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Conversations about PROOF page 1 of 8
Conversations about PROOF
This is a short unit of logical thinking created for Year 9 or 10 classes. It uses a ‘conversation’ method of
informal discussion to deal with learning the process of proof. This should accommodate a range of
readiness for this challenging topic.
The ‘informal discussion’ process requires students to be seated in small groups. Each student gets a copy
of the ‘cards’ cut from the pages that follow, just one at a time. The teacher’s role is to set up and explain the
process, and to encourage clear thinking from all students. Whether or not students are expected to write
anything is up to the teacher; it may slow down the process too much.
As has traditionally been the case, the context for this is ‘elementary’ geometry. It does not include any
formal teaching about logic, relying on ‘common sense’ and a strict adherence to one rule: only use results
already proved, or axioms – facts assumed without proof.
Elementary angle facts, including parallel lines
Group members start with just the Axiom A, and ‘To prove #1’. They have to think through the logic that
allows them to prove #1 (Vertically opposite angles are equal) from Axiom A. The process (introducing angle
c) is described in the top left panel of page 2, and they could be given this information after they have
thought about it for a while.
Then they get a copy of ‘To prove #2’ (alternate angles are equal if the lines are parallel). This requires
Axiom B (corresponding angles are equal if the lines are parallel). They think about how to use Axioms A
and B (or result #1 and Axiom B) to prove that #2 is always true. The top right panel of page 2 provides this
proof, should they need this assistance.
Proof #3 uses Axiom A and result #2. Proof #4 also uses Axiom A and result #2.
Proof #5 uses Axiom A and result #4 and do #7 and #8. Proof #6 uses result #4 only.
For some students getting as far as #8 will have established the idea of proof. However for many students
this will be enough to whet their appetite for more of this logical way of thinking. Rigorous proof is an
essential part of mathematics process. Although it should be admitted that this is not as rigorous as it might
be, this will still serve to give many students an engaging introduction to the ideas of logic involved.
From here on it is not required that the problems be tackled in numerical order.
Proofs using SSS congruency, including isosceles triangle properties and circle theorems
Proofs #9 to #19 require Axiom C, SSS congruency. It might be necessary to spend some time convincing
students that this is true, but for most people it is intuitively obvious. It cannot be proved.
Proofs using the ‘tangent perpendicular to radius’ property
It is possible to prove this, but at this stage it is simpler to take it as axiom D. Proofs #20 to #24 depend on
this fact.
Proofs using ASA congruency, including properties of parallelograms
Proofs #25 to #29 require Axiom E, ASA congruency. As with the other congruency conditions it is intuitively
obvious. but not provable.
Proofs using SAS congruency,
Proofs #30 to 34 require Axiom F, SAS congruency.
Conversations about PROOF page 2 of 8
We start with a fact we are sure about - but cannot
prove. For example AXIOM A is fact we are sure
about, but cannot prove; it is called an axiom.
Adjacent angles (two angles side-by-side) on a
straight line add to 180°. a + b = 180.
AXIOM B says that two corresponding angles are
equal if the lines are parallel. b = f
But we can add the angle c to the diagram. Now we
have already proved that vertically-opposite angles
are equal, so b = c.
We extend one line to make ‘vertically-opposite’
angles. We name one of the new angles c. But now
a + c must also add to 180.
c
a
b
c
b
f
But since a + b = 180 and a + c = 180, it follows
logically that b = c.
So vertically-opposite angles are equal.
a
But b = f, and b = c, so it follows logically that f = c.
Alternate angles are equal if lines are parallel.
Now it is your turn to prove #3, and then continue.
Please discuss these with your group, and insist on
each person being logical in their thinking.
You can use the axioms and any fact you have
previously proved.
AXIOM A
AXIOM B
Adjacent angles (two angles side-by-side) on a
straight line add to 180°.
Corresponding angles on parallel lines are equal.
b=f
a + b = 180
b
f
a
b
To prove #1
To prove #2
Vertically-opposite angles are equal.
Alternate angles are equal if lines are parallel.
b=c
c=f
b
c
c
f
Conversations about PROOF page 3 of 8
To prove #3
To prove #4
Allied (or interior) angles add to 180° for parallel
lines.
Angles in any triangle add to 180°
a + b + c = 180
c=g
c
b
g
a
c
To prove #5
To prove #6
Any exterior angle of any triangle is the sum of
the two interior opposite angles.
The sum of the angles of any quadrilateral is
360°.
z=a+b
a + b + c + g + h + I = 360
b
b
a
g
z
c
a
i
h
To prove #7
To prove #8
The sum of the exterior angles of any triangle is
360°.
The sum of the exterior angles of any
quadrilateral is 360°.
x + y + z = 360
e + f + g + h = 360
x
b
h
c
d
g
c
z
a
a
y
e
b
f
Conversations about PROOF page 4 of 8
AXIOM C
To prove #9
Two triangles having three corresponding sides
equal (SSS) are congruent – identical in all
respects.
