Parallelograms
In this unit, we will focus our studies on quadrilaterals. A quadrilateral is a polygon with four sides. Quadrilaterals can be classified further if they have more properties than just having four sides. Parallelograms are one quadrilateral with some important properties.
In this unit, we will focus our studies on quadrilaterals. A quadrilateral is a polygon with four sides. Quadrilaterals can be classified further if they have more properties than just having four sides. Parallelograms are one quadrilateral with some important properties.
Based on the definition of a parallelogram, we can discover and prove other properties. Before we look at other properties we need the definition of a diagonal of a polygon.
Example 1:
Prove that opposite sides of parallelogram are congruent. Given: Parallelogram \(LUIS\) Prove: \(\overline{LU}\cong\overline{IS}\) Solution: We start with the definition of a parallelogram: the opposite sides are parallel. So \(\overline{LU}\parallel\overline{IS}\) and \(\overline{LS}\parallel\overline{UI}\). If we draw the diagonal \(\overline{US}\) we create two triangles. Let's work at proving those two triangles are congruent. What transformation that is needed to map \(\triangle{LUS}\cong\triangle{ISU}\)? From the parallel lines we have alternate interior angles that are congruent, \(\angle{LUS}\cong\angle{ISU}\) and \(\angle{LSU}\cong\angle{SIU}\). From the reflexive property \(\overline{US}\cong\overline{US}\). As a result\(\triangle{LUS}\cong\triangle{ISU}\) because of \(ASA\). Therefore \(\overline{LU}\cong\overline{IS}\) because corresponding parts of congruent triangles are congruent. |
Therefore, if a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. Though we can write proofs for each property of a parallelogram, we will use the following construction of a parallelogram to manipulate and discover those properties.
a) Measure the length of each side
b) Measure of each angle
c) Find the slope of each side.
In summary, if a quadrilateral is a parallelogram, then the properties of a parallelogram are listed below:
b) Measure of each angle
c) Find the slope of each side.
In summary, if a quadrilateral is a parallelogram, then the properties of a parallelogram are listed below:
Properties of a Parallelogram
- Both pairs of opposite sides are parallel (definition)
- Both pairs of opposite sides are congruent.
- Both pairs of opposite angles are congruent.
- All pairs of consecutive angles are supplementary.
- Diagonals bisect each other. That is, the diagonals split each other in half
Example 2:
Given \(NEIL\) is a parallelogram, find the value of \(x\). Describe the property you used. Solution: If \(NEIL\) is a parallelogram, then its opposite angles must be congruent. Thus, we will set the two expressions equal: \(\begin{align*}\\ m\angle{L} &= m\angle{E}\\ 2x + 5 &= 6x - 25\\ 2x - 6x + 5 &= 6x - 6x - 25\\ -4 x+ 5 - 5 &= -25 - 5\\ -4x &= -30\\ x &= \dfrac{-30}{-4} = \dfrac{15}{2} = 7.5 \end{align*}\) |
Rhombuses
A rhombus is a special type of parallelogram where all four sides are congruent. If the diagonals of a rhombus are drawn, four congruent right triangles are created. Just like the parallelogram, the rhombus has several properties. Manipulate the figure below to discover the properties.
A rhombus is a special type of parallelogram where all four sides are congruent. If the diagonals of a rhombus are drawn, four congruent right triangles are created. Just like the parallelogram, the rhombus has several properties. Manipulate the figure below to discover the properties.
Properties of a Rhombus:
- All four sides are congruent.
- Diagonals are perpendicular.
- Diagonals bisect the opposite angles.
- All properties of a parallelogram (listed above).
Example 3:
Given \(RE = 3\) units, \(EN = 5\) units, and \(BR = 2y\), find the value of \(y\) that will make the following figure a rhombus. Describe the properties you used to find the value of \(y\). Solution: Because this quadrilateral is a rhombus, we can put tick marks to show that all four sides are congruent, the diagonals are perpendicular, and the diagonals bisect each other. \(BE = EN = 5\) units. Now we will focus on \(\triangle BRE\), which is a right triangle where we know two of its side lengths. We can now use the Pythagorean Theorem. \(\begin{align*}\\ (BR)^2 + (RE)^2 &= (BE)^2\\ (2y)^2 + (3)^2 &= (5)^2\\ 4y^2 + 9&= 25\\ 4y^2 &= 16\\ y^2 &= 4\\ y &= \pm 2 \end{align*}\) We will use \(y = 2\) since that will produce a positive side length. |
A common misconception about rhombuses is that their diagonals are always congruent. Use this construction to help you understand that this is not always true.
