Middle Years

Worksheet
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?

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

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

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

10

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

11

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

12

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

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

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.