Is a rectangle a parallelogram? If so, what does this mean about the properties of a rectangle?
Theorems about Quadrilaterals
A quadrilateral is a polygon with four sides. Five special quadrilaterals are shown below, along with their definitions and pictures.
The formal definitions of these quadrilaterals only give some information about them. Each quadrilateral has other properties that can be proved. For example, while a parallelogram is defined as a quadrilateral with two pairs of parallel sides, it can be proved that the opposite sides of a parallelogram must be congruent.
Suppose you are given a quadrilateral and believe that it is a parallelogram. You can prove that it is a parallelogram by showing that it has two pairs of parallel sides; however, you can also use other properties that are unique to parallelograms to prove that the given shape is a parallelogram.
Let's prove that the diagonals of a parallelogram divide the parallelogram into congruent triangles. Then, we'll use this to prove that the opposite sides of a parallelogram are congruent.
1. Start by drawing a generic parallelogram and previewing this proof.
All you can assume in this proof is the definition of a parallelogram. This means that all you know is that the shape is a quadrilateral with two pairs of parallel sides. Your first goal is to prove that the diagonals divide the parallelogram into congruent triangles. You can use the parallel lines to give you congruent angles, which will help you to prove that the triangles are congruent.
Your second goal is to prove that the opposite sides of the parallelogram are congruent. Since the opposite sides of the parallelogram are corresponding parts of the congruent triangles, you can use CPCTC to show that they must be congruent.
Given: Parallelogram \begin{align*}ABCD\end{align*}
Prove: \begin{align*}\overline{AB} \cong \overline{DC}\end{align*} and \begin{align*}\overline{AD} \cong \overline{BC}\end{align*}
Here is a two-column proof:
Statements |
Reasons |
Parallelogram \begin{align*}ABCD\end{align*} |
Given |
\begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*} and \begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*} |
Definition of a parallelogram |
\begin{align*}\angle BAC \cong \angle DCA \\ \angle ABD \cong \angle BDC \\ \angle DAC \cong \angle BCA \\ \angle ADB \cong \angle DBC\end{align*} |
Alternate interior angles are congruent if lines are parallel |
\begin{align*}\overline{BD} \cong \overline{BD} \\ \overline{AC} \cong \overline{AC}\end{align*} |
Reflexive Property |
\begin{align*}\Delta ADB \cong \Delta CBD \\ \Delta ADC \cong \Delta CBA\end{align*} |
\begin{align*}ASA \cong\end{align*} |
\begin{align*}\overline{AB} \cong \overline{DC}\end{align*} and \begin{align*}\overline{AD} \cong \overline{BC}\end{align*} |
CPCTC |
You have now proven two theorems about parallelograms. You can use these theorems in future proofs without proving them again.
Parallelogram Theorem #1: Each diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Parallelogram Theorem #2: The opposite sides of a parallelogram are congruent.
2. Now, let's prove that if a quadrilateral has opposite sides congruent, then its diagonals divide the quadrilateral into congruent triangles. Use this to prove that the quadrilateral must be a parallelogram.
These are the converses of the parallelogram theorems proved in Example A. Draw a generic quadrilateral with two pairs of congruent sides and preview the proof.
Your first goal is to show that the set of triangles created by each diagonal must be congruent. You can use the congruent opposite sides as well as the reflexive property to show that the triangles are congruent with \begin{align*}SSS \cong\end{align*}.
Your second goal is to prove that the opposite sides must be parallel and so therefore the quadrilateral is a parallelogram. You can show that alternate interior angles are congruent and hence lines are parallel for this part of the proof.
Given: Quadrilateral \begin{align*}ABCD\end{align*} with \begin{align*}\overline{AB} \cong \overline{DC}\end{align*} and \begin{align*}\overline{AD} \cong \overline{BC}\end{align*}.
Prove: \begin{align*}ABCD\end{align*} is a parallelogram
Here is a two-column proof:
Statements |
Reasons |
\begin{align*}\overline{AB} \cong \overline{DC}\end{align*} and \begin{align*}\overline{AD} \cong \overline{BC}\end{align*} |
Given |
\begin{align*}\overline{BD} \cong \overline{BD} \\ \overline{AC} \cong \overline{AC}\end{align*} |
Reflexive Property |
\begin{align*}\Delta ADB \cong \Delta CBD \\ \Delta ADC \cong \Delta CBA\end{align*} | \begin{align*}SSS \cong\end{align*} |
\begin{align*}\angle BAC \cong \angle DCA \\ \angle ABD \cong \angle BDC \\ \angle DAC \cong \angle BCA \\ \angle ADB \cong \angle DBC \end{align*} |
CPCTC |
\begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*} and \begin{align*}\overline{AD} \ \| \ \overline{BC} \end{align*} |
If alternate interior angles are congruent then lines are parallel |
\begin{align*}ABCD\end{align*} is a parallelogram |
Definition of a parallelogram |
You have now proven the converses of the first two parallelogram theorems. These are two additional ways to show that a quadrilateral is a parallelogram besides showing that the quadrilateral satisfies the definition of a parallelogram.
