# 222 notes chapter 6 quadrilaterals

```The Distance Formula
• Used to find the distance between two points:
A( x1, y1) and B(x2, y2)
d  ( x2  x1 )   y2  y1 
2
2
You also could just plot the points and use the
Pythagorean Theorem!!
Find the distance between the two points.
1. T(5, 2) and R(-4, -1)
2. A( -2, -3) and B(1, 3)
Take a look at example 2, p. 44
Midpoint Formula
• Find the midpoint coordinates between 2
points
• Find by averaging the x-coordinates and the ycoordinates of the endpoints
(x2, y2)
(x1, y1)
 x1  x2 y1  y2 
M
,

2 
 2
Find the coordinates of the midpoint
QSof
1. Q(3, 5) and S(7, -9)
2. Q( -4, 4) and S(5, -1)
Warm-up
Draw each figure on graph paper if possible. If
not possible explain why.
1. A parallelogram that is neither a rhombus
nor a rectangle
2. An isosceles trapezoid with vertical and
horizontal congruent sides
3. A trapezoid with only one right angle
4. A trapezoid with two right angles
5. A rhombus that is not a square
6. A kite with two right angles
Parallelogram – A quadrilateral with both pairs of opposite sides
parallel.
Rhombus – A parallelogram with four congruent sides.
Rectangle – A parallelogram with four right angles.
Square – A parallelogram with four congruent sides and four right
angles.
and no opposite sides congruent.
Trapezoid – A quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid – A trapezoid whose nonparallel opposite sides
are congruent.
Theorem 6-1
Opposite sides of a parallelogram are congruent.
Theorem 6-2
Opposite angles of a parallelogram are congruent.
(Consecutive angles of a parallelogram are
supplementary, they are same-side interior angles!)
Theorem 6-3
The diagonals of a parallelogram bisect each other.
Theorem 6-4
If three or more parallel lines cut off
congruent segments on one transversal,
then they cut off congruent segments on
every transversal.
TR=12 find VR
QS=10 find VS
Find x and the length of the side
Find all angle measures
110̇
120
30
Proving that a quadrilateral is a parallelogram.
(both pairs of opposite sides are parallel)
Theorem 6-5
If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
Theorem 6-6
If one pair of opposite sides of a quadrilateral is both
congruent and parallel, then the quadrilateral is a
parallelogram.
Theorem 6-7
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Proving that a quadrilateral is a parallelogram.
(Both pairs of opposite sides are parallel)
Theorem 6-5
If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
Theorem 6-6
If one pair of opposite sides of a quadrilateral is both
congruent and parallel, then the quadrilateral is a
parallelogram.
Theorem 6-7
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
Theorem 6-9
Each diagonal of a rhombus bisects two angles of the
rhombus.
Theorem 6-10
The diagonals of a rhombus are perpendicular.
Theorem 6-11
The diagonals of a rectangle are congruent.
Theorem 6-12
If one diagonal of a parallelogram bisects two angles of the
parallelogram, then the parallelogram is a rhombus.
Theorem 6-13
If the diagonals of a parallelogram are perpendicular, then
the parallelogram is a rhombus.
Theorem 6-14
If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
Use the properties of rectangles to
find all missing angle measures, list the
properties you used.
32
6.5: Trapezoids and Kites
Objective:
To verify and use properties of trapezoids
and kites
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to the bases.
The length of the midsegment of a trapezoid is half the
sum of the lengths of the bases.
If you are given the midsegment length and one
base: -double the length of the midsegment
-subtract the other base.
Midsegment (Median) of a Trapezoid
Joins the midpoints of the nonparallel sides
Is parallel to the bases
Its length is &frac12; the sum of the bases
MN || BC
86&deg;
3
108&deg;
4
Find the following:
EF:
mCFE :
mDCB :
DF :
AB :
Find x:
x
8
12
Theorem 6-15
The base angles of an isosceles trapezoid are
congruent.
Theorem 6-16
The diagonals of an isosceles trapezoid are
congruent.
Theorem 6-17
The diagonals of a kite are perpendicular.
Theorem:
The diagonals of a kite are perpendicular.
A kite has exactly one pair of
opposite, congruent angles.
Find the measure of the missing angles.
1
44&deg;
112&deg;
2
What is the sum of the angles in a quadrilateral?
TRAPEZOID
•The 2 parallel sides are the bases
•The 2 non-parallel sides are the legs
BASE ANGLES
B
D
LEG
LEG
A
C
BASE ANGLES
Name the following:
Bases:
Legs:
2 Pairs of Base Angles:
Theorem:
The base angles of an isosceles trapezoid are
congruent
B
D
A  C
B  D
C
A
m A  3 x
mC  x  21
Find x.
Theorem:
The diagonals of an isosceles trapezoid are
congruent.
EXAMPLE:
If BD= 2x+10 and AC=x+15, find x and the length of the
diagonals.
2 angles that share a leg are supplementary because they are
same-side interior angles.
mA  mC  180
The measure of angle A= 110. find the measures of the
other 3 angles.
One side of a kite is 4 cm less than two times the
length of another side. The perimeter of the kite is
58 cm. Find the lengths of the sides of the kite.
Midsegments of trapezoids
• The midsegment of an isosceles trapezoid
measures 14 cm. One of the bases measures
24 cm. Find the length of the other base.
```