[PACKET 5.3: PROVING PARALLELOGRAMS] 1 Parallaragon  taught  us  all  about  the  characteristics  of  a  parallelogram.  But  how  do  we  KNOW  when  a  quadrilateral  is  a  parallelogram?    For  this,  we  must  use  the  converses  of  our  “precious”  theorems:

Theorem: Converse:              

                                                         

Write your questions here!

Name______________________ Must pass MC by:___________

If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

If a quadrilateral is a parallelogram, then its consecutive angles are

supplementary.

If a quadrilateral is a parallelogram, then its opposite angles are

congruent.

If a quadrilateral is a parallelogram, then its

diagonals bisect each other.

If a quadrilateral has 2 pairs of opposite sides ≅,

then the quad. is a ▱.

If a quadrilateral’s diagonals bisect each other,

then the quad. is a ▱.

If a quadrilateral has 2 pairs of opposite ∡′𝑠, then

the quad. is a ▱.

If an ∡ of a quadrilateral is supplementary to both of

its consecutive ∡𝑠, then the quad. is a ▱.

2   PACKET 5.3: PROVING PARALLELOGRAMS  

 

       

       

Statements Reason 1. Quad ABCD w/ diag. BD 1. _______________          𝐵𝐶  ||𝐷𝐴;      𝐵𝐶  ≅ 𝐷𝐴   2. ∡CBD≅ ∡ADB 2. _______________ 3.          𝐵𝐷  ≅ 𝐵𝐷 3. _______________ 4.    ∆BCD≅ ∆DAB 4. _______________ 5. ∡BDC≅ ∡DBA 5. _______________ 6. 𝐴𝐵  ||𝐶𝐷;       6. _______________

7. ABCD is a ▱ 7. _______________  We  have  a  new  theorem!:                          

         

Summary: Proving Quadrilaterals are Parallelograms  

ü Show  that  both  pairs  of  opposite  _______  are  _________________.  (Def  of  ▱)  ü Show  that  both  pairs  of  opposite  ________  are  _____________________.  ü Show  that  both  pairs  of  opposite  ___________  are  ____________________.  ü Show  that  one  angle  is  supplementary  to  _______________________________.  ü Show  that  the  diagonals  _____________  each  other.  ü Show  that  one  pair  of  opposite  sides  are  ____________  and  _______________.    

Write your questions here!

If one pair of opposite sides of a quadrilateral are BOTH congruent and parallel, then the quadrilateral is a parallelogram!

[PACKET 5.3: PROVING PARALLELOGRAMS] 3                                1.                      

2.                  For  what  values  of  x  and  y  is  FLIP  a  parallelogram?                          

3.   For  what  values  of  x  and  y  are  the  following  parallelograms?                       Ahh ohhh… SUBSTITUTION! Check Section 8.2 in Algebra!

Now

, sum

mar

ize

your

not

es h

ere!

Write your questions here!

4   PACKET 5.3: PROVING PARALLELOGRAMS  

Diagonals  of  a  parallelogram  bisect  each  

other.  

Opposite  sides  of  a  parallelogram  are  congruent.  

𝑃𝑅!!!!  is  the  perpendicular  bisector  of  𝑄𝑆!!!!.  

1. Use  the  diagram  at  the  right  and  your  theorems  to  fill  in  the                  ’s.                        2.     For  what  values  of  x  and  y  is  SULY  a  parallelogram?    

   

           3.   a.          Circle  the  reason  𝑃𝑇 ≅ 𝑇𝑅  and  𝑆𝑇 ≅ 𝑇𝑄.        

   

     

b.        Cross  out  the  equation(s)  that  is  (are)  NOT  true:     3(x + 1) – 7 = 2x y = x + 1 3y – 7 = x + 1 3y – 7 = 2x    

c.   Solve  for  x  and  y.                         d.   PT  =  ______     ST  =  ______         PR  =  ______     SQ=  ______    

E

[PACKET 5.3: PROVING PARALLELOGRAMS] 5

Algebra For what values of x and y must each figure be a parallelogram?

4. 5.

6. 7.

8. Developing Proof Complete the two-column proof. Remember, a rectangle is a parallelogram with four right angles.

Given: ABCD, with AC BD≅

Prove: ABCD is a rectangle

Statements Reasons 1) ABCD, with AC BD≅ 2) 3) DC CD≅ 4) 5) ∠ADC and ∠BCD are supplementary. 6) ∠ADC ≅ ∠BCD

7) 8) ∠DAB and ∠CBA are right angles.

9) ______________________

1) Given 2) Opposite sides of a are congruent. 3) 4) SSS 5) 6) CPCTC 7) Congruent supplementary

angles are right angles. 8) 9) Definition of a rectangle

6   PACKET 5.3: PROVING PARALLELOGRAMS  

1. For  what  values  of  x  and  y  is  FLIP  a  parallelogram?                                2. Are  you  given  enough  information  to  determine  if  MATH  is  a  parallelogram?    Explain.          3.      Show  that  A(2,  -­‐1),  B(1,  3),  C  (6,  5)  and  D  (7,  1)  are  vertices  of  a  parallelogram  because  the  opposite  sides  are  congruent  by  using  the  distance  formula.  Be  sure  to  be  very  clear  in  your  work.  (You  should  have  4  clearly  label  distances  computed.)    𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 =   𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐  

4. Now show that  A(2,  -­‐1),  B(1,  3),  C  (6,  5)  and  D  (7,  1)  are  vertices  of  a  parallelogram  because  the  opposite  sides  are  parallel  by  using  the  slope  formula.    𝑺𝒍𝒐𝒑𝒆 =𝒎 = 𝚫𝒚

𝚫𝒙= 𝒚𝟐!𝒚𝟏

𝒙𝟐!𝒙𝟏

5. Is ABCD a rectangle? Tell how you know by examining your work to #4.

[PACKET 5.3: PROVING PARALLELOGRAMS] 5

6. Complete the following proof.

Statements Reasons

 

Alg

ebra

Rev

iew

Solve each equation for x!

1.                                            !

!!!= !"

!

             

2.                                    1- (2x – 5) + 2 = 0

Multiply!   Factor!    

3.                                                    (2x – 1)(2x + 1)  

 

4.                                              (x2 - 9)          

5.      Graph  the  equation:                                  y  =  5x        

6.      Graph  the  equation:                                            y  =  3  +  2x