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Quadrilaterals and Their Properties. Types of Quadrilaterals.

Sides intersect	Extension of a side intersects another side	Three vertices on the same line
	Not quadrilaterals

The segments connecting opposite vertices are called diagonals.

If we consider a quadrilateral as two triangles, the sum of the quadrilateral's angles is 360 degrees.

Types of Quadrilaterals

Among quadrilaterals, two large groups can be distinguished – parallelograms and trapezoids.

Parallelogram

The definition of a parallelogram can be given in several ways (in which case the others become theorems that need to be proven).

A parallelogram is a quadrilateral whose opposite sides are parallel.

To construct a parallelogram in GeoGebra, we place three points A, B, and C, while the fourth vertex is obtained through construction.

We create the sides AB and BC of the quadrilateral	We draw a line through point A parallel to BC	And through point C, we draw a line parallel to AB.
The constructed lines will intersect at point D, the fourth vertex of the quadrilateral	We will hide the lines used for construction	We will draw the segments AD and CD.

After this definition, we can prove that in a parallelogram:

• Opposite sides are equal.
• Opposite angles are equal.
• Adjacent angles sum up to 180 degrees.
• The diagonals bisect each other at their intersection point.

Trapezoid

A trapezoid is a quadrilateral in which two sides are parallel.

The parallel sides AD and BC are called the bases of the trapezoid. AB and CD are its legs.

Arbitrary trapezoid	Right trapezoid	Isosceles trapezoid

We place three points A, B, and D, and construct the two sides AD and AB	Next, we draw a line through point B parallel to AD	Finally, we place point C on this line
We draw the sides BC and CD	Next, we copy the style of segment AB to the new segments	We hide the line, and point C can now be freely moved

Rectangle

A rectangle is a parallelogram in which all the angles are equal. Therefore, the angles of a rectangle are 90 degrees. A rectangle also has the property that its diagonals are equal.

Rhombus

If all sides of a parallelogram are equal, it is a rhombus. A rhombus has the following properties:

• The diagonals of a rhombus intersect at a right angle.
• The diagonals of a rhombus are the bisectors of its angles.
  

Square

I believe everyone knows about a square, but did you know that a square is all these quadrilaterals combined – a parallelogram, a rectangle, and a rhombus (and a trapezoid, although no one really calls it that)?

So, a square has sides equal like a rhombus, and angles equal like a rectangle. The same goes for the diagonals – they are perpendicular like in a rhombus, and equal like in a rectangle.

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Homework

This time, you can choose: either complete 5 tasks or complete 2 tasks.

Easy Task Option:

1. Build a parallelogram, demonstrate its elements and properties.
2. Build a trapezoid, demonstrate its elements and properties.
3. Build a rhombus, demonstrate its elements and properties.
4. Build a rectangle, demonstrate its elements and properties.
5. Build a square, demonstrate its elements and properties.

For all tasks: 1.8 points.
Additional point for aesthetic appearance, accuracy, and neatness of drawings, labels, and annotations: 1 point.
Total: 5 * 1.8 + 1 = 10 points.

Harder Task Option:
Repeat the above.

1 (5point) Build an isosceles trapezoid, similar to (or the same as) the one shown in this animation. I freely move the vertices C and D, and the trapezoid changes its size but remains isosceles. The key is the change in size and proportions of the trapezoid, but it always remains isosceles.

2 (5point) Given a segment – the diagonal of a rectangle. Build a rectangle with the given diagonal.

Describe the construction process in detail, so that others can learn to build as well based on your task.

Rules for Participation

Participants must be verified and active users of the platform.

All images used must belong to the author or be free of copyright. (Don’t forget to credit the source.)

Participants must not use any bot services for voting or engage in vote buying.

Recommend your friends to participate.

Submission Period: From Monday (March 10/2025), to Sunday (March 16/2025).

Your work will be reviewed, commented on, and evaluated by me. Four best works will be selected.

Good luck 🍀 on your task!

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