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You are here: Home Grade 10 Mathematics Euclidean geometry Quadrilaterals

Quadrilaterals

Definition 1: Quadrilateral

A quadrilateral is a closed shape consisting of four straight line segments.

Polygons

Parallelogram

Definition 2: Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Example 1: Properties of a parallelogram

Question

ABCD is a parallelogram with ABDC and ADBC. Show that:

  1. AB=DC and AD=BC

  2. A=C and B=D

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Answer

Connect AC to form ABC and CDA
Use properties of parallel lines to indicate all equal angles on the diagram
Prove ABCCDA

In ABC and CDA,

A2=C3(alt.int.'s,ABDC)C4=A1(alt.int.'s,BCAD)ACisacommonsideABCCDA(AAS)AB=CD  and  BC=DA(1)

Opposite sides of a parallelogram have equal length.

We have already shown A2=C3 and A1=A4. Therefore,

A=A1+A2=C3+C4=C.(2)

Furthermore,

B=D(ABCCDA)(3)

Therefore opposite angles of a parallelogram are equal.

Summary of the properties of a parallelogram:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal in length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

Figure 1
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Example 2: Proving a quadrilateral is a parallelogram

Question

Prove that if both pairs of opposite angles in a quadrilateral are equal, the quadrilateral is a parallelogram.

Image

Answer

Find the relationship between x and y

In WXYZ

W=Y=y(given)Z=X=x(given)W+X+Y+Z=360°(sumofint.'sofquad.)2x+2y=360°x+y=180°W+Z=x+y=180°(4)

But these are co-interior angles between lines WX and ZY. Therefore WXZY.

Find parallel lines

Similarly W+X=180°. These are co-interior angles between lines XY and WZ. Therefore XYWZ.

Both pairs of opposite sides of the quadrilateral are parallel, therefore WXYZ is a parallelogram.

Investigation 1: Proving a quadrilateral is a parallelogram

  1. Prove that if both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

  2. Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

  3. Prove that if one pair of opposite sides of a quadrilateral are both equal and parallel, then the quadrilateral is a parallelogram.

A quadrilateral is a parallelogram if:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal.

  • Both pairs of opposite angles are equal.

  • The diagonals bisect each other.

  • One pair of opposite sides are both equal and parallel.

Exercise 1:

Prove that the diagonals of parallelogram MNRS bisect one another at P.

Image

Hint: Use congruency.

Rectangle

Definition 3: Rectangle

A rectangle is a parallelogram that has all four angles equal to 90°.

A rectangle has all the properties of a parallelogram:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal in length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

It also has the following special property:

Example 3: Special property of a rectangle

Question

PQRS is a rectangle. Prove that the diagonals are of equal length.

Image

Answer

Connect P to R and Q to S to form PSR and QRS
Use the definition of a rectangle to fill in on the diagram all equal angles and sides
Prove PSRQRS

In PSR and QRS

PS=QR(equalopp.sidesofrectangle)SRisacommonsidePSR=QRS=90°(int.ofrectangle)PSRQRS(RHS)ThereforePR=QS(5)

The diagonals of a rectangle are of equal length.

Summary of the properties of a rectangle:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are of equal length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

  • Diagonals are equal in length.

  • All interior angles are equal to 90°.

Figure 2
Image

Exercise 2:

ABCD is a quadrilateral. Diagonals AC and BD intersect at T. AC=BD, AT=TC, DT=TB. Prove that:

  1. ABCD is a parallelogram.

  2. ABCD is a rectangle.

Image

Rhombus

Definition 4: Rhombus

A rhombus is a parallelogram with all four sides of equal length.

A rhombus has all the properties of a parallelogram:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal in length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

It also has two special properties:

Example 4: Special properties of a rhombus

Question

XYZT is a rhombus. Prove that:

  1. the diagonals bisect each other perpendicularly;

  2. the diagonals bisect the interior angles.

