove: ∠B ≅ ∠E
Proof: Either ∠B ≅ ∠E or ∠B ≅ ∠E
Let’s assume ∠B ≅ ∠E
Given: RS ⊥ PQ
F
R
Q
PR ≅ QR
JJJG
Prove: RS does not bisect ∠PRQ
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P
Summary
S
R
Q
JJJG
JJJG
Proof: Either RS does bisect ∠PRQ or RS does not bisect ∠PRQ
JJJG
Assume RS bisects ∠PRQ. Then we can say ∠PRS ≅ ∠QRS.
In your own words describe how to write an indirect proof.
Since RS ⊥ PQ , we know that ∠PSR ≅ ∠QSR , thus, triangle
PSR is congruent to triangle QSR by ASA (Since SR ≅ SR ).
This means PR ≅ QR by CPCTC, but this contradicts the given
information that PR ≅ QR therefore JJJ
the
G assumption must be false,
leaving the only other possibility: RS does not bisect ∠PRQ
Hang in there, one last example…
C
B
A
Given:
O
O
Homework
Worksheet 5.1
AB ≅ BC
Prove: ∠AOB ≅ ∠COB
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Theorem: If two lines are cut by a transversal
such that two alternate interior
angles are congruent, the lines are
parallel.
5.2 Proving That Lines Are Parallel
(short form: Alt. int. ∠s ≅⇒ ll lines.)
Given: ∠3 ≅ ∠6
Prove: a ll b
Objective:
After studying this section, you will be able to apply the exterior angle inequality theorem and use various methods to prove lines are parallel.
Theorem: If two lines are cut by a transversal
such that two alternate exterior
angles are congruent, the lines are
parallel.
(short form: Alt. ext.∠s ≅⇒ ll lines.)
F
Given: ∠1 ≅ ∠8
Prove: a ll b
Remote Interior
angle
Theorem: The measure of an exterior angle of a
triangle is greater than the measure of either
remote interior angle.
Given: exterior angle BCD
Prove: m∠BCD>m∠B
B
m∠BCD > m∠BAC
P
M
A
C
Locate the midpoint, M, of BC.
Draw AP so that AM=MP
Draw CP
Triangles AMB and PMC are congruent because of SAS
This makes angle B and angle MCP congruent (CPCTC)
This also proves that angle BCD is greater than angle B
Extend BC, creating a vertical angle to angle BCD
The following results:
b
Use an indirect proof to
prove that a ll b.
D Exterior angle
E
3
6
An exterior angle of a triangle is formed
whenever a side of the triangle is extended to
form an angle supplementary to the adjacent
interior angle.
Adjacent Interior
angle
a
D
a
1
b
8
This can be proved by use
of Alt. int. ∠s ≅⇒ ll lines.
Theorem: If two lines are cut by a transversal
such that two corresponding
angles are congruent, the lines are
parallel.
(short form: corr.∠s ≅⇒ ll lines.)
Given: ∠2 ≅ ∠6
Prove: a ll b
a
2
b
6
This can be proved by use
of Alt. int. ∠s ≅⇒ ll lines.
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a
Theorem: If two lines are cut by a transversal
such that two interior angles on the
same side of the transversal are
supplementary, the lines are parallel.
Given: ∠4 supplementary to ∠6
Prove: a ll b
Given: a ⊥ c and b ⊥ c
Prove: a ll b
b
1
2
c
a
4
b
6
This can be proved by use
of Alt. int. ∠s ≅⇒ ll lines.
Given: ∠1 ≅ ∠2
Theorem: If two lines are cut by a transversal
such that two exterior angles on the
same side of the transversal are
supplementary, the lines are parallel.
∠MAT ≅ ∠THM
Prove: MATH is a parallelogram
A
1
Given: ∠2 supplementary to ∠8
Prove: a ll b
a
T
3
2
4
M
H
2
b
8
This can be proved by use
of Alt. int. ∠s ≅⇒ ll lines.
