Patterns and Conjectures | Geometry | LevelKite

Add the odd numbers in order, starting from $1$ . Two of them give $1 + 3 = 4 = 2 \cdot 2$ . Three give $1 + 3 + 5 = 9 = 3 \cdot 3$ . Four give $1 + 3 + 5 + 7 = 16 = 4 \cdot 4$ . After these three cases, you may start to suspect that adding the first $n$ odd numbers always gives $n \cdot n$ .

That suspicion is a useful start, not a final answer. Much of geometry begins this way: you look at some cases, notice a pattern and make a claim about all cases. In this lesson, you learn to do that carefully, first with numbers and then with figures.

Key concept

From cases to a conjecture

Inductive reasoning means studying specific cases, finding a pattern and using it to make a general claim. That claim is a conjecture: a statement based on the cases you have seen, not yet proved. A proof is a chain of logical reasons showing that a claim holds in every case. A case that fits the claim is a confirming example, and a case where the claim fails is a counterexample. One counterexample disproves a conjecture; any number of confirming examples never proves it.

You can make a conjecture in four moves:

Collect cases. Use only values that are given or that you can compute.
Describe the pattern. Compare each case with the next. Ask what stays the same and what changes.
Extend the pattern. Use your description to predict the next case.
State the conjecture. Write a sentence that says which cases it covers and what it claims about them.

If a new case fits, you have one more confirming example. That adds evidence, but any number of confirming examples never proves a conjecture. If a new case fails, that one counterexample disproves the conjecture.

Worked example

Extending a number pattern

Each number in a list like $3, 7, 11, 15, \dots$ is called a term. Find the next two terms, then make a conjecture about term number $n$ , where $n$ is a positive whole number.

Compare each term with the next one by subtracting, because subtraction shows how much the list changes from one term to the next.
$7 - 3 = 4, \quad 11 - 7 = 4, \quad 15 - 11 = 4$
Every difference is 4, so the rule "add 4 to get the next term" fits all the given terms. Use that rule twice to extend the pattern.
$15 + 4 = 19, \quad 19 + 4 = 23$
Write the rule using n. Starting at the first term, 3, you add 4 a total of n - 1 times to reach term n: term 1 needs no additions, term 2 needs one, and term 3 needs two. Then use the Distributive Property and combine like terms.
$\text{term } n = 3 + 4(n - 1) = 3 + 4n - 4 = 4n - 1$
Test the rule on the four given terms and the two new ones, because a rule has to fit the cases it came from. All six fit. They are confirming examples, and any number of those never proves a conjecture.
$4(1) - 1 = 3, \; 4(2) - 1 = 7, \; 4(3) - 1 = 11, \; 4(4) - 1 = 15, \; 4(5) - 1 = 19, \; 4(6) - 1 = 23$

Answer

Using the rule "add $4$ ," the next two terms are $19$ and $23$ . Conjecture: term number $n$ of this pattern is $4n - 1$ for every positive whole number $n$ .

Geometric patterns work the same way, except that your cases are figures. Your facts are stated measures and counts that follow from a definition. A drawing can suggest that two angles are equal, but looking equal does not make it a fact, so nothing here is read from a picture. Each example from now on separates what is given from what you claim.

Worked example

Angle measures of triangles

Given: three triangles with these stated angle measures. Triangle 1: $50^\circ, 60^\circ, 70^\circ$ . Triangle 2: $35^\circ, 55^\circ, 90^\circ$ . Triangle 3: $20^\circ, 30^\circ, 130^\circ$ . Make a conjecture about the sum of the three angle measures of a triangle.

Separate the given facts from your claim. The nine stated measures are given, and each is between 0 and 180 degrees, as an angle of a triangle must be. Nothing is given about any other triangle, so anything you say about all triangles is a claim, not a given fact.
Add the three measures in each triangle, because the question is about their sum.
$50 + 60 + 70 = 180, \quad 35 + 55 + 90 = 180, \quad 20 + 30 + 130 = 180$
Describe the pattern. Each sum is 180 degrees, even though the triangles differ: the largest angle is under 90 degrees in the first, exactly 90 degrees in the second and over 90 degrees in the third.
State a claim about every triangle, where m means "the measure of" and A, B and C name the triangle's angles. It stays a conjecture, because three confirming examples, or any number of them, never prove a claim about all triangles.
$m\angle A + m\angle B + m\angle C = 180^\circ$

Answer

Conjecture: in every triangle, the three angle measures have a sum of $180^\circ$ . The three given triangles support this claim but do not prove it.

The same moves work with other figures. Two definitions come first. When two lines intersect, they form four angles. Two of those angles are vertical angles when they share only the point of intersection, which puts them across from each other. A parallelogram is a four-sided figure with both pairs of opposite sides parallel, meaning the lines containing each pair lie in one plane and never meet. Two of its angles are opposite angles when they do not share a side.

In the table, every value is a given fact. Each conjecture rests on only three cases, so each one is still unproved.

