Notes: Proofs Using Special Quadrilaterals
Name_________________________________
Thinking about proofs—We know a lot about when two triangles are congruent. So when we want to proves
that parts of a parallelogram are congruent, we usually construct one or both of the diagonals, to make
triangles.
Let’s prove some of the things we have learned about parallelograms. To prove a theorem, we need to begin
with a diagram that makes the antecedent (the “if” part) true, but does not assume anything else. As you work
your way through the proof, be sure to mark the diagram.
1. If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Given: PQRS is a parallelogram
Prove: 𝑃𝑄 ≅ 𝑅𝑆, 𝑄𝑅 ≅ 𝑃𝑆
Statements
Reasons
1. PQRS is a parallelogram.
1. Given
2. Draw QS.
2. Through any 2 points there exists exactly 1 line.
3. 𝑃𝑄 ∥ 𝑅𝑆, 𝑄𝑅 ∥ 𝑃𝑆
3. Definition of parallelogram.
4. ∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄
4. Alternate Interior Angles Theorem.
5. 𝑄𝑆 ≅ 𝑄𝑆
5. Reflexive Property of Congruence.
6. ∆𝑃𝑄𝑆 ≅ ∆𝑅𝑆𝑄
6. ASA Congruence Postulate.
7. 𝑃𝑄 ≅ 𝑅𝑆, 𝑄𝑅 ≅ 𝑃𝑆
7. Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
2. If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Given: PQRS is a parallelogram
Prove: ∠𝑃 ≅ ∠𝑅, ∠𝑄 ≅ ∠𝑆
Statements
Reasons
1. PQRS is a parallelogram.
1. Given
2. Draw QS.
2. Through any 2 points there exists exactly 1 line.
3. 𝑃𝑄 ∥ 𝑅𝑆, 𝑄𝑅 ∥ 𝑃𝑆
3. Definition of parallelogram.
4. ∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄, ∠𝑃𝑆𝑄 ≅ ∠𝑅𝑄𝑆
4. Alternate Interior Angles Theorem.
5. 𝑄𝑆 ≅ 𝑄𝑆
5. Reflexive Property of Congruence.
6. ∆𝑃𝑄𝑆 ≅ ∆𝑅𝑆𝑄
6. ASA Congruence Postulate.
7. ∠𝑃 ≅ ∠𝑅, ∠𝑄 ≅ ∠𝑆
7. Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
Proofs, page 1
Let’s prove the converse of our first proof.
3. If both pairs of opposite sides in a quadrilateral are congruent, then it is a parallelogram.
P
Given: PQRS is a quadrilateral in which 𝑃𝑄 ≅ 𝑅𝑆, 𝑄𝑅 ≅ 𝑃𝑆
Q
Prove: PQRS is a parallelogram
Statements
1. PQRS is a quadrilateral in which
Reasons
S
R
1. Given
𝑃𝑄 ≅ 𝑅𝑆, 𝑄𝑅 ≅ 𝑃𝑆
2. Draw QS.
2. Through any 2 points there exists exactly 1 line.
3. 𝑄𝑆 ≅ 𝑄𝑆
3. Reflexive Property of Congruence.
4. ∆𝑃𝑄𝑆 ≅ ∆𝑅𝑆𝑄
4. SSS
5. ∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄 and ∠𝑃𝑆𝑄 ≅ ∠𝑅𝑄𝑆
5. Corresponding Parts of Congruent Triangles are Congruent
6. 𝑃𝑄 ∥ 𝑅𝑆, 𝑄𝑅 ∥ 𝑃𝑆
6. Alternate Interior Angles Converse
7. PQRS is a parallelogram
7. Definition of parallelogram.
The converse of the second theorem we proved requires a slightly different approach.
4. If both pairs of opposite angles in a quadrilateral are congruent, then it is a parallelogram.
P
Given: PQRS is a quadrilateral in which ∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄 and ∠𝑃𝑆𝑄 ≅ ∠𝑅𝑄𝑆
Prove: PQRS is a parallelogram
Statements
1. PQRS is a quadrilateral in which
Reasons
S
x°
y°
1. Given
∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄 and ∠𝑃𝑆𝑄 ≅ ∠𝑅𝑄𝑆.
2. 2x  2y  360
2. The interior angle sum of a quadrilateral is 360°.
3. x  y  180
3. Multiplication property of equality
4. ∠𝑃𝑄𝑆 and ∠𝑅𝑆𝑄 are supplementary,
4. Definition of supplementary angles
and ∠𝑃𝑆𝑄 and ∠𝑅𝑄𝑆 are supplementary
5. 𝑃𝑄 ∥ 𝑅𝑆, 𝑄𝑅 ∥ 𝑃𝑆
5. Consecutive Interior Angles Converse
6. PQRS is a parallelogram
6. Definition of parallelogram.
Proofs, page 2
y°
x°
R
Q
For some proofs, we need both diagonals. We also use things we proved on page 1, to make our job easier.
