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Middle Years

7.05 Quadrilaterals

Worksheet
Construct quadrilaterals from triangles
1

Consider following the isosceles triangle:

a

Construct another triangle by reflecting the original across its base.

b

If we join the triangles together, what type of quadrilateral will be formed?

2

Consider following the isosceles triangle:

a

Construct another triangle by rotating the original by 180 \degree around the middle of its base.

b

If we join the triangles together, what type of quadrilateral will be formed?

3

Consider the following obtuse scalene triangle:

a

Construct another triangle by reflecting the original across its shortest side.

b

If we join the triangles together, what type of quadrilateral will be formed?

4

Consider the following acute scalene triangle:

a

Construct another triangle by rotating the original by 180 \degree around the middle of its longest side.

b

If we join the triangles together, what type of quadrilateral will be formed?

5

Consider the following right-angled scalene triangle:

a

Create another by rotating the original by 180 \degree around the middle of its longest side.

b

If we join the triangles together, what type of quadrilateral will be formed?

Diagonals of special quadrilaterals
6

State the types of quadrilaterals that always have:

a

Diagonals that are equal in length.

b

Diagonals that bisect each other.

c

Diagonals that are perpendicular.

d

At least one diagonal that bisects the angles it passes through.

e

Both diagonals that bisect the angles they pass through.

7

Identify the type of quadrilateral that has the following properties for their diagonals:

a
  • Their diagonals are equal in length.

  • Their diagonals bisect each other.

  • Their diagonals are perpendicular.

b
  • Their diagonals are not equal in length.

  • Their diagonals bisect each other.

  • Their diagonals are perpendicular.

c
  • Their diagonals are equal in length.

  • Their diagonals bisect each other.

  • Their diagonals are not perpendicular.

d
  • Their diagonals are not equal in length.

  • One diagonal bisects the other.

  • Their diagonals are perpendicular.

e
  • Their diagonals are not equal in length.

  • Their diagonals bisect each other.

  • Their diagonals are not perpendicular.

8

Consider the following quadrilateral:

a

If the length of PX is 6, state the length of PR.

b

If the length of QS is 16, state the length of QX.

9

Consider the following quadrilateral:

If the length of XT is 6, state the length of YT.

10

Consider the following quadrilateral:

If the length of BT is 12, state the length of AC.

11

Consider the following quadrilateral:

If the size of the highlighted angle \angle KLN is 53 \degree, find the size of \angle KLM.

12

Consider the following quadrilateral:

If the size of \angle SPQ is 78 \degree, find the size of the highlighted angle \angle SPT.

Properties of quadrilaterals
13

Given quadrilateral ABCD is a parallelogram, with \angle ADC = 134\degree, \angle BAC = 22\degree, and \angle CBD = 52\degree, find the size of each of the following:

a
\angle BCD
b
\angle BDC
c
\angle ACD
d
\angle ABD
14

Given quadrilateral ABCD is a rhombus, with \angle ADC = 116\degree, and BD = 10, find the size of each of the following:

a
\angle DBA
b
\angle BCD
c
BM
d
\angle BMC
15

Given quadrilateral ABCD is a square, with AC= 14, find the size of each of the following:

a
BM
b
\angle BCD
c
BD
d
\angle ABD
16

Given quadrilateral ABCD is a rectangle, with \angle ABD = 58\degree, and AC = 20, find the size of each of the following:

a
\angle CBD
b
\angle BCD
c
BD
d
BM
17

The triangles \triangle ABC and \triangle CDA are congruent:

a

State the angle that is equal to the following:

i

\angle CAB

ii

\angle BCA

b

Explain why AD \parallel BC.

c

Explain why AB \parallel DC.

d

What type of quadrilateral is ABCD?

e

Suppose that \angle ACD = 36 \degree and \angle BCA = 44 \degree. Find the size of \angle ABC.

18

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.

a

What can be said about \triangle RST and \triangle RQT

b

State whether the following equalities are correct:

i

\angle RTS = \angle TQR

ii

\angle RTS = \angle TRQ

iii

\angle RTS = \angle RTQ

iv

\angle RTS + \angle RTQ = 180 \degree

v

\angle RTS + \angle RTQ = 90 \degree

vi

\angle RTS + \angle RTQ = 360 \degree

c

State whether the following are true about the parts of this kite:

i

RP bisects SQ.

ii

RP is perpendicular to SQ.

iii

RS is perpendicular to RQ.

iv

SQ bisects RP.

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