Parallelogram Theorem #1 Converse: If each of the diagonals of a quadrilateral divide the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram.
Parallelogram Theorem #2 Converse: If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
3. To get another theorem for parallelograms, let's prove that the opposite angles of a parallelogram are congruent.
Draw a generic parallelogram and preview the proof. Your goal is to prove that the opposite angles are congruent. Remember that you can use the theorems that have already been proven about parallelograms, so you can use the fact that the triangles created by the diagonals must be congruent.
Given: Parallelogram \begin{align*}ABCD\end{align*}.
Prove: \begin{align*}\angle A \cong \angle C \end{align*} and \begin{align*}\angle B \cong \angle D \end{align*}
Here is a paragraph proof:
Because each diagonal of a parallelogram divides the parallelogram into two congruent triangles, \begin{align*}\Delta ADB \cong \Delta CBD \end{align*} and \begin{align*}\Delta ADC \cong \Delta CBA \end{align*}. \begin{align*}\angle A\end{align*} and \begin{align*}\angle C\end{align*} are corresponding parts of \begin{align*}\Delta ADB\end{align*} and \begin{align*}\Delta CBD\end{align*}. Similarly, \begin{align*}\angle B\end{align*} and \begin{align*}\angle D\end{align*} are corresponding parts of \begin{align*}\Delta ADC\end{align*} and \begin{align*}\Delta CBA\end{align*}. This means that \begin{align*}\angle A \cong \angle C\end{align*} and \begin{align*}\angle B \cong \angle D\end{align*} because corresponding parts of congruent triangles are congruent.
You have now proven a third theorem about parallelograms. You can use this theorem in future proofs without proving it again.
Parallelogram Theorem #3: The opposite angles of a parallelogram are congruent.
4. Finally, let's prove that the diagonals of a parallelogram bisect each other.
Draw a generic parallelogram and preview the proof.
What does it mean for the diagonals to bisect each other? If the diagonals bisect each other then each diagonal crosses the other diagonal at its midpoint. This would mean that \begin{align*}\overline{AE} \cong \overline{EC}\end{align*} and \begin{align*}\overline{ED} \cong \overline{BE}\end{align*}.
Your goal is to use the parallelogram definition and theorems to show that \begin{align*}\overline{AE} \cong \overline{EC}\end{align*} and \begin{align*}\overline{ED} \cong \overline{BE}\end{align*}. First, try to prove that the two diagonals divide the parallelogram into four small triangles, and each pair of these triangles is congruent.
Given: Parallelogram \begin{align*}ABCD\end{align*}
Prove: \begin{align*}\overline{AE} \cong \overline{EC} \end{align*} and \begin{align*}\overline{ED} \cong \overline{BE}\end{align*} (the diagonals bisect each other)
Here is a flow diagram proof.
You have now proven a fourth theorem about parallelograms. You can use this theorem in future proofs without proving it again.
Parallelogram Theorem #4: The diagonals of a parallelogram bisect each other.
Examples
Example 1
Earlier, you were asked if a rectangle is a parallelogram.
To prove that a rectangle is a parallelogram, you must prove that it either satisfies the definition of a parallelogram or satisfies any of the theorems that prove that quadrilaterals are parallelograms.
A rectangle is a quadrilateral with four right angles. You can use these angles to show that the opposite sides of a rectangle must be parallel.
Given: Rectangle \begin{align*}ABCD\end{align*}
Prove: \begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*} and \begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*}
Here is a paragraph proof:
A rectangle has four right angles by definition, so \begin{align*}m \angle A=m \angle B=m \angle C=m \angle D=90^\circ\end{align*}. \begin{align*}\angle A\end{align*} and \begin{align*}\angle D\end{align*} are same side interior angles. \begin{align*}m \angle A+m \angle D=180^\circ\end{align*}, which means that \begin{align*}\angle A\end{align*} and \begin{align*}\angle D\end{align*} are supplementary. If same side interior angles are supplementary, then lines are parallel. This means that \begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*}. Similarly, \begin{align*}\angle A \end{align*} and \begin{align*}\angle B \end{align*} are same side interior angles. \begin{align*}m \angle A+m \angle B=180^\circ\end{align*}, which means that \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary. If same side interior angles are supplementary, then lines are parallel. This means that \begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*}.
You have proven that a rectangle is a parallelogram.
Rectangle Theorem #1: A rectangle is a parallelogram.
This means that rectangles have all the same properties as parallelograms. Like parallelograms, rectangles have opposite sides congruent and parallel and diagonals that bisect each other.
Example 2
Prove that the diagonals of a rectangle are congruent.