Q
R
5. If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Given: PQRS is a parallelogram
M
Prove: 𝑀 bisects ̅̅̅̅
𝑄𝑆 and ̅̅̅̅
𝑃𝑅
P
Statements
S
Reasons
1. PQRS is a parallelogram.
1. Given
̅̅̅̅, ̅̅̅̅
2. ̅̅̅̅
𝑃𝑄 ≅ 𝑅𝑆
𝑄𝑅 ≅ ̅̅̅̅
𝑃𝑆
2. If a quadrilateral is a parallelogram, then its opposite sides
are congruent.
3. ∠𝑄𝑃𝑅 ≅ ∠𝑆𝑅𝑃, ∠𝑃𝑄𝑆 ≅ ∠𝑅𝑆𝑄;
3. Alternate Interior Angles Congruence Theorem
∠𝑅𝑃𝑆 ≅ ∠𝑄𝑅𝑃, ∠𝑃𝑆𝑄 ≅ ∠𝑅𝑄𝑆
4. ∆𝑃𝑀𝑄 ≅ ∆𝑅𝑀𝑆, ∆𝑄𝑀𝑅 ≅ ∆𝑆𝑀𝑃
4. ASA
5. ̅̅̅̅̅
𝑄𝑀 ≅ ̅̅̅̅
𝑆𝑀, ̅̅̅̅̅
𝑃𝑀 ≅ ̅̅̅̅̅
𝑅𝑀
5. Corr. parts of ≅ triangles are ≅.
6. 𝑀 bisects ̅̅̅̅
𝑄𝑆 and ̅̅̅̅
𝑃𝑅
6. Definitions of segment bisector.
The converse…
6. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Given: Diagonals JK and KM bisect each other
Prove: JKLM is a parallelogram
Statements
Justifications
1. Diagonals JK and KM bisect each other
1. Given
2. KP  MP and JP  LP
2. Definition of Segment Bisector
3. KPL  MPJ and KPJ  MPL
3. Vertical Angles Congruence Theorem
4. KPL  MPJ and KPJ  MPL
4. SAS
5. KJ  ML and JM  LK
5. Congruent Parts of Congruent Triangles are
Congruent
6. JKLM is a parallelogram
6. Thm. 8.7 – If opposite sides of a quadrilateral
are  then it is a parallelogram.
Proofs, page 3
Homework: Proofs
Name ________________________________________
1. The diagonals of a rectangle are congruent.
Given: Parallelogram ABCD is a rectangle.
A
B
D
C
Prove: AC  BD
Statements
1. Parallelogram ABCD is a rectangle
Justifications
1. Given
2. BC  AD
2. Opposite sides of a parallelogram are 
3. DC  DC
3. Reflexive Property
4. C  D
4. Definition of a Rectangle
5. ACD  BDC
5. SAS
6. AC  BD
6. Corresponding Parts of Congruent Triangles are Congruent
A
B
2. The diagonals of a rhombus are perpendicular
Given: Parallelogram ABCD is a rhombus.
Prove: AC  BD
D
C
Statements
Justifications
1. Parallelogram ABCD is a rhombus
1. Given
2. AB  BC  CD  DA
2. Definition of a rhombus
3. Draw AC and BD
3. Through any two points, there exists a line
4. AM  MC and BM  MD
4. Diagonals of a parallelogram bisect each other
5. AMD  DMC  CMB  BMA
5. SSS
6. AMD  DMC
6. Corresponding Parts of Congruent Triangles are Congruent
7. AC  BD
7. If two lines intersect to form a linear pair of congruent
angles, then the lines are perpendicular.
Proofs, page 4
W
Is this enough information to prove that WXYZ is a parallelogram?
X
Try to prove it on your own.
3. If opposite sides of a quadrilateral are parallel and congruent,
Z
Y
the quadrilateral is a parallelogram.
Given: WX ZY , WX  ZY
Prove: WXYZ is a parallelogram
We now have 5 ways to prove a quadrilateral is a parallelogram. Fill in the blanks. Mark the diagrams to show
the conditions under which we can be sure the quadrilateral is a parallelogram.
1.
Use the definition: Show that both pairs of _________________
sides are __________________.
2.
Use #3 from page 2: Show that both pairs of _________________
sides are ___________________.
3.
Use #4 from page 2: Show that both pairs of _________________
sides are ___________________.
4.
Use #6 from page 4: Show that the ______________________
bisect each other.
5.
Use homework #3 from above: Show that ____________________
sides are ____________________ and ________________________.
Proofs, page 5