Image

Answer

Use the definition of a rhombus to fill in on the diagram all equal angles and sides
Prove XTOZTO
XT=ZT(allsidesequalforrhombus)TOisacommonsideXO=OZ(diag.ofrhombusbisect)XTOZTO(SSS)O1=O4ButO1+O4=180°(sum'sonstr.line)O1=O4=90°(6)

We can further conclude that O1=O2=O3=O4=90°.

Therefore the diagonals bisect each other perpendicularly.

Use properties of congruent triangles to prove diagonals bisect interior angles
X2=Z1(XTOZTO)andX2=Z2(alt.int.'s,XTYZ)Z1=Z2(7)

Therefore diagonal XZ bisects Z. Similarly, we can show that XZ also bisects X; and that diagonal TY bisects T and Y.

We conclude that the diagonals of a rhombus bisect the interior angles.

To prove a parallelogram is a rhombus, we need to show any one of the following:

  • All sides are equal in length.

  • Diagonals intersect at right angles.

  • Diagonals bisect interior angles.

Summary of the properties of a rhombus:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal in length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

  • All sides are equal in length.

  • The diagonals bisect each other at 90°.

  • The diagonals bisect both pairs of opposite angles.

Figure 3
Image

Square

Definition 5: Square

A square is a rhombus with all four interior angles equal to 90°.

OR

A square is a rectangle with all four sides equal in length.

A square has all the properties of a rhombus:

  • Both pairs of opposite sides are parallel.

  • Both pairs of opposite sides are equal in length.

  • Both pairs of opposite angles are equal.

  • Both diagonals bisect each other.

  • All sides are equal in length.

  • The diagonals bisect each other at 90°.

  • The diagonals bisect both pairs of opposite angles.

It also has the following special properties:

  • All interior angles equal 90°.

  • Diagonals are equal in length.

  • Diagonals bisect both pairs of interior opposite angles (i.e. all are 45°).

Figure 4
Image

To prove a parallelogram is a square, we need to show either one of the following:

  • It is a rhombus (all four sides of equal length) with interior angles equal to 90°.

  • It is a rectangle (interior angles equal to 90°) with all four sides of equal length.

Trapezium

Definition 6: Trapezium

A trapezium is a quadrilateral with one pair of opposite sides parallel.

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Kite

Definition 7: Kite

A kite is a quadrilateral with two pairs of adjacent sides equal.

Example 5: Properties of a kite

Question

ABCD is a kite with AD=AB and CD=CB. Prove that:

  1. ADC=ABC

  2. Diagonal AC bisects A and C

Image

Answer

Prove ADCABC

In ADC and ABC,

AD=AB(given)CD=CB(given)ACisacommonsideADCABC(SSS)ADC=ABC(8)

Therefore one pair of opposite angles are equal in kite ABCD.

Use properties of congruent triangles to prove AC bisects A and C
A1=A2(ADCABC)andC1=C2(ADCABC)(9)

Therefore diagonal AC bisects A and C.

We conclude that the diagonal between the equal sides of a kite bisects the two interior angles and is an axis of symmetry.

Summary of the properties of a kite:

Figure 5
Image
  • Diagonal between equal sides bisects the other diagonal.

  • One pair of opposite angles are equal (the angles between unequal sides).

  • Diagonal between equal sides bisects the interior angles and is an axis of symmetry.

  • Diagonals intersect at 90°.

Investigation 2: Relationships between the different quadrilaterals

Heather has drawn the following diagram to illustrate her understanding of the relationships between the different quadrilaterals. The following diagram summarises the different types of special quadrilaterals.

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  1. Explain her possible reasoning for structuring the diagram as shown.

  2. Design your own diagram to show the relationships between the different quadrilaterals and write a short explanation of your design.

Exercise 3:

Use the sketch of quadrilateral ABCD to prove the diagonals are perpendicular to each other.

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Explain why quadrilateral WXYZ is a kite. Write down all the properties of quadrilateral WXYZ.

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