Geometry

Deductive reasoning

A protractor says two angles in a drawing are equal. What would it take to know they are equal in every drawing of that kind, including the ones nobody has drawn?

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Deductive reasoning

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Why this exists

There are several ways to come to believe something in geometry. Someone you trust can tell you. You can try it on a few drawings and see that it works. You can look at a well-made diagram and be convinced by it. Or you can argue from statements already accepted, one justified step at a time, until the claim has no way out.

The first three are how most geometry is first met, and they are useful. But none of them settles a claim about every triangle, or every pair of parallel lines, because there are endlessly many of those and a protractor reads to about a degree. The fourth way does settle it. That way is deductive reasoning, and it is what turns a geometric guess into a theorem.

Definition

Deductive reasoning reaches a conclusion that must be true whenever the statements it starts from are true. Each step is justified by a statement already accepted, and in geometry those come in four kinds:

Givens: what the problem says about this figure, such as l \parallel m.
Definitions: what a word means, such as “a midpoint cuts a segment into two equal parts”.
Postulates: basic facts accepted without proof, such as the segment addition postulate.
Theorems: facts already proved by deduction, such as the Vertical Angles Theorem.

The sorting of statements into these kinds is its own lesson, definitions, postulates and theorems.

The logic underneath is already in place. A step that applies a theorem to the case at hand is modus ponens, which geometry books call the law of detachment: the reason is a conditional, and an earlier line supplies its if-part. Joining two conditionals end to end is hypothetical syllogism, which geometry books call the law of syllogism. A deduction in geometry is these valid steps, run on facts about angles and segments, ending in \therefore (“therefore”) and the conclusion.

The contrast is inductive reasoning: noticing a pattern in examples and proposing that it always holds. Induction produces a conjecture. Only deduction proves one.

Intuition

Look at Figure 1. One angle is a right angle and the other is 88^\circ.

Figure 1One of these is a right angle and the other measures 88^\circ. The drawing cannot tell you which.

You cannot tell which by looking, and a quick measurement might not tell you either. (\angle P is the 88^\circ one.) A drawing is evidence about itself, drawn with a pencil of some thickness. It is never a reason.

Deduction avoids the problem by never asking the drawing anything it cannot answer. It uses only what it is told, such as “\angle 1 and \angle 2 form a linear pair”, and facts that hold for every figure like that. So a deduction about a badly drawn figure is exactly as good as one about a perfect figure: it was never looking.

Results

Deduction carries truth forward. If the statements an argument starts from are true, and every step is a valid use of a given, a definition, a postulate or a theorem, then the conclusion is true.

A deduced conclusion covers every figure that fits the givens. An argument that uses only “l \parallel m and t crosses them” proves its conclusion for every pair of parallel lines and every transversal, not just the pair on the page.

A proved conclusion becomes a reason. Once deduced, a statement is a theorem, and later deductions may cite it exactly as they cite a postulate.

Why it is true

Each step of a deduction has the form of modus ponens: a conditional (“if two angles form a linear pair, they are supplementary”), its if-part established on an earlier line, and its then-part as the new line. Modus ponens is valid, and so is chaining conditionals. A valid argument whose premises are true is sound, and a sound argument’s conclusion is true. That is the first result.

For the second, notice what the argument names. It speaks about \angle 1 and \angle 2 only through the givens, and it never uses a length or an angle read off the page. Take any other figure in which the givens hold: the same lines, word for word, are true there, with the same reasons. So the conclusion holds there too.

The third follows from the first. A deduced statement is true, so using it as a premise keeps later arguments sound.

Worked examples

Example 1

Which of these is deduction?

Decide whether each argument is deductive or inductive.

(a) Six different triangles are drawn and their angles measured. Each time the three angles add to 180^\circ, within a degree. So the angles of every triangle add to 180^\circ.

(b) \angle 1 and \angle 2 form a linear pair, and linear pairs are supplementary. So m\angle 1 + m\angle 2 = 180^\circ.

Solution

(a) Inductive. Six examples support the claim, but a seventh triangle could, as far as this argument knows, behave differently. The conclusion is a conjecture. (It is true, and it is proved by deduction in the triangle angle sum.)

(b) Deductive. “Linear pairs are supplementary” is the Linear Pair Theorem, a conditional, and the first sentence supplies its if-part. Modus ponens gives the conclusion.

(c) Neither a sound deduction nor evidence from examples. The step itself would be valid, but its starting statement, l \parallel m, was read off the look of the drawing rather than given. If l \parallel m were given, the same argument would be a deduction.

