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Year 10AC9M10SP01

Deductive Reasoning and Proof

Apply deductive reasoning to proofs involving shapes in the plane and use theorems to solve spatial problems.

A proof is not a description of what you can see. It is a chain, where every link is a statement and a reason — and the reason is the part being marked.

Builds onYr 7 · Angle Relationships

Method

Every line has two halves
statement  +  reason\text{statement} \;+\; \text{reason}
Write what is true, then why it is true. A statement with no reason earns nothing.
You may only use what is proved
Start from the given information and named theorems. Never from what the diagram looks like.
Looks equal is not a reason
Diagrams are not drawn to scale unless you are told so. They look the same is never accepted.
Name the theorem exactly
Alternate angles, parallel lines is a reason. Angle rule is not.
End by stating what you set out to prove
The last line closes the chain. Without it the marker has to guess where you finished.

Worked example

e.g. Two parallel lines are cut by a transversal. One angle is 70∘. Prove the co-interior angle is 110∘.\text{Two parallel lines are cut by a transversal. One angle is } 70^\circ. \text{ Prove the co-interior angle is } 110^\circ.
  1. Mark on the diagram everything you are given. Nothing else.
    Two parallel lines cut by a transversal with one angle marked 70 degrees70°Only the given angle is marked
  2. Find a theorem that connects what you have to what you want.corresponding angles, parallel lines\text{corresponding angles, parallel lines}
  3. Write the first line as a statement, then its reason.B^=70∘[corresponding angles, AB∥CD]\hat{B} = 70^\circ \qquad [\text{corresponding angles, } AB \parallel CD]
    The corresponding angle on the second parallel line is also 70 degrees70°70°Corresponding angles are equal
  4. Take the next step, again with a reason.C^=180∘−70∘[angles on a straight line]\hat{C} = 180^\circ - 70^\circ \qquad [\text{angles on a straight line}]
  5. Do the arithmetic on its own line so the working is visible.C^=110∘\hat{C} = 110^\circ
  6. Close the chain by stating what you have proved.∴ the co-interior angle is 110∘\therefore \text{ the co-interior angle is } 110^\circ

Practice

Give a reason for every statement. A correct answer with no reasons scores about half.

1
Give the reason: A^=B^ where they are vertically opposite.\text{Give the reason: } \hat{A} = \hat{B} \text{ where they are vertically opposite.}
Answer[vertically opposite angles][\text{vertically opposite angles}]
2
Two angles on a straight line are x and 3x. Find x, with a reason.\text{Two angles on a straight line are } x \text{ and } 3x. \text{ Find } x \text{, with a reason.}
Answer4x=180∘ [angles on a straight line]∴x=45∘4x = 180^\circ \ [\text{angles on a straight line}] \quad\therefore\quad x = 45^\circ
3
A learner writes P^=Q^ because they look the same size. Why is this rejected?\text{A learner writes } \hat{P} = \hat{Q} \text{ because they look the same size. Why is this rejected?}
AnswerDiagrams are not to scale. A reason must name a theorem or given information.\text{Diagrams are not to scale. A reason must name a theorem or given information.}
4
In a triangle, two angles are 54∘ and 61∘. Find the third, with a reason.\text{In a triangle, two angles are } 54^\circ \text{ and } 61^\circ. \text{ Find the third, with a reason.}
Answer180∘−115∘=65∘ [angle sum of a triangle]180^\circ - 115^\circ = 65^\circ \ [\text{angle sum of a triangle}]
5
Why must a proof begin from the given information rather than the diagram?\text{Why must a proof begin from the given information rather than the diagram?}
AnswerThe diagram may be inaccurate or a special case. Only the given facts and proved theorems hold in every case.\text{The diagram may be inaccurate or a special case. Only the given facts and proved theorems hold in every case.}
6
An exterior angle of a triangle is 118∘. Find the sum of the two opposite interior angles.\text{An exterior angle of a triangle is } 118^\circ. \text{ Find the sum of the two opposite interior angles.}
Answer118∘ [exterior angle equals the sum of the opposite interior angles]118^\circ \ [\text{exterior angle equals the sum of the opposite interior angles}]
Next step
Practise Deductive Reasoning and Proof with instant marking
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