MAWA Secondary Convention 2016
TEACHING DEDUCTIVE GEOMETRY
Dr Nathan (Norm) Hoffman
Western Australian Mathematics Problem Solving Program (WAMPSP)
Teaching Deductive Geometry
The subject for this session is ‘Teaching Deductive Geometry’ so I’m going to do just that. I’m going to solve a number of geometric problems. and in the course of doing that I’ll deal with a number of aspects that I consider to be important in teaching and learning about deductive geometry.
1 Terminology
As a preliminary I need to make some comment on alternative terminology and notation used in relation to lines, line segments, and angles.
As another preliminary I need to establish a collection of some of the significant
geometric results (theorems) that I will need to use.
2 Preliminaries
2.1 Parallel lines and related angles.
a) If two lines, l1 and l2 are parallel, then
the alternate angles are equal,
the corresponding angles are equal, and
the co-interior angles are supplementary.
b) If a line crosses two other lines and makes:
a pair of alternate angles equal, or
a pair of corresponding angles equal, or
a pair of co-interior angles supplementary,
then the two other lines are parallel.
2.2 Conditions for congruence of two triangles (What’s so special about congruent triangles?)
Two triangles are congruent:
i) If three sides of one triangle are respectively equal to the three sides of the other triangle (SSS), or
ii) If two side and the included angle of one triangle are equal to two sides and the included angle of the other triangle (SAS), or
iii) If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle (AAS), or
iv) If the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of the other right triangle. (RHS)
2.3 The angle made at the centre of a circle by an arc is twice the angle made by the arc at any other point of the circle (Angle at the centre theorem)
2.4 The angle between a tangent to a circle and a chord from the point of contact of the tangent is equal to any angle in the alternate segment of the circle (Tangent-chord theorem).
2.5 If one side of a triangle is extended, the exterior angle formed is greater than either of the two interior opposite angles.
3 Why study deductive geometry?
Deductive Geometry is Geometry, and it is Deductive
Geometry can be 1-D, 2-D or 3-D (or n-D)
1-D is about points on a line. This is covered in the use of the number line in arithmetic and algebra. I don’t intend to say anything more about this.
2-D is about planes and subsets of the plane: points, lines, line segments, triangles and circles. It’s these geometric figures that I’ll focus on in this session.
3-D is about spheres, prisms and pyramids. Properties of these objects are explored in primary and secondary school programs, but typically not deductive ways. I’ll say no more about 3-D geometry in this session.
Geometric figures have properties. For example an isosceles triangle has two congruent sides and two congruent angles. One way in which these properties might be established is by measurement. But it is not possible to draw ALL possible isosceles triangles, nor is it possible to measure the properties of all of these triangles. To do this we need the processes of Deductive Geometry.
4 Solving geometry problems deductively
Solving deductive geometry problems can involve three separate but related sets of skills and understandings:
i) Translating a problem expressed in words into a geometric diagram. A diagram can be very helpful in providing clues as to how to assemble the geometric facts to get a solution.
ii) Assembling the given facts and the related geometric theorems to get a solution.
iii) Setting down the solution, using standard geometric notation and symbols, so that someone else can come to understand how the problem has been solved.
I now intend to examine each of these three aspects. In so doing I intend to offer advice as to how help students achieve high levels of competence. I also intend to draw attention to common pitfalls and common examples of poor practice.
Example 1: Two circles C1 and C2 meet at the points A and B. CD is a common tangent to these circles where C and D lie on the circumferences of C1 and C2 respectively. CA is the tangent to C2 at A. When extended (produced), DB meets the circumference of C1 at P. Prove that PC is parallel to AD.
Given: Two circles, C1 and C2 intersect at A and B.
CD is a direct common tangent to the two circles
The tangent to C2 at A intersects C1 at C
P is a point on C1 and PB intersects C2 at D
To Prove: PC ∥ AD
Proof: CA is a tangent to C2 at A and AB is a chord in C2
∴ ∠CAB = ∠ADB (Angle between tangent & chord)
But in C1 , ∠CAB = ∠CPB (Same arc CB)
∴ ∠ADB = ∠CPB
But these are alternate angles
∴ PC ∥ AD
Points to note:
i) Avoid drawing a diagram which is more special than specified in the problem.
ii) Statements/conclusions should be justified. The justification can precede or follow the conclusion. In general it is preferable for the justification to precede the conclusion.
