Transcript Applied Geometry - South Harrison County R2

Geometry
Lesson 6 – 2
Parallelograms
Objective:
Recognize and apply properties of the sides of angles of parallelograms.
Recognize and apply properties of the diagonals of parallelograms.
Parallelograms
What is a parallelogram?
A quadrilateral with both pairs of
opposite sides parallel.
To name a parallelogram:
Theorems:
Properties of Parallelograms
Theorem 6.3

If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Theorem
Theorem 6.4
 If
a quadrilateral is a parallelogram, then its
opposite angles are congruent.
Theorem
Theorem 6.5
 If
a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
Theorem
Theorem 6.6

If a parallelogram has one right angle, then
it has four right angles.
In parallelogram ABCD, suppose the
measure of angle A is 55, segment AB
is 2.5 feet, and segment BC is 1 foot.
Find each measure.
Find DC
2.5
m B
180 – 55 = 125
mC
55
Theorems:
Diagonals of Parallelograms
Theorem 6.7

If a quadrilateral is a parallelogram, then its
diagonals bisect each other.
Theorem
Theorem 6.8

If a quadrilateral is a parallelogram, then
each diagonal separates the parallelogram
into two congruent triangles.
If QRST is a parallelogram, find the following.
Find x
5x = 27 Opps. Sides
x = 5.4 equal
Find y
2y – 5 = y + 4 Diagonals bisect
y=9
Find z
3z = 33
z = 11
Alt. Interior angles are congruent.
Find the value of each variable in
the parallelograms.
4x + 2x – 6 = 180
6x – 6 = 180
6x = 186
x = 31
y + 8 = 5y
2=y
3z – 4 = z + 5
2z = 9
z = 4.5
Determine the coordinates of the intersection of
the diagonals of parallelogram FGHJ with
vertices F (-2, 4) G (3, 5) H (2, -3) and J (-3, -4)
What do you know about the diagonals of a
parallelogram?
 Since we know they bisect what is a good point to
find?
 Find the Midpoint of the diagonals:
 xx y y
m idpoint  
,

2 
 2
  2 2 43
m idpoint of FH  
,
  0, 1 2
2 
 2
Double check:m idpoint of GJ   3  3 , 5  4   0, 1
2
2 
 2





Determine the coordinates of the intersection of
the diagonals of RSTU with vertices R (-8, -2)
S (-6, 7) T (6, 7) and U (4, -2)
 8 6  2  7 
m idpoint of RT  
,
   1,2.5
2 
 2
 64 72
m idpoint of SU  
,
   1,2.5
2 
 2
Statements
1.
Reasons
1. Given
2. HJ  PK & PK  ML 2. Opp. Sides congruent.
3. HJ  ML
3. Transitive
Homework
Pg. 403 1 – 6 all, 10 – 22 E,
44 – 60 E