Quadrilaterals

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 sum of the angles of any quadrilateral is always the same, what must the fourth angle measure?
2. b) If the 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 it has 4 equal sides.

3. A parallelogram does not have equal diagonals.

In a graphic design workshop, Marcus 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 reasoning the measure of the other three angles of the logo.

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

1. What is the measure of each of those two angles?

2. What is the measure of the other two remaining angles of the rhomboid?

3. If the longest side of the rhomboid measures $12 \text{ cm}$ and the perimeter is $38 \text{ cm}$, how long is the shortest side?

The top of a classroom table has 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 has the shape of an isosceles trapezoid. 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 lateral sides are $72^\circ$. Indicate the measure of the two angles at the top and justify your answer based on the properties of trapezoids.

In the construction of a ramp for a skatepark, a structure in the shape of a right trapezoid is used. The obtuse angle of the structure measures $125^\circ$. Calculate the measure of all the interior angles of this trapezoid and justify your answer based on the properties of parallel lines intersected by a secant.

In a geometry exam, Lucy is asked to calculate the missing angle in a paper kite that has the shape of a kite (symmetric deltoid). The unequal angles measure $110^\circ$ (the top one) and $40^\circ$ (the bottom one). If the other two angles are equal to each other, what is the size of each of them? Remember to use the property of the sum of the interior angles of a quadrilateral.

In an amusement park they have designed a new play area in the shape of 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 lateral angles are equal to each other. Calculate the measure of these two lateral angles and explain with reasoning why they must be equal in this type of figure.

We are designing the logo for a new eSports team. The main logo has the shape of the quadrilateral shown in the drawing. 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, trapezoids or irregular quadrilaterals? Justify your answer by looking at the parallelism of its sides.

Look at the following statements about quadrilaterals and state whether they are True or False, justifying your answer:

1. "A square is, at the same time, a rectangle and a rhombus".

2. "A right trapezoid has exactly four right angles".

3. "In a parallelogram, the diagonals are equal and perpendicular".

4. "Any parallelogram has two pairs of parallel sides".

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

Identify which of these drawings 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:

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

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

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

Find out which quadrilateral we are talking about 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 drawing a small sketch of the diagonals for each case.

Draw a square with a side length of $4\text{ cm}$ in your notebook.

1. a) Draw all its lines of symmetry. How many have you found in total?
2. b) Do all the lines intersect at the same point? What is this point called?

The screen of a new smartphone model has the shape of a rectangle. The manufacturer indicates that its base measures $7\text{ cm}$ and its height is $15\text{ cm}$.

1. a) Calculate the perimeter of the smartphone (the length of the metal frame surrounding the screen).
2. b) If we want to fit a screen protector that covers the entire surface, what will the area of this protector be in $\text{cm}^2$?

A mobile phone company wants to design a screen for a new smartphone model in the shape of a rectangle. The designer has established that the base must measure twice the height. If the total perimeter of the screen must be $36\text{ cm}$, what will be the dimensions (base and height) of the screen?

The screen of a new smartphone model has a rectangular shape. The manufacturer indicates that the total perimeter of the screen is $48 \text{ cm}$ and that the base measures exactly twice the height.

1. Calculate the dimensions (base and height) of the screen.

2. If you want to place a screen protector that costs $0,50 \text{ \$}$ for every $\text{cm}^2$ of surface area, how much will it cost to protect the entire screen?

Calculate the area of the shaded region on the following grid, knowing that each small square has a side length of $1\text{ cm}$. To make it easier, divide the shape into a rectangle and a triangle.

A square garden with a side of $20 \text{ meters}$ has a square fountain in its center. We know that the distance from any side of the garden to the parallel side of the fountain is $6 \text{ meters}$.

1. How long is the side of the central fountain?

2. Calculate the area of the walking zone (the region of the garden not occupied by the fountain).

An architect is designing a window in the shape of a rhombus. For the structure to be stable, they have placed two wooden slats that coincide with the diagonals of the rhombus. The major diagonal measures $160\text{ cm}$ and the minor diagonal measures $120\text{ cm}$. When they cross, the diagonals divide the rhombus into four equal triangles.

1. What type of triangles are these? (according to their angles).

2. If the side of the rhombus measures $100\text{ cm}$, calculate the total perimeter of the window.

An artist is designing a mosaic with pieces in the shape of a rhombus. They know that the two diagonals of the rhombus add up to $28 \text{ cm}$ and that the major diagonal measures $4 \text{ cm}$ more than the minor diagonal.

1. Calculate the measure of both diagonals.

2. Calculate the area of a single piece of the mosaic.

3. If the complete mosaic uses $50$ identical pieces, how many square decimeters ($\text{dm}^2$) of surface area will it cover in total?

Draw any quadrilateral with no parallel sides in your notebook. Find the midpoint of each of its four sides and join them in order around the edge.

1. What special geometric figure is always formed inside?

2. If the sides of the original quadrilateral measure $8\text{ cm}$, $10\text{ cm}$, $12\text{ cm}$ and $14\text{ cm}$, what is the perimeter of the original quadrilateral?

A graphic design company is creating a logo. They have drawn a random quadrilateral and marked the midpoints of each of its sides. By joining these midpoints in order, a new quadrilateral is formed inside.

1. What type of quadrilateral is always formed inside, regardless of the shape of the original?

2. If the area of the outer quadrilateral is $120 \text{ cm}^2$, what will be the area of the figure formed by joining the midpoints?