Question

Three angles of a quadrilateral are equal. The fourth angle measures \[120^\circ \]. What is the measure of each of the equal angles?

Answer 1

Hint: In the above question, we are given a quadrilateral whose three interior angles are equal. Also, the fourth angle is given here as \[120^\circ \] . We have to find the measure of each of the three other angles which are equal. In order to approach the solution, first we have to find the sum of all the four interior angles of a quadrilateral.

Complete step by step answer:
Given that, a quadrilateral whose three interior angles are equal.And the measure of the fourth interior angle is \[120^\circ \]. Since we know that the sum of all interior angles of a n-sided polygon is given by the formula,
\[ \Rightarrow \left( {n - 2} \right)180^\circ \]
Where \[n\] is the number of sides of that polygon.
Here the polygon is a quadrilateral, that means it has \[4\] sides.
Therefore, the sum of \[4\] angles of a quadrilateral is,
\[ \Rightarrow \left( {4 - 2} \right)180^\circ \]
\[ \Rightarrow 2 \times 180^\circ \]
Hence,
\[ \Rightarrow 360^\circ \]

Now, let the three equal angles be \[x\]. Therefore, the sum of three equal angles and the fourth angle \[120^\circ \] is,
\[ \Rightarrow x + x + x + 120^\circ = 360^\circ \]
That gives us the equation,
\[ \Rightarrow 3x + 120^\circ = 360^\circ \]
Subtracting \[120^\circ \] from both the sides, we get
\[ \Rightarrow 3x = 240^\circ \]
Dividing both sides by \[3\] , we get
\[ \Rightarrow x = \dfrac{{240^\circ }}{3}\]
Hence,
\[ \therefore x = 80^\circ \]

Therefore, the measure of each of the equal angles is \[80^\circ \].

Note: A quadrilateral is a closed two-dimensional figure that has 4 sides, 4 angles, and 4 vertices. A few examples of quadrilaterals are square, rectangle and trapezium. It is a closed shape that is formed by joining four points among which any three points are non-collinear. There are different types of quadrilaterals that are identified on the basis of their unique properties. For example, square, rectangle, parallelogram, rhombus, kite, trapezium, isosceles trapezium are all categorized under quadrilaterals.