Question 16

We will show that a quadrilateral with four equal sides and one right angle must have all its angles as right angles, making it a square.

Step 1 — Set up the quadrilateral

Let us consider a quadrilateral named ABCD. We are given that all its sides are equal in length. So, AB = BC = CD = DA. We are also given that one of its angles is 90 degrees. Let us assume $\angle DAB = 90^\circ$.

Step 2 — Draw a diagonal and analyze triangles

Let us draw a diagonal from point B to point D. This diagonal divides the quadrilateral into two triangles. These triangles are $\triangle ABD$ and $\triangle CBD$.

Step 3 — Examine triangle ABD

In $\triangle ABD$, we know that side AB equals side AD. This is because all sides of the quadrilateral are equal. So, $\triangle ABD$ is an isosceles triangle. Also, we know that $\angle DAB$ is 90 degrees. The sum of angles in any triangle is 180 degrees. So, $\angle ABD + \angle ADB + \angle DAB = 180^\circ$. $\angle ABD + \angle ADB + 90^\circ = 180^\circ$ $\angle ABD + \angle ADB = 180^\circ - 90^\circ$ $\angle ABD + \angle ADB = 90^\circ$ In an isosceles triangle, angles opposite to equal sides are equal. So, $\angle ABD = \angle ADB$. Let us call this angle 'x'. $x + x = 90^\circ$ $2x = 90^\circ$ $x = \frac{90^\circ}{2}$

$\boxed{x = 45^\circ}$ So, $\angle ABD = 45^\circ$ and $\angle ADB = 45^\circ$.

Step 4 — Examine triangle CBD

Now, let us look at $\triangle CBD$. We know that side CB equals side CD. This is because all sides of the quadrilateral are equal. So, $\triangle CBD$ is also an isosceles triangle. This means that $\angle CBD = \angle CDB$.

Step 5 — Compare the two triangles

Let us compare $\triangle ABD$ and $\triangle CBD$. We know that AB = CB (given equal sides). We know that AD = CD (given equal sides). The side BD is common to both triangles. So, by the SSS (Side-Side-Side) congruence rule, the two triangles are congruent. This means $\triangle ABD \cong \triangle CBD$. When two triangles are congruent, their corresponding angles are equal. So, $\angle BCD$ must be equal to $\angle BAD$. We know $\angle BAD = 90^\circ$.

$\boxed{\angle BCD = 90^\circ}$ Also, $\angle CBD$ must be equal to $\angle ABD$. We found $\angle ABD = 45^\circ$. So, $\angle CBD = 45^\circ$. And $\angle CDB$ must be equal to $\angle ADB$. We found $\angle ADB = 45^\circ$. So, $\angle CDB = 45^\circ$.

Step 6 — Calculate all angles of the quadrilateral

Now we can find all the angles of the quadrilateral ABCD. We are given $\angle DAB = 90^\circ$. We found $\angle BCD = 90^\circ$. Let us find $\angle ABC$. $\angle ABC = \angle ABD + \angle CBD$ $= 45^\circ + 45^\circ$

$\boxed{\angle ABC = 90^\circ}$ Let us find $\angle ADC$. $\angle ADC = \angle ADB + \angle CDB$ $= 45^\circ + 45^\circ$ $\boxed{\angle ADC = 90^\circ}$ So, all four angles of the quadrilateral ABCD are 90 degrees. Since all sides are equal and all angles are 90 degrees, ABCD is a square.

Construction and Measurement

Step 1 — Draw the first side and angle

Let us draw a line segment AB. Let its length be 5 cm. At point A, we will construct an angle of 90 degrees. We can use a protractor or a compass for this.

Step 2 — Mark the other vertices

Along the 90-degree line from A, mark point D. Make sure the length AD is also 5 cm. Now, from point D, draw an arc with a radius of 5 cm. From point B, draw another arc with a radius of 5 cm. The point where these two arcs meet is point C.

Step 3 — Complete the quadrilateral

Join point B to C and point C to D. We now have quadrilateral ABCD.

Step 4 — Measure sides and angles

Let us measure all the sides of ABCD. We will find that AB = BC = CD = DA = 5 cm. Let us measure all the angles of ABCD using a protractor. We will find that $\angle DAB = 90^\circ$ (by construction). We will also find that $\angle ABC = 90^\circ$. And $\angle BCD = 90^\circ$. And $\angle CDA = 90^\circ$.

Answer

Yes, a quadrilateral with four equal sides and one angle of 90° will be a square.

Geometric Reasoning: We proved that if a quadrilateral ABCD has four equal sides and one angle of 90°, then all its angles must be 90°. Since all sides are equal and all angles are 90°, it is a square.

Construction and Measurement: By constructing such a quadrilateral with sides of 5 cm and one angle of 90°, we measured the other angles and found them all to be 90°. This confirms that the figure is a square.