Chapter 13 · Two-dimensional figures
Identify and Draw Quadrilaterals by Their Properties
Name trapezoids, parallelograms, rectangles, rhombuses and squares by checking sides and angles. Then draw a quadrilateral from a list of properties and check it.
A quadrilateral is a closed shape with four straight sides and four angles. Each corner is called a vertex. Two or more are called vertices.
Quadrilaterals have family names, such as rectangle and rhombus. You do not choose a name by how a shape looks. You choose it by the shape's properties: facts about its sides and angles. In this lesson you check three kinds of facts:
- which sides are parallel: they stay the same distance apart, so they would never meet even if you made them longer;
- which angles are right angles: square corners, where two perpendicular sides meet;
- which sides are equal in length.
Every fact must come from what the problem tells you, or from the coordinates of the vertices on a grid. In a pair such as , the first number is the -coordinate: it counts units across, along the horizontal -axis. The second number is the -coordinate: it counts units up, along the vertical -axis.
Each name is a promise about parallel sides, right angles or equal sides. The names build up, one promise at a time.
- A trapezoid is a quadrilateral with at least one pair of parallel sides.
- A parallelogram is a quadrilateral with two pairs of parallel sides. Each side is parallel to its opposite side, the side across from it.
- A rectangle is a parallelogram with four right angles.
- A rhombus is a parallelogram with four equal sides.
- A square is a parallelogram with four right angles and four equal sides.
Notice what the promises leave out. A trapezoid may have a second pair of parallel sides. A rectangle may have four equal sides. A rhombus may have four right angles. Each of these five names tells you what a shape must have. It never tells you what a shape must not have.
Key concept
A shape gets every name it earns
To name a quadrilateral, check its parallel sides, its right angles and its equal sides. Use only facts that are stated or that follow from the coordinates. Then give the shape every name whose properties it has.
- Two pairs of parallel sides count as at least one pair. So every parallelogram is also a trapezoid.
- Every rectangle and every rhombus is a parallelogram. So each one is also a trapezoid.
- Every square is a rectangle and a rhombus. So a square has all five names.
Worked example
Name a quadrilateral from its vertices
The figure above shows a quadrilateral with vertices , , and , joined in that order to make a closed shape. Which of the five names does it have?
Check the bottom and top sides. The bottom side joins (2, 1) and (6, 1). The top side joins (2, 5) and (6, 5). Each has the same y-coordinate at both ends, so both are horizontal. Their y-coordinates are different, 1 and 5, so these two sides are parallel. One pair of parallel sides is enough for a trapezoid.
Check the left and right sides. The left side has x-coordinate 2 at both ends. The right side has x-coordinate 6 at both ends. Both are vertical with different x-coordinates, so they are parallel too. Two pairs of parallel sides make a parallelogram. So far the shape is a trapezoid and a parallelogram.
Check the angles. At each of the four vertices, a horizontal side meets a vertical side. Those sides are perpendicular, so every angle is a right angle. A parallelogram with four right angles is a rectangle.
Check the side lengths. Each horizontal side runs from x-coordinate 2 to x-coordinate 6, so subtract to get 4 units. Each vertical side runs from y-coordinate 1 to y-coordinate 5, so subtract to get 4 units. All four sides are equal. A parallelogram with four equal sides is a rhombus.
Put the checks together. The shape is a parallelogram with four right angles and four equal sides. That is the definition of a square.
Answer
The quadrilateral has all five names. It is a trapezoid, a parallelogram, a rectangle, a rhombus and a square.
Worked example
Name a quadrilateral with slanted sides
The figure above shows a quadrilateral with vertices , , and , joined in that order to make a closed shape. Two of its sides are slanted, which means they are not horizontal or vertical. You are told that each slanted side is 5 units long. Which of the five names does it have?
Check the bottom and top sides. The bottom side has y-coordinate 1 at both ends, and the top side has y-coordinate 5 at both ends. Both are horizontal with different y-coordinates, so they are parallel. That one pair makes the shape a trapezoid.
