Post on 11-Jan-2015
description
Section 6-3Tests for Parallelograms
Wednesday, April 11, 2012
Essential Questions
How do you recognize the conditions that ensure a quadrilateral is a parallelogram?
How do you prove that a set of points forms a parallelogram in the coordinate plane?
Wednesday, April 11, 2012
Theorems6.9 - OPPOSITE SIDES:
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Wednesday, April 11, 2012
Theorems6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Wednesday, April 11, 2012
Theorems6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Wednesday, April 11, 2012
Theorems6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Wednesday, April 11, 2012
Theorems6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.12 - PARALLEL CONGRUENT SET OF SIDES: IF ONE PAIR OF OPPOSITES SIDES OF A QUADRILATERAL IS BOTH CONGRUENT AND PARALLEL, THEN THE QUADRILATERAL IS A PARALLELOGRAM
Wednesday, April 11, 2012
Example 1DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.
JUSTIFY YOUR ANSWER.
Wednesday, April 11, 2012
Example 1DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.
JUSTIFY YOUR ANSWER.
BOTH PAIRS OF OPPOSITE SIDES HAVE THE SAME MEASURE, SO EACH OPPOSITE PAIR IS CONGRUENT, THUS MAKING IT A
PARALLELOGRAM.
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
Wednesday, April 11, 2012
Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
5 = y
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
= 4
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
= 4 m(TO ) =
−1− 3−2 − (−1)
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
= 4 m(TO ) =
−1− 3−2 − (−1)
=−4−1
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
= 4 m(TO ) =
−1− 3−2 − (−1)
=−4−1
= 4
Wednesday, April 11, 2012
Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO
IS A PARALLELOGRAM.
m(TA) =
1− 33 − (−1)
=−24
= −12
m(CO ) =−1− (−3)−2 − 2
=2−4
= −12
m(AC ) =
−3 − 12 − 3
=−4−1
= 4 m(TO ) =
−1− 3−2 − (−1)
=−4−1
= 4
SINCE EACH SET OF OPPOSITE SIDES HAVE THE SAME SLOPE, THEY ARE PARALLEL. WITH EACH SET OF OPPOSITE SIDES BEING PARALLEL, TACO IS
A PARALLELOGRAM
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180− 72
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180− 72 = 108
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180− 72 = 108
8y + 8 = 108
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180− 72 = 108
8y + 8 = 108
8y = 100
Wednesday, April 11, 2012
Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180− 72 = 108
8y + 8 = 108
8y = 100
y = 12.5
Wednesday, April 11, 2012
Check Your Understanding
REVIEW #1-8 ON P. 413
Wednesday, April 11, 2012
Problem Set
Wednesday, April 11, 2012
Problem Set
P. 414 #9-23 ODD, 27, 51, 53
“I AM ALWAYS DOING THAT WHICH I CAN NOT DO, IN ORDER THAT I MAY LEARN HOW TO DO IT." – PABLO PICASSO
Wednesday, April 11, 2012