How to Write a Proof — Practice

A proof is not a calculation. It is a chain of statements where every link carries a reason.

What these questions 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.

Problem 1

What is always the first line of a two-column proof?

The given
What you are proving
A definition
A theorem

Worked answer

The given

The given is your starting material.

The prove statement is the last line.

Problem 2

A statement in a proof with no reason beside it is

not acceptable, even if true
fine if obvious
acceptable once per proof
a definition

Worked answer

not acceptable, even if true

Every statement needs a justification.

A true but unjustified line is still a gap.

Problem 3

Write the converse of "if it rains, the ground is wet".

Worked answer

if the ground is wet, it rains

Original: if P then Q

Converse: if Q then P

Problem 4

How many counterexamples are needed to disprove a claim?

Worked answer

One case that fails makes the claim false.

Proving needs every case; disproving needs one.

Problem 5

You know M is the midpoint of AB. What reason lets you write AM = MB?

Worked answer

definition of midpoint

A midpoint cuts a segment into two congruent parts.

That is the definition.

Problem 6

Is the converse of "if two angles are vertical, they are congruent" true? Answer yes or no.

Worked answer

Converse: if two angles are congruent, they are vertical.

The base angles of an isosceles triangle are congruent but not vertical.

So the converse is false.

Problem 7

In a proof, two triangles share a side. What reason lets you say that side is congruent to itself?

Worked answer

reflexive property

BD ≅ BD

That is the reflexive property.

Problem 8

CPCTC may be used

only after proving two triangles congruent
at any point in a proof
to prove triangles congruent
only in coordinate proofs

Worked answer

only after proving two triangles congruent

It is what congruence gives you.

Using it to prove congruence would be circular.

Problem 9

Give a counterexample to "all quadrilaterals with four equal sides are squares".

Worked answer

a rhombus

A rhombus has four equal sides.

Its angles need not be right angles.

Problem 10

You are given that AB ∥ CD. What angle relationship does a transversal give you?

Worked answer

congruent alternate interior angles

Alternate interior angles are congruent.

Corresponding angles are congruent too.

Problem 11

If "if P then Q" is true, which of these is also always true?

If not Q then not P
If Q then P
If not P then not Q
None of these

Worked answer

If not Q then not P

The contrapositive is always true with the original.

The converse and inverse are not.

Problem 12

A proof says "∠ A is a right angle" because the diagram looks square. Is that valid? Answer yes or no.

Worked answer

Diagrams are not drawn to scale.

A right angle must be given or marked.

Problem 13

The converse of the Pythagorean theorem is true. What does it let you conclude?

Worked answer

the triangle is right-angled

If a² + b² = c² with c longest, then the triangle has a right angle.

It is a genuine theorem, proved separately.

Problem 14

Write the contrapositive of "if a shape is a square, then it has four right angles".

Worked answer

if it does not have four right angles, it is not a square

Original: if P then Q

Contrapositive: if not Q then not P

Problem 15

In a proof you want to show AD ≅ CD. What is usually the step before?

Worked answer

prove the triangles containing them are congruent

Prove △ ABD ≅ △ CBD.

Then CPCTC gives the segments.

Problem 16

Disprove: "if two triangles have the same perimeter, they are congruent."

Worked answer

a 3-4-5 and a 4-4-4 triangle both have perimeter 12

3-4-5 has perimeter 12.

So does 4-4-4.

Different shapes, same perimeter.

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