Example 2

A chain of reasons along a line

In Figure 2, A, B, C and D lie on a line in that order, and AB = CD. Deduce that AD = 2AB + BC.

Figure 2A, B, C, D in order on a line, with AB = CD.

Solution

Each line is justified by a statement already accepted, or by an earlier line.

Statements	Reasons
1. AB = CD	given
2. AD = AB + BD	segment addition postulate: B is between A and D
3. BD = BC + CD	segment addition postulate: C is between B and D
4. AD = AB + (BC + CD)	substitution: line 3 into line 2
5. CD = AB	symmetric property of equality: line 1
6. AD = AB + (BC + AB)	substitution: line 5 into line 4
7. AD = 2AB + BC	simplify: line 6

Nothing was measured, so the argument holds however long the segments are. With AB = CD = 4 and BC = 7, for instance, AD = 4 + 7 + 4 = 15 and 2(4) + 7 = 15.

Example 3

Carrying an angle between parallel lines

In Figure 3, l \parallel m and m\angle 2 = 64^\circ. Deduce m\angle 6, m\angle 5 and m\angle 3, giving the reason for each step.

Figure 3Parallel lines l and m cut by t. At each crossing the angles are numbered upper left, upper right, lower left, lower right.

Solution

The angles asked for are spread over two crossings, so no single fact reaches them all. Build a chain.

\angle 2 and \angle 6 sit in the same position at the two crossings, above their line and to the right of t, so they are corresponding angles. The lines are parallel, so by the Corresponding Angles Theorem, m\angle 6 = 64^\circ.

\angle 5 and \angle 6 form a linear pair along m. By the Linear Pair Theorem, m\angle 5 = 180^\circ - 64^\circ = 116^\circ.

\angle 3 sits opposite \angle 2 at the crossing on l, so they are vertical angles. By the Vertical Angles Theorem, m\angle 3 = 64^\circ.

So m\angle 6 = 64^\circ, m\angle 5 = 116^\circ and m\angle 3 = 64^\circ. Notice that \angle 3 and \angle 6 came out equal.

Example 4

From one case to every case

In Figure 3, l \parallel m and m\angle 2 = x^\circ, with x unknown. What can be deduced about \angle 3 and \angle 6?

Solution

Run the same chain with x in place of 64. By vertical angles, m\angle 3 = x^\circ. By corresponding angles, m\angle 6 = x^\circ. So

m\angle 3 = m\angle 6

whatever x is. \angle 3 and \angle 6 lie between l and m on opposite sides of t, which makes them alternate interior angles, and nothing in the chain used anything but l \parallel m and theorems already accepted. The other pair follows at once: \angle 4 and \angle 5 are the linear-pair partners of \angle 3 and \angle 6, so each is 180^\circ - x^\circ. So this is a proof that alternate interior angles along parallel lines are congruent, for every pair of parallel lines and every transversal, deduced from the Corresponding Angles Theorem. The numerical Example 3 found three angles; this one proved a theorem.

Example 5

Finding the faulty step

In Figure 3, with l \parallel m and m\angle 2 = 64^\circ, someone argues:

“\angle 4 is vertical to \angle 1, and \angle 1 forms a linear pair with \angle 2, so m\angle 4 = 180^\circ - 64^\circ = 116^\circ. From Example 3, m\angle 6 = 64^\circ. So \angle 4 and \angle 6 are supplementary. Linear pairs are supplementary, so \angle 4 and \angle 6 form a linear pair.”

Which step fails?

Solution

The first three sentences are sound: each uses a theorem forwards, and 116 + 64 = 180.

The last step does not. The Linear Pair Theorem says: if two angles form a linear pair, then they are supplementary. The argument has the then-part and claims the if-part, which is affirming the consequent. The conclusion is false: \angle 4 has its vertex on l and \angle 6 on m, and a linear pair shares a vertex. What is true is that \angle 4 and \angle 6 are same-side interior angles, which are supplementary for another reason.

Example 6

From a pattern to a proof

Measuring quadrilaterals always gives angle sums near 360^\circ, and pentagons near 540^\circ. Each extra side seems to add 180^\circ, which suggests that a polygon with n sides has angle sum 180(n - 2) degrees. Prove this for a convex polygon, and use it for a hexagon.

Solution

The pattern came from induction, so it is a conjecture. Here is a deduction.

Figure 4Two diagonals from one vertex cut a pentagon into three triangles.

