Q.Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?

The universal set $U$ contains every possible triangle in the plane. The complement $A'$ consists of all triangles in $U$ that are not in $A$.

Set $A$ includes all triangles having at least one angle different from $60°$. So $A'$ must contain triangles where no angle is different from $60°$ — meaning every angle equals $60°$.

A triangle has three angles summing to $180°$. If all three angles are equal, each measures $\frac{180°}{3} = 60°$. This describes exactly the equilateral triangles. …

Understanding the Universal Set and Complement

When we work with sets, the complement $A'$ (or $A^c$) consists of everything in the universal set $U$ that is not in $A$. Here $U$ is the set of all triangles in the plane, and $A$ is defined by a condition on angles. To find $A'$, we need to identify which triangles fail to satisfy the condition that defines $A$.

The key is to recognize what "at least one angle different from $60°$" actually means, and then negate it carefully.

Step-by-Step Reasoning

1. Parse the condition for set $A$

   A triangle belongs to $A$ if at least one of its three angles is different from $60°$. In other words, $A$ contains every triangle that has one or more angles $\neq 60°$.

2. Negate the condition to find $A'$

   The complement $A'$ consists of triangles that do not have at least one angle different from $60°$. By the logic of negation:

So a triangle is in $A'$ if and only if every one of its angles equals $60°$.

3. Recognize the geometric object …

Method: Complement of a Set (Set Difference)

This problem uses the complement rule:

If $U$ is the universal set and $A \subseteq U$, then

Step 1: Identify the universal set

Step 2: Understand set $A$

$A$ = set of all triangles with at least one angle different from 60°.

This means:

• A triangle belongs to $A$ if not all its angles are 60°.
• In other words, $A$ contains all triangles except equilateral triangles (where each angle = 60°).

Step 3: Find $A'$ using the complement

$A'$ = set of all triangles not in $A$.

Since $A$ contains all triangles that are not equilateral,

$A'$ must contain only those triangles that are not in $A$ — i.e., triangles where every angle is 60°.

--- …

Let’s break this down step-by-step, starting with the concept.

1. Understanding the Universal Set

The universal set $U$ is all triangles in a plane.

That includes every possible triangle — equilateral, isosceles, scalene, acute, obtuse, right-angled — everything.

2. Understanding Set A

Set $A$ is defined as:

All triangles with at least one angle different from 60°.

This means:

• If a triangle has any angle that is not exactly 60°, it belongs to $A$.
• The only triangles excluded from $A$ are those where every angle is 60°.

3. What is $A'$ (Complement of A)?

By definition:

So $A'$ contains all triangles not in $A$.

Since $A$ contains all triangles except those with all angles 60°, $A'$ must contain only those triangles where every angle is 60°.

That is exactly the set of equilateral triangles.

Common Mistakes & How to Avoid Them

✗ Mistake 1: Thinking $A'$ is “triangles with all angles different from 60°”

Why it’s wrong:

The complement is the opposite condition of the entire description of $A$, not just a word swap.

$A$ says “at least one angle ≠ 60°”. Its negation is “no angle is different from 60°”, i.e., all angles = 60°.

✓ How to avoid:

Always write the condition in precise logical form:

• $A$: $\exists$ an angle $\neq 60^\circ$
• $A'$: $\forall$ angles $= 60^\circ$

Then translate back to words.

✗ Mistake 2: Confusing “at least one” with “exactly one”

Why it’s wrong:

“At least one angle different from 60°” includes triangles with 1, 2, or 3 angles not equal to 60°.

Students sometimes think $A$ means “exactly one angle ≠ 60°”, which is a much smaller set.

✓ How to avoid:

Remember: “at least one” = “one or more”.

Draw a quick mental picture:

• A scalene triangle (all angles different) is in $A$.
• An isosceles triangle with angles 70°, 70°, 40° is in $A$.
• Only the equilateral triangle (60°, 60°, 60°) is not in $A$.

✗ Mistake 3: Thinking $A'$ is empty or “no such triangle” …