Theorem: If two coplanar lines are
perpendicular to a third line, they are
parallel.
Given: ∠2 ≅ ∠3
∠1 is supplementary ∠3
Prove: FROM is a TRAPEZOID
S
a
b
R
Given: a ⊥ c and b ⊥ c
Prove: a ll b
F 2
3
O
1
M
c
This can be proved by use
of corr. ∠s ≅⇒ ll lines.
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Summary
Name the different ways to prove lines are parallel.
Homework: worksheet
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5.3 Congruent Angles Associated With Parallel Lines
Theorem: If two parallel lines are cut by a
transversal, then any pair of angles
formed are either congruent or
supplementary.
Objective:
After studying this section, you will be able to:
a. apply the parallel postulate, b. identify the pairs of angles formed by a transversal cutting parallel lines, and c. apply six theorems about parallel lines.
x
Theorem: If two parallel lines are cut by a
transversal, each pair of alternate
exterior angles are congruent.
Postulate: Through a point not on a line there is
exactly one parallel to the given line.
(short form: ll lines ⇒ alt. ext. ∠s ≅ .)
Given: a ll b
Prove: ∠1 ≅ ∠8
P
a
1
b
8
Theorem: If two parallel lines are cut by a
transversal, each pair of alternate
interior angles are congruent.
(short form: ll lines ⇒ alt. int. ∠s ≅ .)
1
2
Theorem: If two parallel lines are cut by a
transversal, each pair of
corresponding angles are congruent.
(short form: ll lines ⇒ corr. ∠s ≅ .)
Given: a ll b
Prove: ∠1 ≅ ∠5
a
1
b
5
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Theorem: If two parallel lines are cut by a
transversal, each pair of interior
angles on the same side of the
transversal are supplementary.
Theorem: If two lines are parallel to a third line,
they are parallel to each other.
(Transitive property of Parallel Lines)
a
Given: a ll b
Prove: ∠4 supplementary to ∠6
4
b
a
Given: a ll b, b ll c
Prove: a ll c
b
6
c
Theorem: If two parallel lines are cut by a
transversal each pair of exterior
angles on the same side of the
transversal are supplementary.
If c ll d, find m ∠1
a
Given: a ll b
Prove: ∠2 supplementary to ∠8
2
b
c
1
d
2x + 10
3x + 5
8
Theorem: In a plane, if a line is perpendicular to
one of two parallel lines, it is
perpendicular to the other.
a
b
Given: a ll b
a⊥c
Prove: b ⊥ c
Given: FA & DE
FA ≅ DE
AB ≅ CD
Prove:∠F ≅ ∠E
F
A
B
C
D
E
c
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Given:
g&h
Prove: ∠1 supplementary ∠2
1
g
2
h
If a ll b, find m ∠1
a
3
100
Hint: use the parallel postulate to start
1
b
C
40
Summary
If lines are parallel, name the different ways angles are congruent or supplementary.
Homework: worksheet
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So, polygons must have:
5.4 Four Sided Polygons
Segments
Consecutive sides intersect only at endpoints
Nonconsecutive sides should not intersect
Objective:
a.
b.
c.
d.
e.
After studying this section, you will be able to:
Recognize polygons, Understand how polygons are named, Recognize convex polygons,
Recognize diagonals of polygons, and
Identify special types of quadrilaterals.
Each vertex must belong to only two sides
Segments must meet at a vertex
Consecutive sides must be noncollinear
Polygons are plane figures.
To name a polygon begin at any vertex and
proceed either clockwise or counterclockwise
around the figure.
A
B
C
F
E
D
Definition: A convex polygon is a polygon in
which each interior angle has a
measure less than 180.
Non-polygons
Convex
a
e
b
Not convex (concave)
c
d
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Definition: A diagonal of a polygon is any
segment that connects two
nonconsecutive (nonadjacent)
vertices of the polygon.