Pick one vertex, A. Its two neighbors are already joined to it by sides, which leaves n - 3 other vertices. Drawing a diagonal from A to each of them, as in Figure 4 for n = 5, cuts the polygon into n - 2 triangles, since each diagonal adds one piece to the one piece you started with.

Convexity is what makes this work: in a convex polygon every diagonal from A runs inside it, so the pieces really are triangles that fill the polygon without overlapping.

The angles of each triangle add to 180^\circ, by the Triangle Angle Sum Theorem. And the angles of the triangles together make up exactly the angles of the polygon: at A the diagonals split the polygon’s angle into pieces that add back to it, by the angle addition postulate, and at every other vertex the same holds. No triangle has an angle anywhere else. So

\text{angle sum} = 180(n - 2)

degrees. For a hexagon, n = 6 and the sum is 180 \times 4 = 720^\circ.

In a polygon with a dent (a concave polygon), a diagonal from a badly chosen vertex can pass outside the polygon, and the fan of triangles breaks. The sum is still 180(n - 2) degrees, but the proof needs a different way of cutting the polygon into n - 2 triangles.

This was adapted from Mathematics Vision Project, Geometry, Task 4.5 “Measured Reasoning” (CC BY 4.0). The general result is the polygon interior angle sum.

Boundaries

Deduction cannot make its starting points true. A valid chain from a false given ends wherever the false given sends it. The conclusion is certain relative to what was accepted, and no more.

The chain has to start somewhere. Every theorem rests on earlier ones, and those on postulates and definitions, which are accepted without proof. On this map the corresponding angles fact comes first, proved from the parallel postulate in Corresponding angles (some courses simply take it as a postulate), and alternate interior angles are deduced from it, as in Example 4 and Alternate and same side angles. Using each to prove the other would be circular, and would prove neither.

The diagram may be read for position, not for size. That points lie on a line, or in a certain order, is usually read from the figure. That two lengths or angles are equal, or that lines are parallel, must be given or deduced. The details are in what a diagram may assume.

Deduction does not find the claim. The idea of 180(n - 2) came from examples. Induction proposes, deduction proves, and a single counterexample ends a false conjecture before any proof is attempted.

Each reason must fit exactly. A theorem used backwards, or applied when part of its if-part is missing, gives no conclusion (Example 5).

Where this goes next

Every proof in geometry is a deduction written down. Given, prove and the diagram is about the first move, sorting what is assumed from what must be shown. Two-column and paragraph proofs are the standard ways of writing the chain, with each statement beside its reason as in Example 2. The algebra steps inside a proof get named reasons of their own in properties of equality as reasons.

The first theorems proved this way are about angles, in proofs about angle pairs, starting with proving vertical angles congruent. From there deduction reaches parallel lines, triangles and circles, and every result it reaches becomes a reason for the next.

Exercises

Deductive or inductive?

Say whether each argument is deductive or inductive, or neither, and why.

Five paper triangles are torn up and their three corners placed side by side each time; the corners always make a straight line. So every triangle’s angles add to 180^\circ.
\angle A and \angle B are vertical angles, and vertical angles are congruent. So \angle A \cong \angle B.
The diagonals of three different kites were drawn, and each pair crossed at a right angle. So the diagonals of every kite are perpendicular.
In the figure, M looks like the midpoint of \overline{AB}, so AM = MB.

Naming the reason

Give the definition, postulate or theorem that justifies each step.

\angle 1 and \angle 2 form a linear pair, so m\angle 1 + m\angle 2 = 180^\circ.
B is between A and C, so AB + BC = AC.
M is the midpoint of \overline{AB}, so AM = MB.
l \parallel m, and \angle 3 and \angle 6 are alternate interior angles, so \angle 3 \cong \angle 6.
B lies in the interior of \angle AOC, so m\angle AOB + m\angle BOC = m\angle AOC.

Deducing measures

Exercises 10 to 12 use Figure 3, with l \parallel m. Give a reason for each step.

m\angle 8 = 125^\circ. Find m\angle 5 and m\angle 4.
m\angle 3 = (2x + 10)^\circ and m\angle 6 = (3x - 22)^\circ. Find x and m\angle 6.
m\angle 2 = (4y + 6)^\circ and m\angle 5 = (6y - 16)^\circ. Find y and both angles.
A, B, C and D lie on a line in that order. AB = 2x + 1, BC = 7, CD = x + 5, and AC = BD. Find x and AD.

Finding the flaw

Each argument has a step that is not a valid deduction. Find it and say what is wrong.

