proving quadrilaterals are parallelograms lesson 6.3 chapter 6 section 6.3 proving quadrilaterals...
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Proving Quadrilaterals are Parallelograms Lesson 6.3
Chapter 6Section 6.3
Proving Quadrilaterals Are Parallelograms
Proving Quadrilaterals are Parallelograms Lesson 6.3
Some Properties of ParallelogramsQuick Review
If a quadrilateral is a parallelogram, then …
Theorem 6.2
Theorem 6.3
Theorem 6.4
Theorem 6.5
both pairs of opposite sides are congruent
both pairs of opposite angles are congruent
consecutive angles are supplementary
diagonals bisect each other
Definition: A parallelogram is a quadrilateral with both pairs of opposite sides congruent
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem
Theorem 6.6
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Quadrilateral PQRS with both pairs of opposite sides congruent
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem
Theorem 6.7
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Quadrilateral PQRS with both pairs of opposite angles congruent
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem
Theorem 6.8
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram
mS + mP = 180 and mS + mR = 180
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem
Theorem 6.9
If the diagonals of a quadrilateral bisect each other,then the quadrilateral is a parallelogram
PQRS is a parallelogram
QStssecbiPR and PRtssecbiQS
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem
Theorem 6.10
If one pair of opposite sides of a quadrilateral are congruentand parallel, then the quadrilateral is a parallelogram
PQRS is a parallelogram
SR//PQ and SRPQ
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Yes, one pair of opposite sides congruent and parallel
Yes, diagonals bisect each other
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Yes, both pairs of opposite sides congruent No, same pair of opposite sides
must be parallel and congruent
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Yes, could show both pairs of opposite sides are parallel
Yes, Consecutive angles are supplementary
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
Theorem 6.9: ECAE
Theorem 6.8: mCDA + mDCB = 180
OR
mDAB + mABC = 180
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram opposite sides must be congruent
3x = 6 x + 2 = y - 1x = 2 2 + 2 = y – 1
4 = y – 1
5 = y
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram opposite angles must be congruent
2x = 70 3x + 5 = x + 3yx = 35 3(35) + 5 = 35 + 3y
110 = 35 + 3y
75 = 3y
25 = y
Proving Quadrilaterals are Parallelograms Lesson 6.3
Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram diagonals must bisect each other
3x = 12 x + y = 5yx = 4 4 + y = 5y
4 = 4y
1 = y
Proving Quadrilaterals are Parallelograms Lesson 6.3
Use both slope and distance formula to show one pair of opposite side is congruent and parallel
• Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(-6,2), and D(0,7) is a parallelogram
2222 )72()06()23()15( AB = CD
2222 )5()6()5()6(
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6161 €
3 − −2
5 − −1//
7 − 2
0 − −6
AB // CD
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5
6//
5
6€
3+ 2
5 +1//
7 − 2
0 + 6
Proving Quadrilaterals are Parallelograms Lesson 6.3
HWPg 342 #’s 9, 10, 12, 17-19 all, & 25-26
Quiz on Thursday!!