What is a
Sketch
Parallelogram A quad with 2 sets of parallel sides.
Rhombus A parallelogram in which at least two
consecutive sides are congruent.
What is a quadrilateral?
Sketch
What is a
It is a four-sided polygon.
Special Quadrilaterals
What is a …
Trapezoid A quadrilateral with exactly 1 pair of parallel sides (bases).
Isosceles Trapezoid
Kite A trapezoid in which the non‐
parallel sides (legs) are congruent. (lower and upper
base angles)
A quad with 2 disjoint pairs
of consecutive congruent
sides.
Rectangle A parallelogram with at least one right angle.
Square A parallelogram that is both
a rectangle and a rhombus.
Quadrilateral
Sketch
Parallelogram
Trapezoid
Isosceles
Trapezoid
Square
Rhombus
Rectangle
Kite
a. Trapezoid
b. Rectangle
e. Rhombus
f. Isosceles Trapezoid
C. Square
D. Quadrilateral
g. Parallelogram h. Kite
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Summary
Describe the difference between convex and concave. Explain how you will remember the special quadrilaterals.
Homework: worksheet
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Properties of Kites
5.5 Properties of Quadrilaterals
Objective:
After studying this section, you will be able to identify some properties of:
parallelograms, rectangles, kites,
rhombuses, squares, and
isosceles triangles.
a.
b.
c.
d.
e.
f.
Properties of Parallelograms
2
1
3
1. Two disjoint pair of consecutive sides are
congruent by definition
2. Diagonals are perpendicular
3. One diagonal is the perpendicular bisector of
the other
4. One of the diagonals bisects a pair of
opposite angles
5. One pair of opposite angles are congruent
Properties of Rhombuses
4
1. Opposite sides are parallel by definition
2. Opposite sides are congruent
3. Opposite angles are congruent
angles 1 and 3 are congruent;
angles 2 and 4 are congruent
4. Diagonals bisect each other
5. Any pair of consecutive angles are supplementary
1. All properties of a parallelogram apply by
definition
2. All properties of a kite apply
3. All sides are congruent (equilateral)
4. Diagonals bisect the angles
5. Diagonals are perpendicular bisectors of each
other
6. Diagonals divide the rhombus into four congruent
right triangles
Properties of Rectangles
Properties of Squares
1. All the properties of a parallelogram apply by
definition.
2. All angles are right angles
1. All the properties of a rectangle apply by
definition
2. All the properties of a rhombus apply by
definition
3. Diagonals form four isosceles right triangles
3. Diagonals are congruent
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Properties of Isosceles Trapezoid
A
Given: ABCD is a parallelogram
3
2
D
E
Prove: AC and BD bisect each other
1
B
1. Legs are congruent by definition
2. Bases are parallel by definition
3. Lower base angles are congruent
4. Upper base angles are congruent
5. Diagonals are congruent
6. Any lower base angle is supplementary to any
upper base angle
E
D
C
Given: ABCD is a parallelogram
∠GHA ≅ ∠FEC
HB ≅ DE
Conclusion:
A
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
H
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
C
Summary
Draw each of the figures and apply the properties in a foldable.
B
Homework: worksheet
A
Given: VRZA is a parallelogram
AV = 2x - 4
VR = 3y + 5
RZ = 1/2x + 8
ZA = y + 12
Find: The perimeter of VRZA
1.
F
G
GH ≅ EF
1.
4
V
Z
R
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E
F
D
Given: ACDF is a parallelogram
∠AFB ≅ ∠ECD
5.6 Proving That a Quadrilateral is a Parallelogram
Prove: FBCE is a parallelogram
A
Objective:
After studying this section, you will be able to prove that a quadrilateral is a parallelogram.
Methods to prove quadrilateral
ABCD is a parallelogram
A
D
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
10.
10.
