Geometry 6 3 Proving Quadrilaterals are Parallelograms Goal

Geometry 6. 3 Proving Quadrilaterals are Parallelograms

Goal n Use properties of parallelograms to prove a quadrilateral is a parallelogram. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 2

Parallelogram Property 1. Opposite Sides are Parallel 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 3

Parallelogram Property 2. Opposite Sides are Congruent 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 4

Parallelogram Property 3. Opposite Angles are Congruent 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 5

Parallelogram Property 4. Consecutive Angles are Supplementary (180 – x) x 12/30/2021 x (180 – x) Geometry 6. 3 Proving Quadrilaterals are Parallelograms 6

Parallelogram Property 5. Diagonals Bisect Each Other 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 7

12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 8

Converse n n n If A then B. The converse is… If B then A. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 9

Example n n n If MESA is a parallelogram then M S and E A. (opposite angles are congruent) Converse: If M S and E A, then MESA is a parallelogram. (because opposite angles are congruent) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 10

A quadrilateral is a parallelogram if… n n n Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 11

To show a quadrilateral is a parallelogram only one of these is needed. n n n Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 12

And one other way… n If one pair of sides is parallel and congruent, then the quadrilateral is a parallelogram. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 13

Proof 2 s by SAS opp. sides means it’s . 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 14

Six ways to prove a quadrilateral is a parallelogram. opp. sides || x 180 - x consec. s supp. opp. s diags. bis. 12/30/2021 pair sides || and Geometry 6. 3 Proving Quadrilaterals are Parallelograms 15

Is this a parallelogram? 110 70 YES! 70 110 Reason: Opposite angles congruent Or, consecutive angles supplementary. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 16

Is this a parallelogram? 10 cm 7 cm YES! 7 cm 10 cm Reason: Opposite sides congruent. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 17

Is this a parallelogram? 4 6 6 YES! 4 Reason: Diagonals bisect each other. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 18

Is this a parallelogram? 7 cm NO! 7 cm Reason: One pair of opp. sides ||, but not . 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 19

Is this a parallelogram? 7 cm NO! 7 cm Reason: One pair of opp. sides ||, but not . 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 20

Is this a parallelogram? 9 cm YES! 9 cm Reason: One pair of opp. sides || & . 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 21

Show RSTV is a parallelogram. T S R V Since RT = RT, RST TVR by ASA. Using CPCTC, RS = TV and ST = RV. RSTV is a parallelogram since opposite sides are congruent. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 22

Now some algebra… n n We will determine if a quadrilateral on the coordinate plane is a parallelogram. Use the distance formula or the slope formula. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 23

Coordinate Problem 1 n n n Graph the quadrilateral with points A(2, 4) B(4, – 3) C(9, – 6) D(7, 1) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 24

You should have this graph: A (2, 4) D(7, 1) B (4, -3) C (9, -6) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 25

Choices n n n Choose a method using the given information that will prove ABCD is a parallelogram: Show opposite sides congruent. Or, show they are parallel. Or, one pair of sides congruent and parallel. Or even the diagonals bisect each other. But choose the easiest if possible! 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 26

Method: Show opposite sides congruent. Compute the length of segment AB. A (2, 4) D(7, 1) B (4, -3) C (9, -6) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 27

Now find the length of DC. A (2, 4) D(7, 1) AB = DC B (4, -3) C (9, -6) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 28

Now find lengths AD & BC. A (2, 4) D(7, 1) B (4, -3) AD = BC C (9, -6) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 29

A (2, 4) D(7, 1) Both pairs of opposite sides are congruent, so ABCD is a parallelogram. B (4, -3) C (9, -6) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 30

Another Example n n n On another graph, plot the points: R(– 5, 5) S(6, 4) T(2, – 5) U(– 9, – 4) Then connect the points to make a quadrilateral. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 31

Your graph should be… R(-5, 5) U(-9, -4) 12/30/2021 S(6, 4) T(2, -5) Find the slopes of segments RS and UT. Slopes are equal, sides are parallel. Geometry 6. 3 Proving Quadrilaterals are Parallelograms 32

Find the slopes of RU & ST. R(-5, 5) U(-9, -4) 12/30/2021 S(6, 4) T(2, -5) Slopes are equal, sides are parallel. Geometry 6. 3 Proving Quadrilaterals are Parallelograms 33

RSTU is a parallelogram. R(-5, 5) Reason: S(6, 4) Both pairs of opposite sides are parallel. U(-9, -4) 12/30/2021 T(2, -5) Geometry 6. 3 Proving Quadrilaterals are Parallelograms 34

Try this. A, B, and C are three vertices of a parallelogram. A(0, 1) C(-2, -1) B(1, -1) 12/30/2021 Find the possible coordinates of the fourth vertex. Geometry 6. 3 Proving Quadrilaterals are Parallelograms 35

D could be (-3, 1) D(-3, 1) A(0, 1) C(-2, -1) B(1, -1) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 36

Where else could D be? A(0, 1) D(3, 1) C(-2, -1) B(1, -1) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 37

Anywhere else? A(0, 1) C(-2, -1) B(1, -1) D(-1, -3) 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 38

What you’ve learned… n n n You can prove a quadrilateral is a parallelogram if any one of six conditions is true. You MUST know these six methods. You can also show a quadrilateral on the plane is a parallelogram using the distance formula or slope formula. 12/30/2021 Geometry 6. 3 Proving Quadrilaterals are Parallelograms 39

Six ways to prove a quadrilateral is a parallelogram. opp. sides || x 180 - x consec. s supp. opp. s diags. bis. 12/30/2021 pair sides || and Geometry 6. 3 Proving Quadrilaterals are Parallelograms 40