Given measures and the conjecture each pattern suggests

Pair being compared	Case 1	Case 2	Case 3	Conjecture
Vertical angles formed by two intersecting lines	$40^\circ$ and $40^\circ$	$75^\circ$ and $75^\circ$	$110^\circ$ and $110^\circ$	Vertical angles have equal measures.
Opposite angles of a parallelogram	$70^\circ$ and $70^\circ$	$55^\circ$ and $55^\circ$	$120^\circ$ and $120^\circ$	Opposite angles of a parallelogram have equal measures.
Opposite sides of a parallelogram	$5$ cm and $5$ cm	$8$ cm and $8$ cm	$12$ cm and $12$ cm	Opposite sides of a parallelogram have equal lengths.

Some geometric patterns come from counting, not measuring. A polygon is a closed plane figure made of three or more straight segments, called sides. Each side meets exactly two other sides, one at each endpoint, and sides that meet do not lie on the same line. The endpoints are the vertices. Two vertices are adjacent when a side joins them. A diagonal is a segment joining two vertices that are not adjacent.

This definition lets you list diagonals without a drawing. Name the vertices in order around the polygon. Then each vertex is adjacent to the names just before and after it, and the first and last names are adjacent too. Every remaining pair of vertices gives a diagonal.

Diagonals listed from the definition

Polygon	Sides, n	Vertices in order	Diagonals	Number of diagonals
Triangle	$3$	$A, B, C$	none	$0$
Quadrilateral	$4$	$A, B, C, D$	$AC, BD$	$2$
Pentagon	$5$	$A, B, C, D, E$	$AC, AD, BD, BE, CE$	$5$
Hexagon	$6$	$A, B, C, D, E, F$	$AC, AD, AE, BD, BE, BF, CE, CF, DF$	$9$

Worked example

Counting diagonals of polygons

Given: by the definition of a diagonal, polygons with $3, 4, 5$ and $6$ sides have $0, 2, 5$ and $9$ diagonals, as listed in the table. Predict the number of diagonals of a heptagon, a polygon with $7$ sides. Then make a conjecture about a polygon with $n$ sides, where $n$ is a whole number and $n \geq 3$ .

Compare each count with the next one by subtracting, as in the first example. The differences are not constant, but they have their own pattern: each one is 1 more than the one before.
$2 - 0 = 2, \quad 5 - 2 = 3, \quad 9 - 5 = 4$
Extend the pattern. If each difference keeps growing by 1, the next difference is 5, so add 5 to the hexagon's count to predict the heptagon's count.
$9 + 5 = 14$
Look for a rule that uses n. In the table, count only the diagonals that start at A. There are 0, 1, 2 and 3 of them, and each of those counts is 3 less than the number of sides.
$3 - 3 = 0, \quad 4 - 3 = 1, \quad 5 - 3 = 2, \quad 6 - 3 = 3$
Each polygon in the table has n vertices, so try multiplying n by n - 3, then compare each product with the number of diagonals. In all four polygons, the product is twice the number of diagonals.
$3 \cdot 0 = 2 \cdot 0, \quad 4 \cdot 1 = 2 \cdot 2, \quad 5 \cdot 2 = 2 \cdot 5, \quad 6 \cdot 3 = 2 \cdot 9$
Write that pattern as an equation, then divide both sides by 2 using the Division Property of Equality, which is valid because 2 is not zero. Since n is at least 3, n - 3 is not negative, so the rule never gives a negative count.
$2 \cdot \text{diagonals} = n(n - 3) \quad \Rightarrow \quad \text{diagonals} = \frac{n(n - 3)}{2}$
Test the rule at n = 7. It gives a whole number, as a count must be, and it matches your prediction. Listing the diagonals of a heptagon with vertices A to G in order, by the definition of a diagonal, also gives 14: AC, AD, AE, AF, BD, BE, BF, BG, CE, CF, CG, DF, DG, EG. These matches add evidence, but any number of confirming examples never proves a conjecture.
$\frac{7(7 - 3)}{2} = \frac{7 \cdot 4}{2} = \frac{28}{2} = 14$

Answer

A heptagon has $14$ diagonals. Conjecture: a polygon with $n$ sides, where $n$ is a whole number and $n \geq 3$ , has $\frac{n(n - 3)}{2}$ diagonals.

You can often write a conjecture as a conditional, an if-then statement, such as "If two angles are vertical angles, then they have equal measures." The if part is the hypothesis, and the then part is the conclusion. A conditional is false only when its hypothesis is true and its conclusion is false. So a $30^\circ$ angle and a $50^\circ$ angle that are not vertical angles do not disprove this conjecture, because the hypothesis is false for them.

Take care when you reword a conditional. Its contrapositive swaps the two parts and negates both (states the not version of each), and it always has the same truth value as the original: both are true or both are false. Its converse swaps the two parts, and its inverse negates both parts. The converse and the inverse share a truth value with each other and may differ from the original. Here the converse is "If two angles have equal measures, then they are vertical angles." That claim is false: two separate $30^\circ$ angles have equal measures but are not vertical angles. A reversed conjecture is a new claim, so it needs its own evidence.

Here is the central idea: inductive reasoning turns specific cases into a conjecture. You collect cases from stated facts and definitions, describe the pattern, extend it and state a claim that says which cases it covers. A conjecture is a claim with evidence, not a proved fact: one counterexample disproves it, and any number of confirming examples never proves it. Next, you will find counterexamples to disprove conjectures and decide whether a conjecture is true or false.