Transcript A B C D - mendeleev08-09

Quadrilaterals
What are quadrilaterals?
These are polygons with four-sides. This
figures are used in shapes of common objects
around us. Like buildings, furniture and other
objects.
The bricks are
quadrilaterals
The monitor is a
quadrilateral
The screen is a
quadrilateral
The following are terms used in quadrilaterals:
B
• Consecutive vertices- vertices
A
at the end of a side. Ex. A and B
• Adjacent sides- two sides with a
common endpoint. Ex. BA and
C
AC
• Opposite sides- two sides that do not have a common
endpoint. Ex. AB and DC
• Consecutive angles- two angles whose vertices are the
endpoints of a side. Ex. A and B
• Opposite angles- two angles which do not have a common
side. Ex. A and D
D
THE TYPES OF
QUADRILATERALS
Parallelograms
A quadrilateral in which both pairs of
opposite sides are parallel
A
D
B
C
ABCD is a parallelogram.
Angles of a Parallelogram
Theorem
• In a parallelogram, any
two consecutive angles
are supplementary and
opposite angles are equal.
A
Given: Parallelogram ABCD
Prove: A and B are supplementary,
A =C, D=B
D
Proof:
Statements
Reasons
1. AB||DC,AD||BC
1. Definition of parallelogram
2. With AD as
transversal , A and
D are
supplementary
2. Interior angles, AB||DC
3. Similarly, with AB
as transversal ,A
and B are
supplementary
3. Interior angles, AD ||BC
4.  D=B
4. Supplements theorem
5. Similarly, A = C
5. Steps 2 to 4
B
C
Diagonal of a Parallelogram
Theorem
• A diagonal divides a
parallelogram into two
congruent triangles.
Given: Parallelogram ABCD with
diagonal AC
~ ∆CBA
Prove: ∆ADC=
Statements
1.ABCD is a parallelogram
2.AB is parallel to DC,
AD is parallel to BC
In ∆ADC, ∆CBA
3. 1=2, 3= 4
4.AC=AC
5. ∆ADC  ∆CBA
A
D
B
C
Reasons
Given
Def. of Parallelogram
AIP
Identity
ASA
Corollary
• Opposite sides of a
parallelogram are
equal
Diagonal of a Parallelogram
Theorem
• The diagonals of a
parallelogram bisect
each other
Given: Parallelogram ABCD with
Diagonals AC and DB meeting at O
Prove: AO=OC, DO=OB
D
Statements
In ∆AOB, ∆COD
1. 1= 2
2. 3= 4
3. AB=DC
4. ∆AOB= ∆COD
5. AO=OC, DO=OB
A
B
C
Reasons
AIP
Vertical Angles Theorem
Opp. Sides of a parallelogram
SAA
CPCTE
Properties of a Parallelogram
In a parallelogram,
1.Both pairs of opposite sides are parallel.
2.A diagonal divides it into two congruent
triangles
3.Both pairs of opposite sides are equal.
4. Both pairs of opposite angles are equal.
5. Any two consecutive angles are
supplementary.
6.The diagonals bisect each other.
Distance Between Parallel
Lines
• The perpendicular distance from one
point on line to the other line.
Theorem:
• Two parallel lines are
everywhere equidistant.
Given: m II n, AC ┴ n, BD ┴ n
Prove: AC=BD
Statements
Reasons
1. AC ┴ n, BD ┴ n
Given
2. AC II BD
Lines Perp. to
the same line
3. m II n
Given
4. ABCD is a parallelogram
Def. of
parallelogram
5. AC = BD
Opp. sides of a
parallelogram
Theorem:
• If a pair of opposite
sides of a quadrilateral
are pair and equal, then
it is a parallelogram
Given: ABCD, AB II DC, AB=DC
Prove: ABCD is a II gram
Construction: Join AC.
Statements
In ∆ADC, ∆CBA
1.AB II DC
2. 1= 2
3.AB=DC
4.AC=AC
5. ∆ADC = ∆CBA
6. 3= 4
7.AD II BC
8. ABCD is a II gram
Reasons
Given
AIP
Given
Identity
SAS
CPCTE
Alternate Interior Angles are equal
Definition of parallelogram
Other Properties:
• If two pairs of opposite sides of a
quadrilateral are equal, then it is a
parallelogram
• If each diagonal divides a quadrilateral
into two congruent triangles, then it is a
parallelogram
• If diagonals of a quadrilateral bisect
each other, then it is a parallelogram.
