[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