Every isosceles triangle (with two equal sides)
will have two equal angles.
The triangles may be rotated or flipped over
(reflected). If any isometric transformation can
place one triangle on top of the other then they
are congruent.
(Hint: draw a line from the apex to the midpoint of the
base.)
Only one triangle can be constructed with three
given side lengths.
a
b
To prove #10
To prove #11
The bisector of the apex angle of every isosceles
triangle will bisect the base at right angles.
Any quadrilateral with both pairs of opposite
sides equal has both pairs of opposite sides
parallel, and hence is a parallelogram.
d = e = 90
f g
h
i
j
k
To prove #12
To prove #13
Two circles intersect at A and B. The angles
formed from their centres are equal.
From one vertex (C) of any triangle (CAT) a line
(CD) is drawn parallel to the side opposite the
vertex (AT) and equal to CA.
x=y
Prove that line AD bisects angle CAT.
x=y
A
D
C
x
y
x
B
T
y
A
Conversations about PROOF page 5 of 8
To prove #14
To prove #15
Any angle in a semi-circle (standing on a
diameter) is a right angle.
In the circle, the angle at the centre is twice the
angle at the circumference.
b + c = 90
e = 2b
b
b c
a
e
To prove #16
To prove #17
In the circle, the angle at the centre is twice the
angle at the circumference.
In the circle all angles standing on the same arc
are equal.
e + f = 2(b + c)
a=b=c
b
b c
a
e
f
a
c
d
To prove #18
To prove #19
For any quadrilateral drawn inside a circle, pairs
of opposite angles add to 180°.
In the diagram, four vertices lie on a circle.
(This is the ’converse’ of #16.)
g + h = 180 and also j + k = 180
a
g
k
j
h
c
Conversations about PROOF page 6 of 8
AXIOM D
To prove #20
For any circle, a tangent (a lie that just touches
the circle at one point) is perpendicular to the
radius to that point.
Two tangents from any point outside a circle are
equal. The figure forms a kite.
The perpendicular (the radius) give the
shortest distance from the circle centre to the
tangent line.
To prove #21
To prove #22
For any quadrilateral (CUTE) drawn to touch a
circle at four points (L, A, M and B), the sums
of the lengths of pairs of opposite sides area
equal.
For the tangent touching one end of a diameter the
angle between the tangent and chord (h) the angle
in the alternate segment (e).
h=e
CU + ET = CE + UT.
E
e
B
f
C
M
g
T
h
L
A
U
To prove #23
To prove #24
For any tangent the angle between the tangent
and chord (h) the angle in the alternate
segment (e).
In a circle with centre O, two equal chords (SH and
(TU) extend to meet at I. Prove that OI bisects angle
SIT.
h=e
S
H
e
O
b
I
U
c
T
h
Conversations about PROOF page 7 of 8
AXIOM E
To prove #25
Two triangles having two corresponding
angles and the side between them equal (ASA)
are congruent – identical in all respects.
A diagonal of any parallelogram (with both pairs of
opposite sides parallel) divides the parallelogram
into two congruent triangles.
Both pairs of opposite angles are equal,
and both pairs of opposite sides are equal.
Only one triangle can be constructed with two
sides and the angle between them.
e = f and a + d = b + c
b
e
c
a
f
d
To prove #26
To prove #27
In any parallelogram, the diagonals bisect each
other.
Any quadrilateral
with one pair of opposite sides parallel
and one pair of opposite angles equal
is a parallelogram.
a
b
e
f
d
c
To prove #28
To prove #29
In the square lines are drawn from one vertex
to midpoints of opposite sides. Prove that the
added lines are equal.
In any triangle the angle bisectors intersect at a
point equi-distant from the sides. It is called the
incentre, as it is the centre of a circle that may be
drawn inside the triangle.
D
a a
A
T
Y
c
G
c
b
C
b
O
Conversations about PROOF page 8 of 8
AXIOM F
To prove #30
Two triangles having two corresponding sides
and the angle between them equal (SAS) are
congruent – identical in all respects.
Any quadrilateral
with one pair of opposite sides
both parallel and equal
is a parallelogram.
The triangles may be rotated or flipped over
(reflected). If any isometric transformation can
place one triangle on top of the other then they
are congruent.
Only one triangle can be constructed with two
angles and the side between them.
To prove #31
To prove #32
Any quadrilateral with both pairs of opposite
angles equal is a parallelogram.
For a rhombus (a parallelogram with equal sides)
the diagonals bisect each other at right angles.
c d
a
e
b
f
g h
To prove #33
To prove #34
Two circles intersect at A and B. The line joining
their centres bisects AB and is perpendicular to
it.
In any triangle, any median (a line joining a
vertex to the midpoint of the opposite side) is
extended an equal length. Prove that x = y.
A
O
y
P
x
B