∠A, ∠B, ∠C are three angles of a triangle. If ∠A -∠B =18°, ∠B – ∠C = 45°, then, ∠B, ∠A and ∠C areChoices:-

1. Break Down the Problem

We know that the sum of the angles in a triangle is 180°. We have two relations:

• $\angle A - \angle B = 18^\circ$ (1)
• $\angle B - \angle C = 45^\circ$ (2)

We can express $\angle A$ and $\angle C$ in terms of $\angle B$.

2. Relevant Concepts

Using the relations above:

• From (1), we can express $\angle A$ as: $\angle A = \angle B + 18^\circ$

• From (2), we find $\angle C$ as: $\angle C = \angle B - 45^\circ$

3. Analysis and Detail

Now, substituting $\angle A$ and $\angle C$ in terms of $\angle B$ into the triangle angle sum equation:

$\angle A + \angle B + \angle C = 180^\circ$

Substituting the expressions for $\angle A$ and $\angle C$: $(\angle B + 18^\circ) + \angle B + (\angle B - 45^\circ) = 180^\circ$

This simplifies to: $3\angle B - 27^\circ = 180^\circ$

Adding $27^\circ$ to both sides: $3\angle B = 207^\circ$

Dividing by 3: $\angle B = 69^\circ$

Now substituting back to find $\angle A$ and $\angle C$: $\angle A = 69^\circ + 18^\circ = 87^\circ$ $\angle C = 69^\circ - 45^\circ = 24^\circ$

4. Verify and Summarize

Now we verify the angles:

• $\angle A = 87^\circ$
• $\angle B = 69^\circ$
• $\angle C = 24^\circ$

Checking the sum: $87^\circ + 69^\circ + 24^\circ = 180^\circ$

This is correct.

Final Answer

• $\angle A = 87^\circ$
• $\angle B = 69^\circ$
• $\angle C = 24^\circ$