In general, the proof for a geometric theorem or problem should contain the following features/headings:
1 A neat, labelled diagram.
2 Given: a full, but concise statement of what was given and from which the diagram was constructed.
3 To Prove or To Find: a concise statement of what is to be done.
4 Proof (if the previous heading is To Prove) or Solution (if the previous heading was To Find).
i) The Proof/Solution should be set, out step by step, in logical order.
ii) It is good practice to leave a line between each section.
iii) Wherever possible statements/conclusions should be justified,
These features are illustrated in the proof (below) of the Isosceles Triangle Theorem.
Prototype for Proofs in Geometry:
The Isosceles Triangle Theorem
Given: ∆PQR with PQ = PR
To Prove: <PQR = <PRQ
Proof: Draw PT to bisect <QPR and meet QR at T
In ∆s PQT and PRT
PQ = PR (Given)
PT = PT (Common)
Incl <QPT = incl <RPT (Constructed)
∴ ∆PQT ≅ ∆PRT (SAS)
∴ <PQT = <PRT
Question: Why prove this theorem when the result is so obvious?
An even more general question is ‘What might be the criteria for determining which results in mathematics need to be proved?’
The best answer I know to this question is the following:
Results worthy of proof or needing to be proved are those where the result is surprising or unexpected, or those where the method of proof is an important method in its own right.
The theorem concerning the angle at the centre of a circle is an example of the first criterion – The result is surprising, and needs to be proved. The Isosceles Triangle Theorem is an example of the second criterion. The method of congruent triangles is so useful that it needs to be understood and mastered by students of geometry.
5 Direct and Indirect proofs
Deductive proofs in Geometry can be classified as either Direct Proofs or Indirect Proofs. My emphasis in this session will be on direct proofs.
In mathematics, important results are known as theorems. One well known theorem is the Theorem of Pythagoras which states that in a right triangle the square of the side opposite the right angle is equal to the sum of the squares of the other two sides.
The statement of a theorem can be rephrased so that it reads as ‘If (hypothesis), then (conclusion)’. As an example of this, the Theorem of Pythagoras can be rephrased as:
‘If a triangle has one of its angles of size one right angle, then the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides’.
The hypothesis in this theorem consists of two conditions:
i) There is a triangle.
ii) One of its angles is a right angle.
The conclusion follows whenever the conditions of the hypothesis are satisfied.
In this theorem, the conclusion is that the square of the hypotenuse is equal to the sum of the other two sides.
A theorem can also be represented in the form:
If p then q, where p is the Hypothesis and q is the Conclusion
In symbols this is written as p ⟹ q , and read as ‘p implies q’, or ‘if p then q’, and the statement is also known as an Implication.
Every theorem has a Converse. This is a statement in which the hypothesis and conclusion of the theorem are interchanged. It is important to realise that the converse of a theorem is not necessarily true. Example: There is a simple theorem in arithmetic which states, “If a positive integer has a factor of 6, then it has a factor of 2”. The converse of this states “If a positive integer has a factor of 2, then it has a factor of 6”. This converse is clearly false, as is shown by considering the positive integer, 10.
But many theorems have converses which are true. Consider the theorem that states: “Angles subtended by the same chord are equal”. This can also be stated as: “If a chord of a circle subtends angles at two points then these angles are equal”. The not-so-obvious converse of this can be worded as: “If a line segment subtends equal angles at two points on the same side of the line segment, then these two points and the two ends of the line segment are concyclic”.
Given: Line segment PQ. A and B are on the same side of PQ and ∠A = ∠B
To Prove: The points P, Q, A, and B are concyclic.
Proof: By Contradiction
A circle can be drawn through the 3 non-collinear points P, Q and A.
Assume that this circle DOES NOT pass through B
Assume that the circle cuts PB at X
∠PXQ and ∠PAQ are subtended by the same chord, PQ
∴ ∠PXQ = ∠PAQ
But ∠PAQ = ∠PBQ (Given)
∴ ∠PXQ = ∠PBQ …………………………………1)
But for ∆BQX:
∠PXQ is an exterior angle and ∠PBQ is an interior opposite angle
∴ ∠PXQ > ∠PBQ …………………………………2)
But statement 2) contradicts 1)
So the assumption that the circle through P, Q, and A does not pass through B is false.
∴ the circle through P, Q, and A does pass through B.
ie the points P, Q, A, and B are concyclic.