Middle Years

Lesson

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.

Each mode always makes a kind of special 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 mode.

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.

Can you make a rectangle? What about a square?

#### Practice questions

##### Question 1

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?

1. It will always be a parallelogram but may not be a rectangle.

A

It will always be a rectangle but may not be a square.

B

It will always be a kite but may not be a parallelogram.

C

It will always be a square.

D

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. We could make the kite below by reflecting $\triangle DAB$DAB across its side $DB$DB, and this side of the triangle becomes one of the diagonals of the kite we make. Drawing in the other diagonal $AC$AC splits the kite into four triangles, and we call the point where the diagonals cross $X$X:

Since $\triangle DAB\equiv\triangle DCB$DABDCB, we know that $\angle DBA=\angle DBC$DBA=DBC. For the same reason we also know that $BA=BC$BA=BC.

Thinking about $\triangle ABX$ABX and $\triangle CBX$CBX, the side $BX$BX is common to both.

Putting this all together means we can conclude that $\triangle ABX\equiv\triangle CBX$ABXCBX 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 $\angle BXA=\angle BXC$BXA=BXC.

Since these angles are supplementary, we know that $\angle BXA+\angle BXC=180^\circ$BXA+BXC=180°.

So the two angles $\angle BXA$BXA and $\angle BXC$BXC are equal, and they also add to $180^\circ$180°. 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. This time we rotate the triangle $\triangle DAB$DAB around the midpoint $X$X of the side $DB$DB, so like before this side becomes a diagonal of the parallelogram we create. Drawing in the other diagonal $AC$AC splits the parallelogram into four triangles:

Since $\angle AXB$AXB and $\angle CXD$CXD are vertically opposite, they must be equal. The sides $AB$AB and $CD$CD are corresponding, so they must be equal too. These sides are also parallel, which means $\angle ABX$ABX and $\angle CDX$CDX are alternate angles on parallel lines.

Putting this all together tells us that $\triangle AXB\equiv\triangle CXD$AXBCXD by the AAS test.

Since $AX$AX and $CX$CX are corresponding sides in these congruent triangles, they must be equal. In other words, $X$X is not only the midpoint of the diagonal $BD$BD, it is also the midpoint of the diagonal $AC$AC. In other words,

The diagonals of a parallelogram bisect each other.

Finally, let's look at a rectangle. When we draw in the diagonals, we create many congruent triangles:

Let's look at the triangles $\triangle DBA$DBA and $\triangle CAB$CAB. We know that the side $AB$AB is common to both triangles, and we know $DA=CB$DA=CB as well. The angles $\angle DAB$DAB and $\angle CBA$CBA are both right angles, and are therefore equal.

We can therefore conclude that $\triangle DBA\equiv\triangle CAB$DBACAB by the SAS test.

The sides $DB$DB and $CA$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.

Summary

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

#### Practice questions

##### Question 2

The triangles $\triangle ABC$ABC and $\triangle CDA$CDA are congruent.

1. Fill in the blanks, to state pairs of equal angles.

$\angle CAB=\angle\editable{}$CAB=

$\angle BCA=\angle\editable{}$BCA=

2. Which two of the following are true about the quadrilateral $ABCD$ABCD?

$AD\parallel BC$ADBC
Because $\angle ACD=\angle CAB$ACD=CAB

A

$AD\parallel BC$ADBC
Because $\angle DAC=\angle BCA$DAC=BCA

B

$AB\parallel DC$ABDC
Because $\angle ACD=\angle CAB$ACD=CAB

C

$AB\parallel DC$ABDC
Because $\angle DAC=\angle BCA$DAC=BCA

D
3. Which one of the following is true of the quadrilateral $ABCD$ABCD?

$ABCD$ABCD must be a rhombus, but is not necessarily a square.

A

$ABCD$ABCD must be a parallelogram, but is not necessarily a rhombus.

B

$ABCD$ABCD must be a square.

C

$ABCD$ABCD must be a rectangle, but is not necessarily a parallelogram.

D
4. Suppose that $\angle ACD=36^\circ$ACD=36° and $\angle BCA=44^\circ$BCA=44°. Find the size of $\angle ABC$ABC.

##### Question 3

The diagonals of a quadrilateral have these properties:

• They are equal in length
• They bisect each other
• They are not perpendicular

1. Kite

A

Parallelogram

B

Square

C

Rectangle

D
##### Question 4

The diagonals of this kite intersect at $T$T, splitting the kite into four triangles.

The diagonal $PR$PR bisects the angles $\angle QRS$QRS and $\angle SPQ$SPQ.

1. Complete the following:

$\triangle RST$RST and $\triangle\editable{}$ are congruent triangles.

2. Which one of the following is true?

$\angle RTS=\angle TQR$RTS=TQR

A

$\angle RTS=\angle TRQ$RTS=TRQ

B

$\angle RTS=\angle RTQ$RTS=RTQ

C
3. Which one of the following is true?

$\angle RTS+\angle RTQ=180^\circ$RTS+RTQ=180°

A

$\angle RTS+\angle RTQ=90^\circ$RTS+RTQ=90°

B

$\angle RTS+\angle RTQ=360^\circ$RTS+RTQ=360°

C
4. Which two of the following are true about the diagonals of this kite?

$RP$RP bisects $SQ$SQ.

A

$RP$RP is perpendicular to $SQ$SQ.

B

$RS$RS is perpendicular to $RQ$RQ.

C

$SQ$SQ bisects $RP$RP.

D