Check the slanted sides. Count the steps from the lower end to the upper end. Left side, from (1, 1) to (4, 5): 3 right and 4 up. Right side, from (6, 1) to (9, 5): 3 right and 4 up. Opposite sides with the same steps slant the same way, so they are parallel. Now the shape has two pairs of parallel sides, so it is also a parallelogram.
Check the angles. The bottom side is horizontal. A side that makes a right angle with a horizontal side must be vertical. The left side moves 3 units right as it goes up, so it is not vertical. The angle at (1, 1) is not a right angle. A rectangle and a square each need four right angles, so the shape is neither.
Check the side lengths. The bottom side runs from x-coordinate 1 to x-coordinate 6, so it is 5 units long. The top side runs from x-coordinate 4 to x-coordinate 9, so it is 5 units long. You are told that each slanted side is 5 units long. So all four sides are equal. A parallelogram with four equal sides is a rhombus.
Answer
The quadrilateral is a trapezoid, a parallelogram and a rhombus. It is not a rectangle or a square, because the angle at is not a right angle.
Now turn the question around. Start with a name and list the properties that define it. These are the properties a shape must have to earn that name.
The table below lists them for all five names. "No rule" means the name does not ask for that property, so a shape with that name may have it or not.
A shape earns the name in every row whose rules it meets. The first example met all five rows. The second example met the trapezoid, parallelogram and rhombus rows. It missed the rectangle and square rows, because both ask for four right angles. A shape can also stop early. With two pairs of parallel sides, but without four right angles and without four equal sides, it earns two names: parallelogram and trapezoid.
| Name | Pairs of parallel sides | Four right angles | Four equal sides |
|---|---|---|---|
Trapezoid | at least 1 | no rule | no rule |
Parallelogram | 2 | no rule | no rule |
Rectangle | 2 | must have | no rule |
Rhombus | 2 | no rule | must have |
Square | 2 | must have | must have |
Worked example
Draw a quadrilateral from a list of properties
Draw a quadrilateral on a coordinate grid that has all three properties: (1) one pair of parallel sides, (2) two other sides that are not parallel, (3) two right angles. Check each property on your drawing. Then name the shape.
Plan the sides so each property gets built in. Draw a bottom side from (1, 1) to (7, 1) and a top side from (1, 5) to (4, 5). Both are horizontal, to make the parallel pair. Join (1, 1) to (1, 5) with a vertical side, to make right angles. Join (7, 1) to (4, 5) with a slanted side, 3 left and 4 up, so it will not be parallel to the vertical side. The figure below shows the drawing.
Check property 1. The bottom side has y-coordinate 1 at both ends, and the top side has y-coordinate 5 at both ends. Both are horizontal with different y-coordinates, so they are parallel. Property 1 holds.
Check property 2. The left side has x-coordinate 1 at both ends, so it is vertical. A side parallel to a vertical side must be vertical too. The right side goes from x-coordinate 7 to x-coordinate 4, so it is not vertical. The left and right sides are not parallel. Property 2 holds.
Check property 3. At (1, 1) and at (1, 5), the vertical left side meets a horizontal side, so both angles are right angles. At (7, 1) and at (4, 5), a horizontal side meets the slanted side. A right angle there would need a vertical side, and the slanted side is not vertical. So the shape has two right angles. Property 3 holds.
Name the shape. It has at least one pair of parallel sides, so it is a trapezoid. It does not have two pairs, so it is not a parallelogram. A rectangle, a rhombus and a square are all parallelograms, so it is none of those.
Answer
The quadrilateral with vertices , , and has all three properties. It is a trapezoid, and it has none of the other four names. Other drawings can work too, as long as every property checks out.
Takeaway: a quadrilateral's names come from its properties, not from how it looks. Check its parallel sides, its right angles and its equal sides, using only stated facts and coordinates. Then give the shape every name it earns. That is why a square is also a rectangle, a rhombus, a parallelogram and a trapezoid. To draw a quadrilateral from a list of properties, plan the sides so each property gets built in. Then check every property on your drawing.