Quadrilaterals

We are designing the logo for a new eSports team. The main logo is in the shape of the quadrilateral shown in the diagram. Look at its sides and angles, and answer:

1. a) What is the precise name of this quadrilateral?
2. b) Which group does it belong to: parallelograms, trapeziums or trapezoids? Justify your answer by looking at the parallelism of its sides.

In a technical drawing class, a student has drawn a quadrilateral and measured three of its interior angles: $110^\circ$, $70^\circ$ and $110^\circ$.

1. a) Knowing that the angle sum of any quadrilateral is always the same, how much must the fourth angle measure?
2. b) If opposite sides are equal in pairs, what shape is it?

State whether the following statements about parallelograms are true or false, briefly justifying your answer:

1. Every square is also a rectangle because it has 4 right angles.

2. A rhombus is a square because its 4 sides are equal.

3. A rhomboid does not have equal diagonals.

In a graphic design workshop, Marcos is creating a logo in the shape of a rhomboid (non-regular parallelogram). If one of the interior angles of the rhomboid measures $65^\circ$, calculate with clear reasoning the size of the other three angles of the logo.

In a parallelogram, the sum of two of its opposite angles is $130^\circ$.

1. How much does each of these two angles measure?

2. How much do the remaining two angles of the parallelogram measure?

3. If the longer side of the parallelogram measures $12 \text{ cm}$ and the perimetre is $38 \text{ cm}$, how long is the shorter side?

The top of a classroom table is in the shape of a quadrilateral. If you look closely, it has two parallel sides of different lengths and the other two sides are equal to each other.

1. a) How is this quadrilateral classified?
2. b) Is it a parallelogram? Explain why.

The cross section of a new skate ramp is in the shape of an isosceles trapezium. We know that the top base measures $3\text{ m}$, the bottom base measures $7\text{ m}$ and the angles formed by the bottom base with the side edges are $72^\circ$. State the sizes of the two top angles and justify your answer using the properties of trapeziums.

In a theme park, they have designed a new play area shaped like a kite. We know that the top angle measures $\hat{A} = 110^\circ$ and the bottom angle measures $\hat{C} = 40^\circ$. The other two side angles are equal to each other. Calculate the measure of these two side angles and explain with reasons why they must be equal in this type of shape.

Look at the diagonals of the following quadrilaterals. One of the most important properties is whether they are perpendicular (forming $90^\circ$) and whether they intersect at their midpoint.

Identify which of these diagrams represents the diagonals of a rhombus and which represents those of a rectangle. Justify your choice.

Classify the following quadrilaterals based exclusively on the description of their diagonals:

• Shape A: Its diagonals are of different lengths, are perpendicular and intersect at their midpoint.

• Shape B: Its diagonals are equal, are not perpendicular and intersect at their midpoint.

• Shape C: Its diagonals are equal, are perpendicular and intersect at their midpoint.

Work out which quadrilateral is being described in each case, based on its diagonals:

1. "My diagonals are equal and intersect perpendicularly at their midpoint."

2. "My diagonals are equal, intersect at their midpoint, but are not perpendicular."

3. "My diagonals are not equal, but they are perpendicular and intersect at their midpoint."

4. "I have two parallel sides and two equal diagonals, but they do not intersect at their midpoints."

Justify your answer by making a small sketch of the diagonals for each case.