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.
Round your answer to the nearest tenth.
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
Special Quadrilaterals
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.
Kite – A quadrilateral with two pairs of adjacent sides congruent
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 ½ the sum of the bases
MN || BC
MN || AD
MN = ½(BC+AD)
86°
3
108°
4
Find the following:
EF:
mBAD :
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°
112°
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.