C
Given: +CAR
is isosceles, with base CR
AC ≅ BK
∠C ≅ ∠K
B
1. If both pairs of opposite sides of a
quadrilateral are parallel, then the quadrilateral is
a parallelogram (reverse of the definition).
2. If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram (converse of a
property).
3. If one pair of opposite sides of a quadrilateral
are both parallel and congruent, then the
quadrilateral is a parallelogram.
4. If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram
(converse of a property).
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
B
A
C
Prove: BARK is a parallelogram
C
B
R
K
)
(x )
Q
⎡3 x
⎣
Given: Quadrilateral QUAD with
angles as shown
Show that QUAD is a parallelogram
(
(x
⎡ x
⎢⎣
2
)
D
− 5 x ⎤⎦
2
5
⎤
⎥⎦
(3 x
10
3
− 15x
2
)
A
U
5. If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram (converse of a
property).
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Given: NRTW is a parallelogram
V
W
NX ≅ TS
WV ≅ PR
T
X
S
Prove: XPSV is a parallelogram
N
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
10.
10.
P
R
Summary
Using one of the methods to prove quadrilaterals are parallelograms, create your own problem and show how it is a parallelogram. Homework: worksheet
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Proving that a quadrilateral is a
rhombus
J
O
K
5.7 Proving That Figures Are Special Quadrilaterals
M
Show that the quadrilateral is a parallelogram first then
apply either of the following methods.
Objective:
1. If a parallelogram contains a pair of consecutive sides that
are congruent, then it is a rhombus (reverse of the definition).
After studying this section, you will be able to prove that a quadrilateral is:
a. A rectangle
b. A kite
c. A rhombus
d. A square
e. An isosceles triangle
2. If either diagonals of a parallelogram bisects two angles of
the parallelogram, then it is a rhombus.
You can prove that a quadrilateral is a rhombus without first
showing that it is a parallelogram
3. If the diagonals of a quadrilateral are perpendicular
bisectors of each other, then the quadrilateral is a rhombus.
Proving that a quadrilateral is a
rectangle
E
H
F
G
Show that the quadrilateral is a parallelogram first then
use one of the methods to complete the proof.
1. If a parallelogram contains at least one right angle, then it
is a rectangle (reverse of the definition).
Proving that a quadrilateral is a
square
N
S
P
R
The following method can be used to prove the NPRS is a
square.
1. If a quadrilateral is both a rectangle and a rhombus, then
it is a square (reverse of the definition).
2. If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
You can prove that a quadrilateral is a rectangle without first
showing that it is a parallelogram
3. If all four angles of a quadrilateral are right angles, then it
is a rectangle.
Proving that a quadrilateral is a
kite
K
E
I
T
To prove that a quadrilateral is a kite, either of the
following methods can be used.
1. If two disjoint pairs of consecutive sides of a quadrilateral
are congruent, then it is a kite (reverse of the definition).
2. If one of the diagonals of a quadrilateral is the
perpendicular bisector of the other diagonal, then the
quadrilateral is a kite.
Proving that a trapezoid is an
isosceles
A
D
B
C
1. If the nonparallel sides of a trapezoid are congruent, then
it is isosceles (reverse of the definition).
2. If the lower or the upper base angles of a trapezoid are
congruent, then it is isosceles.
3. If the diagonals of a trapezoid are congruent, then it is
isosceles.
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Summary
What is the most descriptive name for
quadrilateral ABCD with vertices
A = (-3, -7), B = (-9, 1), C = (3, 9), and D = (9, 1)?
Write a description of each of three special quadrilaterals without using the names of the quadrilaterals. Each description should include sufficient properties to establish the quadrilateral’s identity. Homework: worksheet
A
Given: ABCD is a parallelogram
D
BD bisects ∠ADC and ∠ABC
B
Prove: ABCD is a rhombus
Given: GJMO is a parallelogram
C
M
O
OH ⊥ GK
MK is an altitude of +MKJ
Prove: OHKM is a rectangle
G
H
J
K
2