Download Given: Parallelogram ABCD with diagonal

Geometry Notes Q – 1: Quadrilaterals
Definition: A quadrilateral is a polygon with
Theorem: The sum of the interior angles in a quadrilateral is
A
D
Given: Quadrilateral ABCD
Prove: mA + mB + mC + mD = 360
Statement
Reason
B
C
Ex: In quadrilateral CRAB, CR  BC , mR =  x 2  , mA = 7x and mB = (2x + 50). Find the measures of
all four angles of CRAB.
Review of Coordinate Geometry
Formulas
Slope formula:
Distance Formula:
Midpoint Formula:
Four basic proofs
To prove two segments congruent:
To prove two segments parallel:
To prove two segments perpendicular:
To prove two segments bisect each other:
Parts of a coordinate geometry proof
Graph or diagram (optional but usually very helpful)
Labeled calculations. For example:
Appropriate conclusions and justifications.
For example:
Geometry Notes Q – 2: Parallelograms I
Definition: A parallelogram is a quadrilateral with
Properties of Parallelograms
1.
2.
3.
4.
A
B
Given: Parallelogram ABCD.
Prove: 2, 3 and 4 above.
D
Note: We have also proved the following theorem: A diagonal divides a parallelogram into
C
Ex: In parallelogram TOAD, TO = x2 – 1, OA = x + y, AD = 9x – 3y and DT = 4x – 8.
a. Find the perimeter of TOAD.
b. Find all the possible values of the height of TOAD measured from AD .
c. If the height of TOAD measured from AD is 9, what is the area of TOAD.
Ex: In parallelogram PIKA, mP = (3x + 2y), mI = (2x + y) and K = (3y + 24). Find the numerical
measure of A.
Ex: In quadrilateral HARD, mH =  x 2  , mA= (x2 + x + 10), mR = (160 – 9x) and D = (190 – 10x). Is
HARD a parallelogram?
Geometry Notes Q – 3: Parallelograms II
Theorem: The diagonals of a parallelogram
A
B
Given: Parallelogram ABCD, diagonals
AC and BD intersect at E.
Prove: AC and BD
E
D
C
E
Ex: Parallelogram EASY has diagonals intersecting at R. Find
the lengths of the diagonals.
A
x
2y – 8
Y
x+4
R
y–2
S
Geometry Notes Q – 5: Proving quadrilaterals are parallelograms
Properties of Parallelograms: If a quadrilateral is a parallelogram, then
1.
2.
3.
4.
5.
6.
Proving a quadrilateral is a parallelogram: A quadrilateral is a parallelogram if
1.
or 2.
or 3.
or 4.
or 5.
Given: Quadrilateral ABCD with A  C and B  D.
Prove: Quadrilateral ABCD is a parallelogram.
A
D
B
C
Ex: Determine if each of the following is a parallelogram and give a reason.
a. Given: a > b
a2  b2
A
B
25
25
D
(a + b)(a  b)
b. x > 3
C
1
 x  3
2
A
B
40
40
D
c.
x3
2
C
A
x+1
 20

 x  10 
 3

x2  13
D
B
3x  3
1
(4x + 2)
2x  4
C
Geometry Notes Q – 6: Rectangles
Theorem: If two supplementary angles are congruent, then
Given: A supplementary to B; A  B
Prove: A and B are right angles
Theorem: If two angles are supplementary and one is a right angle, then
Given: A supplementary to B; A is a right angle
Prove: B is a right angle
Definition: A rectangle is a quadrilateral with
Note: A rectangle is a
Properties of rectangles:
1.
2.
3.
Ex: In rectangle RECT, the diagonals intersect at M, RT = 6 and RM = 5. Find the perimeter of TMC.
Theorem: A parallelogram with one right angle is a rectangle.
A
B
D
C
A
B
D
C
Given: Parallelogram ABCD, A is a right angle
Prove: ABCD is a rectangle
Theorem: A parallelogram with congruent diagonals is a rectangle.
Given: Parallelogram ABCD, AC  BD
Prove: ABCD is a rectangle
Proving a quadrilateral is a rectangle: A quadrilateral is a rectangle if
1.
or 2.
or 3.
Ex: Determine if each of the following is a rectangle and give a reason.
a.
b.
A
B
D
C
A
25
B
65
D
c.
C
A
D
B
AC  BD
C
Geometry Notes Q – 7: Rhombi & Squares
Definition: A rhombus is a quadrilateral with
Note: A rhombus is a
Properties of rhombi:
1.
2.
A
3.
B
4.
Given: Rhombus ABCD with diagonal AC
Prove: AC bisects BAD and BCD
D
C
A
Given: Rhombus ABCD; diagonals AC and BD intersect at E.
Prove: AC  BD
B
E
D
C
Ex: The diagram at right shows a rhombus with both diagonals drawn. Find the
values of x, y and z.
25
y
x
z
Proving a quadrilateral is a rhombus: A quadrilateral is a rhombus if
1.
or 2.
or 3.
Ex: Quadrilateral RHOM has vertices R(2, 4), H(–5, 5), O(0, 0) and M(7, –1). Prove using coordinate geometry
that RHOM is a rhombus.
y
H
R
O
x
M
Square
Definition: A square is a quadrilateral
Note: A square is a
Properties of squares: All of the properties of
Geometry Notes Q – 10: Proofs practice
Ex: Given: Quadrilateral WXYZ, diagonals WY and XZ intersect at V,
WX || YZ , V is the midpoint of XZ , WZ  YZ .
X
W
Prove: WXYZ is a rectangle
V
Z
Y
Ex: Given: Rhombus RSTV, VTX , STW , RS  SX , RV  VW
Prove: TX  TW
R
S
V
T
W
X
Geometry Notes Q - R: Review and Practice
Quadrilaterals
1.
C
General:
B
A
2.
D
B
Trapezoid:
C
A
D
2.1. Isosceles trapezoid:
B
C
A
3.
Parallelogram:
B
A
3a. Proving a quadrilateral is a parallelogram
D
C
D
3.1 Rectangle:
B
C
A
D
3.1a Proving a quadrilateral is a rectangle
3.2 Rhombus:
C
B
A
D
B
C
A
D
3.2a Proving a quadrilateral is a rhombus
3.3 Square:
3.3a Proving a quadrilateral is a square