Question

In ΔABC, ∠A – ∠B = 24° and ∠B – ∠C = 30°. Find angle A.

Answer 1

Given:

• △ABC
• ∠A – ∠B = 24°
• ∠B – ∠C = 30°

We need to find ∠A.

Step 1: Express the angles in terms of one variable

Let:

• ∠A = x
• ∠B = y
• ∠C = z

From the given information, we can set up the following equations:

1. x – y = 24  ...(Equation 1: ∠A – ∠B = 24°)
2. y – z = 30  ...(Equation 2: ∠B – ∠C = 30°)

Step 2: Use the fact that the sum of angles in a triangle is 180°

In any triangle, the sum of the interior angles is always 180°.

Therefore: ∠A + ∠B + ∠C = 180°

Substitute the variables for the angles:

x + y + z = 180°  ...(Equation 3)

Step 3: Solve the system of equations

From Equation 1, we have:

x = y + 24

From Equation 2, we have:

y = z + 30

Substitute y = z + 30 into the expression for x:

x = (z + 30) + 24 = z + 54

Now, substitute x = z + 54 and y = z + 30 into Equation 3:

(z + 54) + (z + 30) + z = 180

Simplify the equation:

3z + 84 = 180

Subtract 84 from both sides:

3z = 96

Now, divide by 3:

z = 32°

Step 4: Find ∠A

Now that we know z = 32°, we can substitute this back into the equation for x:

x = z + 54

= 32 + 54

= 86°

Thus, ∠A = 86°.