Ways to Prove That Quadrilaterals are Parallelograms Chapter 5 Section 2 Ways to Prove That Quadrilaterals are Parallelograms

Warm-Up: Tuesday April 1 Get your homework out to be checked from last night. Solve the following quadratics using ANY method of your choice! Today you may give decimal approximations for your answers.

Homework Answers: April 1 1.) CR, CE 2.) REC, WCE, WEC 3.) ER, RC, CW 4.) CRW, CWR, EWR 5.) a = 8, b = 10, x = 118, y = 62 6.) a = 8, b = 15, x =80, y = 70 7.) a = 5, b = 3, x = 120, y = 22 8.) a = 9, b = 11, x = 33, y = 27 9.) a = 8, b = 8, x = 56, y = 68 10.) a = 10, b = 4, x = 90, y = 45 11.) 60 12.) ST = 14, SP = 13 Homework Answers: April 1 p. 169 #’s 1 – 12, 19-28

19. ) x=3, y=5 20. ) x=7, y=18 21. ) x=13, y=5 22. ) x=65, y=20 23 19.) x=3, y=5 20.) x=7, y=18 21.) x=13, y=5 22.) x=65, y=20 23.) x=5, y=4 24.) x=10, y=16 25.) 5,2 26.) 6,3,42 27.) 10,70 28.) 36,49 p. 169 #’s 1 – 12, 19-28

22.) 85 130 x - y x + y

Proving Quadrilaterals Are Parallelograms Remember all the properties of parallel lines? What about congruent triangles?

What do Parallel Lines Make? Congruent Corresponding Angles Congruent Alternate Interior Angles Supplementary Same Side Interior Angles Remember… FUZZ!! 122

Recall… Ways to Prove Triangles Congruent Side-Side-Side Side-Angle-Side Angle-Side-Angle Angle-Angle-Side Hypotenuse Leg Use CPCTC to state corresponding angles or sides are congruent.

What Are We Trying To Do Again? We want to prove that any quadrilateral is a parallelogram. GOAL: (use properties from Chapter 3) Get back to the definition of a parallelogram: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Theorem 5.4: If both pairs of opposite sides of a quadrilateral are , then the quadrilateral is a parallelogram. What would we need to show that ABCD is a parallelogram? congruent A D C B

Statements Reasons Proof of Theorem 5.4 Given: TS  QR; TQ  SR Prove: Quad QRST is a parallelogram Q R Statements Reasons

Theorem 5.5: congruent parallel If one pair of opposite sides of a quadrilateral are both and , then the quadrilateral is a parallelogram. What would we need to show that ABCD is a parallelogram? congruent parallel A D C B

Statements Reasons Proof of Theorem 5.5 Given: AB  CD; AB || CD Prove: Quad ABCD is a parallelogram D C Statements Reasons

Theorem 5.6: If both pairs of opposite angles of a quadrilateral are , then the quadrilateral is a parallelogram. What would we need to show that ABCD is a parallelogram? congruent A D C B

Statements Reasons Proof of Theorem 5.6 Given: mA = mC = x B Given: mA = mC = x mB = mD = y Prove: Quad ABCD is a parallelogram D C Statements Reasons

Theorem 5.7: diagonals bisect If the of a quadrilateral each other, then the quadrilateral is a parallelogram. What would we need to show that WXYZ is a parallelogram? diagonals bisect W Z Y X

Class Practice Look at the marking on each figure and decide if each quadrilateral must be a parallelogram. If the answer is yes, state the definition or property that applies.

Both pairs of opposite sides are congruent! Is it a Parallelogram? 14 6 YES Both pairs of opposite sides are congruent!

Is it a Parallelogram? 4 YES One pair of opposite sides are both parallel and congruent!

The diagonals bisect each other! Is it a Parallelogram? YES The diagonals bisect each other!

None of the properties are satisfied! Is it a Parallelogram? 105 70 NO! None of the properties are satisfied!

Consecutive Angles are Supplementary! Is it a Parallelogram? YES Consecutive Angles are Supplementary!

Is it a Parallelogram? 3 YES One pair of opposite sides are both parallel and congruent!

p. 173 Classroom Exercises #’s 1 - 11 Practice Exercises p. 173 Classroom Exercises #’s 1 - 11

Cool Down: Wednesday, March 26th Summarize the 5 ways to prove parallelograms 1.) 2.) 3.) 4.) 5.) Solve for x and y. y2 x 3x - 40 Y + 30 Cool Down Part 1 Cool Down Part 2

Statements Reasons A X C B F E D Given: ABCX is a parallelogram DXFE is a parallelogram Prove: B  E Statements Reasons