Explain in as much detail as possible if you agree with the following statements:
1.
A Rectangle has two pairs of parallel sides
Prompt: - think about the symmetry of the shape
A
B
D
C
Possible questions:
• What is the definition of a line of symmetry?
• Does a line of symmetry help to define the angle properties of a shape?
The lines of symmetry must be an equal distance from the sides of the rectangle,
and the sides must therefore be parallel.
Using the vertical line of symmetry angle A = angle B and angle D = angle C.
Using the horizontal line of symmetry angle A = angle D and angle C = angle B.
Interior angles of any quadrilateral = 360o and now all four angles are equal.
Each angle must therefore be 90o, and the lines AB and BC are perpendicular.
The lines AB and DC are parallel to each other, as are the lines AD and BC.
2. The Diagonals of a rectangle are congruent.
Prompt:
- Draw in one diagonal on your rectangle
- What angle properties of parallel lines could you use to help define
the angles in your two triangles?
- What can you now say about the two triangles?
- How could we calculate the length of the diagonal?
A
B
A
B
D
C
D
C
Having established that the two triangles are congruent we could use SAS, AAS
or Pythagoras’ Theorem to confirm that the diagonals are congruent.
3. The diagonals of a Rectangle bisect each other.
Prompt: By drawing the diagonals of a rectangle we create 4 triangles.
What kind of triangles do the diagonals create?
Is there any similarity or congruency?
Which parallel line properties could we use to help prove congruency?
Will the rectangle’s lines of symmetry help to determine where the diagonals
intersect?
A
B
A
B
E
D
C
D
C
The simplest method to prove this is to introduce the vertical or horizontal line of
symmetry, as in 1 above.
Extend the task by suggesting that students define the properties of
triangles CDE and ABE.
CE = AE and DE = BE so the diagonals must therefore bisect each other.
Angles BDC and DBA are alternate and equal as are angles ACD and DBA. The
vertical and horizontal lines of symmetry make all these angles equal to each other
so the triangles ABE and CED are isosceles
A
and congruent . AE = EC and DE = EB and
with congruency AC = BD
D
B
C
4. The diagonals of a Rectangle are angle bisectors?
Prompt:
-
- Draw in one diagonal on your rectangle and label up the angles of the
two triangles.
Are the two triangles similar/congruent? Which parallel line properties
could we use to help determine congruency?
What do you notice about the two angles that the diagonal has created?
Are they the same?
If these two angles are the same what can you say about the two
triangles created by the diagonal?
What type of special rectangle do we have?
A
B
A
B
D
C
D
C
Provided that AB ≠ BC then the diagonals are not an angle bisector.
When AB = BC the diagonal creates two right angled isosceles triangles.
The properties of the isosceles triangle, the alternate angles and the symmetries of
the shape show that the diagonal is an angle bisector i.e. ABCD is a square.
5. Are the diagonals of a Rectangle Perpendicular?
Building on 3 above
- Under what circumstances would the 4 triangles become congruent?
- How many isosceles triangles would we have if the diagonals were
perpendicular? What other feature would these isosceles triangles have?
If the diagonals AC and BD are perpendicular then each
of the angles AED, DEC, CEB and BEA are all 90o.
A
The vertical and horizontal lines of symmetry prove
that BDC, DBA, ACD and DBA are all equal.
By a similar argument angles BDC, CAD, ACB and
DBC are all equal. If the diagonals are perpendicular
D
then angle AED = angle AEB. This may only happen
When angles BDA and CAD = angles CAB and DBA.
This will only happen when AD = AB i.e. when the rectangle is a square.
B
C
A
B
A
B
D
C
D
C