One angle of a quadrilateral is 108° and the remaining three angles are equal. Find each of the three equal angles :

Question

Accepted Answer

To solve the problem of finding each of the three equal angles in a quadrilateral where one angle is 108°, we need to use the properties of a quadrilateral. The sum of all angles in a quadrilateral is always 360°.

Given:
- One angle = 108°
- The remaining three angles are equal.

Let's denote each of the three equal angles as $ x $.

According to the properties of a quadrilateral:
$$ 108° + 3x = 360° $$

Now, solve for $ x $:
$$ 3x = 360° - 108° $$
$$ 3x = 252° $$
$$ x = \frac{252°}{3} $$
$$ x = 84° $$

Therefore, each of the three equal angles is 84°, which corresponds to Option 3.

**Explanation for why Option 3 is correct:**
- The calculation shows that when you subtract the given angle (108°) from the total sum of angles in a quadrilateral (360°), you get 252°. Dividing this by 3 gives 84°, confirming that each of the three equal angles is indeed 84°.

**Explanation for why other options are incorrect:**
1. **Option 1 (63°):** If each angle were 63°, the sum of the three angles would be $ 3 \times 63° = 189° $. Adding the given angle of 108° would result in $ 189° + 108° = 297° $, which is less than 360°.
   
2. **Option 2 (74°):** If each angle were 74°, the sum of the three angles would be $ 3 \times 74° = 222° $. Adding the given angle of 108° would result in $ 222° + 108° = 330° $, which is less than 360°.

3. **Option 4 (96°):** If each angle were 96°, the sum of the three angles would be $ 3 \times 96° = 288° $. Adding the given angle of 108° would result in $ 288° + 108° = 396° $, which exceeds 360°.

Thus, Option 3 (84°) is the only correct choice that satisfies the condition of the sum of angles in a quadrilateral being 360°.