Rectangles
A rectangle is a special type of parallelogram where all four angles congruent. This means that all the interior angles of a rectangle are right angles. Just like the parallelogram, the rectangle has several properties. Manipulate the figure below to discover the properties.
A rectangle is a special type of parallelogram where all four angles congruent. This means that all the interior angles of a rectangle are right angles. Just like the parallelogram, the rectangle has several properties. Manipulate the figure below to discover the properties.
Properties of a Rectangle:
- All four angles are congruent (right angles).
- Diagonals are congruent.
- All properties of a parallelogram (listed above).
Squares
A square is a special type of parallelogram that combines all properties of rhombuses and rectangles into one shape. So you can argue that a square is ALWAYS a rhombus (since it always has all of the properties of a rhombus), and a square is ALWAYS a rectangle. You can also say that a rhombus is SOMETIMES a rectangle: the only time when a single quadrilateral has the properties of a rhombus and a rectangle is when it’s a square.
Please note that one of the major differences between squares and rhombuses is that when you draw in both diagonals in a square, it creates four congruent \(45-45-90\) triangles. Manipulate the figure below to see the properties.
A square is a special type of parallelogram that combines all properties of rhombuses and rectangles into one shape. So you can argue that a square is ALWAYS a rhombus (since it always has all of the properties of a rhombus), and a square is ALWAYS a rectangle. You can also say that a rhombus is SOMETIMES a rectangle: the only time when a single quadrilateral has the properties of a rhombus and a rectangle is when it’s a square.
Please note that one of the major differences between squares and rhombuses is that when you draw in both diagonals in a square, it creates four congruent \(45-45-90\) triangles. Manipulate the figure below to see the properties.
Each of the above quadrilaterals is connected because of its properties. You can use the illustration below to see how they are connected.
Example 5:
Fill in the blank with always, sometimes, or never:
a) A rhombus is __?__ a square.
b) A square is __?__ a rectangle.
Solution:
For these type of problems, it is helpful to think about each figure as a list of properties. This problem is asking: “When does all of the properties of a rhombus have all the properties of a square?”
a) To classify a quadrilateral as a rhombus, we must prove that it is a parallelogram. Then, we must prove that its sides are all congruent. Now to further classify it as a square, we must prove that the angles are right or that the diagonals are congruent. Since this isn’t always the case, our answer to the problem is sometimes. A rhombus sometimes has the properties of a square - when it can be classified as a square.
b) This problem is asking: “When does all of the properties of a square have all the properties of a rectangle?” To classify a quadrilateral as a square, you must prove that it is a parallelogram that has all the properties of a rhombus AND the properties of a rectangle. Because of this, a square will always have all properties of a rectangle.
Fill in the blank with always, sometimes, or never:
a) A rhombus is __?__ a square.
b) A square is __?__ a rectangle.
Solution:
For these type of problems, it is helpful to think about each figure as a list of properties. This problem is asking: “When does all of the properties of a rhombus have all the properties of a square?”
a) To classify a quadrilateral as a rhombus, we must prove that it is a parallelogram. Then, we must prove that its sides are all congruent. Now to further classify it as a square, we must prove that the angles are right or that the diagonals are congruent. Since this isn’t always the case, our answer to the problem is sometimes. A rhombus sometimes has the properties of a square - when it can be classified as a square.
b) This problem is asking: “When does all of the properties of a square have all the properties of a rectangle?” To classify a quadrilateral as a square, you must prove that it is a parallelogram that has all the properties of a rhombus AND the properties of a rectangle. Because of this, a square will always have all properties of a rectangle.
Quick Check
1) Quadrilateral \(EDGA\) is a rhombus. Answer the following questions and describe any properties that you may have used. a)What is the \(m\angle{DRG}\)? b) Given \(m\angle{RED}= 36^{\circ}\), what is \(m\angle{RGD}\)? c) Given \(AD = 10\) units and \(AG = 8.5\) units, what is the length of \(\overline{EG}\)? 2) Fill in the blanks with always, sometimes, or never. a) A rectangle is __?__ a parallelogram. b) A parallelogram is __?__ a quadrilateral. c) A rhombus __?__ has opposite angles congruent. d) A rhombus __?__ has congruent diagonals. Quick Check Solutions |