• Both pairs of opposite angles of a
quadrilateral are equal, then it is a
parallelogram.
• In a quadrilateral, if any two
consecutive angles are supplementary,
then it is a parallelogram.
Rectangle
• A parallelogram one of whose angles is a
right angle.
To summarize, to prove that a given
parallelogram is a rectangle, show
anyone of the following is satisfied.
• All angles of a rectangle are right
angles.
• The diagonals of a rectangle are equal
• If the diagonals of a parallelogram are
equal, then the parallelogram is a
rectangle.
A
Given: ABCD, AC and BD are diameters
Prove: ABCD is a rectangle
B
O
Statements
Reasons
D
C
1. AC and BD are diameters
Given
2. AO=OC, BO=OD
A diameter equals two radii
and all radii are equal
3. ABCD is a II gram
Diagonals bisect each other
4. AC=BD
Diameters of a circle are equal
5. ABCD is a rectangle
The diagonals are equal
Rhombus
• A parallelogram with a pair of adjacent sides
equal
To summarize, to prove that a given
parallelogram is a rhombus, show
anyone of the following is satisfied.
• All sides of a rhombus are equal.
• The diagonals of a rhombus are
perpendicular to each other.
• The diagonals of a rhombus bisect
the angles at the vertices.
• If the diagonals of a quadrilateral
bisect each other at right angles,
then it is a rhombus.
• A quadrilateral with all sides equal
is a rhombus.
• A quadrilateral in which the
diagonals bisect the angles at the
vertices is a rhombus.
A
Given: ABCD is a rhombus
Prove: AC bisects A and C,
BD bisects B and D
1
D
B
3
4
2
C
Statements
Reasons
In ∆ADB, ∆CDB
1. AB=BC, AD=DC
All side of a rhombus are equal
2. DB = DB
Identity
3. ∆ADB=∆CDB
SSS
4. 1= 2, 3= 4
CPCTE
5.DB bisects ADC and Definition of angle bisector
ABC
6. Similarly, if diagonal AC
is used, we can prove that
AC bisects A and  C
A
Given: ABCD, 1=2, 3=
4, 5=6, 7=8
Prove: ABCD is a rhombus
5
3
D
4
6
1
7
B
2
8
C
Statements
Reason
In ∆ADB, ∆CDB
1. 1=2, 3=4
Given
2. DB=DB
Identity
3. ∆ADB ∆CDB
ASA
4. AB=BC, AD=DC
CPCTE
5. Similarly, by using AC we can prove
∆CBA  ∆CBA and DC=CB
6. AB=BC=DC=AD
Substitution
7. ABCD is a rhombus
All sides are equal
Square
• A rhombus with a right angle
• A rectangle with a pair of adjacent sides equal
Given: In the figure, ABCD is
an inscribed quadrilateral, AC,
BD are diameters of circle O
and AB=BC
Prove: ABCD is a square
A
D
0
Statements
Reasons
C
1. AC, BD are diameters
Given
2. AO=OC,BO=OD
Radii of circle are equal
3. ABCD is II gram
Diagonals bisect each other
4. AC=BD
All diameters are equal
5. ABCD is a rectangle
Diagonals are equal
6. AB=BC
Given
7. ABCD is a square
Definition of a square
B
Trapezoid
• Is a quadrilateral with only one pair of parallel sides
(bases)
denotes parallel
lines
• An isosceles trapezoid is one in which the nonparallel
sides are equal
Base angles of an isosceles trapezoid are equal
The diagonals of an isosceles trapezoid are equal
Given: ABCD, with AB II DC,
AD=BC
Prove: D= C, A= B
Construction: Through A, draw AX II BC
so that X is on DC.
Statements
1. AX II BC
2. AB II DC
3. ABCX is a II gram
4. AX=BC
5. AD=BC
6. AD=AX
7. In ∆ADX, 1=2
8. 1= C
9. B+C=180°,A+D=180°
10. A= B
A
D
B
1
X
Reasons
Construction
Given
Definition of II gram
Opp. Sides of II gram
Given
Substitution
s opp. Equal sides
Corr. s, AX II BC
Int. s, AB II DC
Supplements of equal angles
C
Proponents:
Maureen Mae Eronico
Milbert Capistrano
• Kenji Sahagun