How to Write a Proof — Drills
How to Write a Proof has 5 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 17 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Base — The number being multiplied. In 53 the base is 5.
- Scale — What one grid step is worth — and where the axis starts.
- Converse — If a2 + b2 = c2, then the triangle must be right-angled.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Transversal — A line that crosses two or more other lines.
- Corresponding angles — Same position at each crossing. Equal when the lines are parallel.
- Alternate interior angles — Between the parallels, on opposite sides of the transversal. Equal.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Parallel — Same slope, different intercept.
- Segment — Two endpoints and everything between them. Written AB.
- Prove — The statement you must arrive at. Always the last line.
- Two-column proof — Statements on the left, the reason for each on the right.
- Counterexample — A single case that makes a claim false. One is enough.
- CPCTC — Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
- Reflexive property — Anything is congruent to itself. A shared side is congruent to itself.
- Rhombus — A parallelogram with four congruent sides.
Structure
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
What is always the first line of a two-column proof?
- The given
- A theorem
- What you are proving
- A definition
Worked answer
The given
You start from what you were handed.
So the answer is: The given.
Problem 2
A statement in a proof with no reason beside it is
- acceptable once per proof
- a definition
- not acceptable, even if true
- fine if obvious
Worked answer
not acceptable, even if true
A true statement is not the same as a proved one.
So the answer is: not acceptable, even if true.
Converse
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is the converse of "if a shape is a square, it has four right angles" true? Answer yes or no.
Worked answer
no
Converse: if a shape has four right angles, it is a square.
Counterexample: a non-square rectangle.
Problem 2
Is the converse of "if two lines are parallel, they never meet" true? Answer yes or no.
Worked answer
yes
Converse: if two lines never meet, they are parallel.
No counterexample exists — in a plane this one does hold.
Counter
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Give a counterexample to "if two triangles have the same perimeter they are congruent".
Worked answer
a 3-4-5 and a 4-4-4 triangle
A 3-4-5 and a 4-4-4 triangle satisfies the condition but not the conclusion.
Disproving needs one case; proving needs an argument.
Problem 2
Give a counterexample to "all quadrilaterals with four equal sides are squares".
Worked answer
a rhombus
A rhombus satisfies the condition but not the conclusion.
Disproving needs one case; proving needs an argument.
Reasoning it through
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
You know M is the midpoint of AB. What reason lets you write that AM = MB?
Worked answer
definition of midpoint
AM = MB.
Reason: definition of midpoint.
Problem 2
You know two triangles share a side. What reason lets you write that that side is congruent to itself?
Worked answer
reflexive property
That side is congruent to itself.
Reason: reflexive property.
Logic
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
If "if P then Q" is true, which of these is also always true?
- If not Q then not P
- If Q then P
- None of these
- If not P then not Q
Worked answer
If not Q then not P
The contrapositive is always true alongside the original.
The converse (Q → P) and inverse are not.