Draw a rectangle with its diagonals and preview the proof. To prove that the diagonals are congruent, you will first want to prove that \begin{align*}\Delta ADC \cong \Delta BCD\end{align*}. These are two right triangles and their hypotenuses are the diagonals of the rectangle. If you can prove that these two right triangles are congruent, then you can prove that the diagonals are congruent.
Given: Rectangle \begin{align*}ABCD\end{align*}
Prove: \begin{align*}\overline{AC} \cong \overline{BD}\end{align*}
Here is a flow diagram proof:
You have proven that a rectangle has congruent diagonals. You can now use this theorem in future proof.
Rectangle Theorem #2: A rectangle has congruent diagonals.
Example 3
Prove that if a quadrilateral has diagonals that bisect each other, then it is a parallelogram.
This is the converse of parallelogram theorem #4 from Example C. Draw a quadrilateral with diagonals that bisect each other and preview the proof.
Your goal will be to show that there are two pairs of congruent triangles created by these diagonals. Then, you can show that alternate interior angles are congruent and therefore the opposite sides must be parallel.
Given: \begin{align*}\overline{AE} \cong \overline{EC}\end{align*} and \begin{align*}\overline{ED} \cong \overline{BE}\end{align*}
Prove: \begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*} and \begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*} (\begin{align*}ABCD\end{align*} is a parallelogram)
Here is a flow diagram proof:
You have now proven the converse of the fourth parallelogram theorem. This is one additional way to show that a quadrilateral is a parallelogram.
Parallelogram Theorem #4 Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Example 4
Prove that a rhombus is a parallelogram. What does this tell you about the properties of a rhombus?
Is a rhombus a parallelogram? This would mean that a rhombus has opposite sides that are parallel. To prove that a rhombus is a parallelogram, you must prove that it either satisfies the definition of a parallelogram or satisfies any of the theorems that prove that quadrilaterals are parallelograms.
Here is a paragraph proof:
A rhombus is a quadrilateral with four congruent sides, therefore opposite sides of a rhombus are congruent. Parallelogram theorem #2 converse states that “if the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram”. Therefore, a rhombus is a parallelogram.
You have proven that a rhombus is a parallelogram.
Rhombus Theorem #1: A rhombus is a parallelogram.
This means that rhombuses have all the same properties as parallelograms. Like parallelograms, rhombuses have opposite sides parallel, opposite angles congruent and diagonals that bisect each other.
Review
1. State the definition of a parallelogram and three additional properties of a parallelogram.
2. State the definition of a rectangle and four additional properties of a rectangle.
3. Use the rhombus below to prove that a rhombus has perpendicular diagonals.
- Use the definition of a rhombus and the theorems about parallelograms to prove that \begin{align*}\Delta AED \cong \Delta AEB\end{align*}.
- Prove that \begin{align*}m \angle AED=m \angle AEB\end{align*}.
- Prove that \begin{align*}\angle AED\end{align*} is a right angle.
- Prove that \begin{align*}AC \perp DB\end{align*}.
4. Use the rhombus below to prove that a rhombus has diagonals that bisect its angles.
- Use the definition of a rhombus and the theorems about parallelograms to prove that \begin{align*}\Delta ADC \cong \Delta ABC\end{align*} and \begin{align*}\Delta ADB \cong \Delta CDB\end{align*}.
- Prove that there are congruent angles and therefore the diagonals have bisected the angles.
5. State the definition of a rhombus and five additional properties of a rhombus.
6. Is a square a rectangle? Is a square a rhombus? Explain.
7. Is a square a parallelogram? Explain.
8. State the definition of a square and five additional properties of a square.
Use the kite below for the proofs in #9-#13.
9. Prove that one diagonal of a kite divides it into two congruent triangles.
10. Prove that a kite has one pair of opposite angles congruent. Hint: Use the result to #9!
11. Prove that one diagonal of a kite bisects its angles. Hint: Draw the diagonal that bisects its angles. Use the result of #9 and then CPCTC.
12. Prove that one diagonal of a kite is bisected by another diagonal. Hint: Draw both diagonals. Prove that two of the four small triangles are congruent and then use CPCTC. Use the result of #11 to help.
13. Prove that the diagonals of a kite are perpendicular. Hint: Use the result of #11 and a similar method to the one that was used in #3!
14. State the definition of a kite and four additional properties of a kite.
15. Use the picture below with \begin{align*}\overline{AC} \cong \overline{BD}\end{align*} to help prove that a parallelogram with congruent diagonals is a rectangle.
16. Use the picture below to help prove that a quadrilateral with opposite angles congruent is a parallelogram. Hint: Use the fact that the sum of the interior angles of a quadrilateral is \begin{align*}360^\circ\end{align*}.
Review (Answers)
To see the Review answers, open this PDF file and look for section 4.7.