Vertical angles are congruent, and \angle A \cong \angle B. So \angle A and \angle B are vertical angles.
In the figure, \angle PQR looks like a right angle, so \overline{PQ} is perpendicular to \overline{QR}.
Lines l and m are cut by a transversal t, and \angle 3 and \angle 6 are alternate interior angles. So \angle 3 \cong \angle 6.

Reasoning

Explain why measuring the angles of a thousand triangles would still not prove that every triangle’s angles add to 180^\circ, while one deduction does.
Prove that if \angle 1 and \angle 2 are vertical angles and \angle 1 is a right angle, then \angle 2 is a right angle. Give a reason for each step.
Use deduction to find the angle sum of a convex polygon with 10 sides, and name the accepted facts your argument uses.
Explain why a course may take either the corresponding angles fact or the alternate interior angles fact as its starting point, but may not use each to prove the other.

Answers

Inductive. Five triangles are examples; they cannot rule out a sixth that behaves differently.
Deductive. The Vertical Angles Theorem is a conditional, and the first clause supplies its if-part, so modus ponens gives the conclusion.
Inductive. Three kites support a conjecture. (It is true, but these drawings do not prove it.)
Neither: it is not a sound deduction and not evidence from examples. “M is the midpoint” was read off the look of the figure, not given. If it were given, AM = MB would follow from the definition of midpoint.
The Linear Pair Theorem: angles that form a linear pair are supplementary.
The segment addition postulate.
The definition of the midpoint: it cuts a segment into two equal parts.
The Alternate Interior Angles Theorem, which needs l \parallel m.
The angle addition postulate.
m\angle 5 = 125^\circ and m\angle 4 = 125^\circ. \angle 5 is vertical to \angle 8 (Vertical Angles Theorem). \angle 4 and \angle 5 lie between the lines on opposite sides of t, so with l \parallel m they are congruent (Alternate Interior Angles Theorem).
x = 32 and m\angle 6 = 74^\circ. Alternate interior angles are congruent, so 2x + 10 = 3x - 22, giving x = 32. Check: 2(32) + 10 = 74 and 3(32) - 22 = 74.
y = 19, m\angle 2 = 82^\circ and m\angle 5 = 98^\circ. \angle 2 \cong \angle 6 (Corresponding Angles Theorem), and \angle 5 and \angle 6 form a linear pair, so m\angle 2 + m\angle 5 = 180^\circ: (4y + 6) + (6y - 16) = 180, so 10y = 190. Check: 82 + 98 = 180.
x = 4 and AD = 25. By segment addition, AC = (2x + 1) + 7 = 2x + 8 and BD = 7 + (x + 5) = x + 12. Setting them equal, x = 4. Then AB = 9 and CD = 9, so AD = 9 + 7 + 9 = 25.
The theorem is used backwards: congruent is its then-part, vertical its if-part. This is affirming the consequent. In Figure 3, with l \parallel m, \angle 3 \cong \angle 6, yet they are at different crossings and are not vertical angles.
The only reason offered is how the angle looks, and Figure 1 shows an 88^\circ angle that looks right. Nothing given says m\angle PQR = 90^\circ, so nothing follows.
The Alternate Interior Angles Theorem needs l \parallel m, and that is not given. If the lines are not parallel, alternate interior angles are not equal.
A thousand triangles are a thousand examples, and there are endlessly many triangles; any one not measured could differ, and each measurement is only good to a degree or so. A deduction uses nothing but “this is a triangle”, so every step holds for every triangle at once.
(1) \angle 1 and \angle 2 are vertical angles (given). (2) \angle 1 is a right angle (given). (3) \angle 1 \cong \angle 2 (Vertical Angles Theorem: line 1). (4) m\angle 1 = m\angle 2 (definition of congruent angles: line 3). (5) m\angle 1 = 90^\circ (definition of right angle: line 2). (6) 90^\circ = m\angle 2 (substitution: line 5 into line 4). (7) m\angle 2 = 90^\circ (symmetric property of equality: line 6). (8) \angle 2 is a right angle (definition of right angle: line 7). Each step applies an accepted fact to what is already on the page, which is deduction.
1440^\circ. From one vertex, 10 - 3 = 7 diagonals cut the polygon into 8 triangles. Each has angle sum 180^\circ (Triangle Angle Sum Theorem), and together their angles make up the polygon’s angles (angle addition postulate), so the sum is 180 \times 8 = 1440^\circ.
A deduction must start from statements accepted before it. If each fact were proved from the other, neither would rest on anything already accepted, so the pair would prove nothing. One of them has to be accepted first, as a postulate or from some other theorem; then the other can be deduced from it.