Explain why all the angles of this new 4-sided figure are right angles and so it is a square.

Question

Homi · Accepted Answer

The central 4-sided figure has all its interior angles equal to $90^\circ$ , and all its sides are equal in length, making it a square.

Step 1 — Identify the side lengths of the central figure

Let us look at the central 4-sided figure in the diagram. The diagram explicitly labels the length of each of its four sides as $b-a$ . This means that all four sides of the central figure are equal in length.

Diagram 1

Step 2 — Calculate the interior angles of the central figure

Let us examine the angles at each corner (vertex) of the central 4-sided figure. Consider the top-right corner of the central figure. At this corner, three angles meet on a straight line. One angle is $x$ , from the top-right triangle. Another angle is $90-x$ , from the top-left triangle. The third angle is the interior angle of the central figure at that corner. The sum of angles on a straight line is $180^\circ$ . Let the interior angle of the central figure at this corner be $\angle C$ .

$x + (90-x) + \angle C = 180^\circ$

$90^\circ + \angle C = 180^\circ$

$\angle C = 180^\circ - 90^\circ$

$\boxed{\angle C = 90^\circ}$

The diagram also explicitly shows a right angle symbol at the top-left corner of the central figure, meaning its interior angle is $\mathbf{90^\circ}$ . Similarly, if we examine the bottom-right and bottom-left corners, we will find that their interior angles are also $90^\circ$ . This is because the same pattern of angles $x$ and $90-x$ meeting on a straight line occurs at those corners too. So, all four interior angles of the central figure are $\mathbf{90^\circ}$ .

Step 3 — Conclude that the figure is a square

A square is a special type of quadrilateral (a 4-sided figure) where all four sides are equal in length and all four interior angles are right angles ( $90^\circ$ ). From Step 1, we found that all four sides of the central figure are equal in length (each side is $b-a$ ). From Step 2, we found that all four interior angles of the central figure are $90^\circ$ . Since both conditions are met, the central 4-sided figure is a square.

Answer

All four interior angles of the central 4-sided figure are $90^\circ$ because at each vertex, the sum of the two adjacent angles from the surrounding triangles ( $x$ and $90-x$ ) and the interior angle of the central figure forms a straight line ( $180^\circ$ ). Since $x + (90-x) = 90^\circ$ , the interior angle must be $180^\circ - 90^\circ = 90^\circ$ . The diagram also explicitly labels all four sides of this central figure as $b-a$ , meaning all its sides are equal. Therefore, a figure with four equal sides and four right angles is a square.

Question 19

Answer

More questions in IT