Question 1

We will construct a quadrilateral with four equal sides to see if it must be a square.

Step 1 — Starting with two equal sides

The problem asks us to start with two equal sides, AD and AB, that are not perpendicular. The given diagram shows these sides forming an angle of $50^\circ$ at vertex A. We will use these as the first two sides of our quadrilateral.

Step 2 — Constructing the third side

To make all four sides of our quadrilateral equal, the side DC must have the same length as AD and AB. We use a compass for this. We place the compass point at D and set its radius to the length of AD. Then, we draw an arc. This arc shows all possible positions for vertex C such that DC is equal to AD.

Step 3 — Constructing the fourth vertex

The fourth side, BC, must also be equal to AB and AD. We place the compass point at B and set its radius to the length of AB. We then draw a second arc. The point where this arc intersects the arc from Step 2 is our fourth vertex, C. This ensures that BC = AB and DC = AD.

Step 4 — Completing the quadrilateral

Now, we connect point C to point D and point C to point B using straight lines. This completes our quadrilateral, which we can call ABCD.

Step 5 — Analyzing the constructed quadrilateral

By our construction, all four sides of the quadrilateral ABCD are equal in length (AB = BC = CD = DA). However, the angle at vertex A is $50^\circ$, which is not a right angle ($90^\circ$). A quadrilateral with all four sides equal is called a rhombus. A square is a special type of rhombus where all angles are $90^\circ$. Since our constructed quadrilateral has equal sides but an angle of $50^\circ$, it is a rhombus but not a square.

Answer

No, squares are not the only quadrilaterals that have equal side lengths. We constructed a rhombus (a quadrilateral with all four sides equal) where the angle between two adjacent sides is $50^\circ$. Since $50^\circ$ is not $90^\circ$, this rhombus is not a square. This construction shows that other quadrilaterals, like rhombuses, can also have equal side lengths without being squares.