We have already learned about the properties of various special quadrilaterals:
Many of these features can be explained by constructing the quadrilateral from a pair of congruent triangles.
If we want to make a quadrilateral from two congruent triangles we need to connect a pair of corresponding sides together. This makes sure that the points match up exactly.
Another way to think about this construction is to start with one triangle, and make a copy of the triangle with a rotation or a reflection.
Try making some of your own quadrilaterals with the applet below:
Each type of transformation always makes a special kind of quadrilateral.
If we use a reflection to make the second triangle, the quadrilateral we form always has two pairs of equal adjacent sides. This means a reflection always makes a kite.
If we use a rotation to make the second triangle, the quadrilateral we form always has two pairs of equal opposite sides. This means rotation always makes a parallelogram.
There are some special quadrilaterals that we can make by using either transformation.
If we start with an isosceles triangle, then reflecting it across its base produces the same quadrilateral as rotating it around the midpoint of its base. The result is always a rhombus, which explains why rhombuses are both parallelograms and kites at the same time.
Suppose we have a right-angled scalene triangle, then create another triangle by rotating the original around the middle of its longest side.
If we join the triangles together, which of the following must be true of the resulting quadrilateral?
By tranforming triangles we can make different types of quadrilaterals.
A reflection always makes a kite.
A rotation always makes a parallelogram.
When we made our special quadrilaterals by joining two congruent triangles together, the sides of the triangles that were joined together formed one of the diagonals of the shape.
If we draw in the other diagonal of a quadrilateral we will make four triangles in total. Sometimes all four of these triangles will be congruent to each other, and sometimes there will only be two pairs of two. But in either case we can use these four triangles to investigate some interesting properties of the diagonals.
Let's have a look at the diagonals of a kite first.
Putting this all together means we can conclude that \triangle ABX \equiv \triangle CBX by the SAS test.
Now that we know that these triangles are congruent, we can match up another pair of corresponding angles and conclude that \angle BXA = \angle BXC.
Since these angles are supplementary, we know that \angle BXA + \angle BXC = 180\degree.
So the two angles \angle BXA and \angle BXC are equal, and they also add to 180\degree. This means they are both right angles. In other words: the diagonals of a kite are perpendicular to each other.
Now let's investigate parallelograms.
Since \angle AXB and \angle CXD are vertically opposite, they must be equal. The sides AB and CD are corresponding, so they must be equal too. These sides are also parallel, which means \angle ABX and \angle CDX are alternate angles on parallel lines.
Putting this all together tells us that \triangle AXB \equiv \triangle CXD by the AAS test.
Since AX and CX are corresponding sides in these congruent triangles, they must be equal. In other words, X is not only the midpoint of the diagonal BD, it is also the midpoint of the diagonal AC. In other words: the diagonals of a parallelogram bisect each other.
Finally, let's look at a rectangle.
We can therefore conclude that \triangle DBA \equiv \triangle CAB by the SAS test.
The sides DB and CA are corresponding sides, and must be equal. But these are also the diagonals of the rectangle. In other words: the diagonals of a rectangle are equal in length.
There are other properties we could prove as well - at least one diagonal of a kite bisects the other, some of the quadrilaterals have diagonals that bisect the angles they pass through, and so on.
All of these properties can also be proved using the congruent triangles that form from drawing the diagonals.
The triangles \triangle ABC and \triangle CDA are congruent.
Fill in the blanks, to state pairs of equal angles.
\angle CAB = \angle ⬚ \\ \angle BCA = \angle ⬚
Which two of the following are true about the quadrilateral ABCD?
Which one of the following is true of the quadrilateral ABCD?
Suppose that \angle ACD = 36\degree and \angle BCA = 44\degree. Find the size of \angle ABC.
The diagonals of this kite intersect at T, splitting the kite into four triangles.
The diagonal PR bisects the angles \angle QRS and \angle SPQ.
Which triangle is congruent to \triangle RST?
Which one of the following is true?
Which one of the following is true?
Which two of the following are true about the diagonals of this kite?
These are the properties of the diagonals of special quadrilaterals:
Kite | Parallelogram | Rectangle | Rhombus | Square | |
---|---|---|---|---|---|
Diagonals are perpendicular | Yes | Yes | Yes | ||
One diagonal bisects the other | Yes | Yes | Yes | Yes | Yes |
Both diagonals bisect each other | Yes | Yes | Yes | Yes | |
One diagonal bisects the angles it passes through | Yes | Yes | Yes | ||
Both diagonals bisect the angles they pass through | Yes | Yes | |||
Diagonals are equal in length | Yes | Yes |