Triangles

In a triangle, we know the measure of two of its interior angles: $\hat{A} = 45^\circ$ and $\hat{B} = 75^\circ$. Calculate the measure of the third angle $\hat{C}$ knowing that the sum of the angles in any triangle is always $180^\circ$.

In the logo of a new sportswear brand, two rectangles intersect forming several triangles. In the central triangle $ABC$, we know that the angle at $A$ measures $45^\circ$ and that the exterior angle at vertex $B$ measures $115^\circ$.

1. Calculate the three interior angles of triangle $ABC$.

2. Classify the triangle by its sides and by its angles.

To decorate her room, Lucía has cut out a fabric pennant shaped like a triangle. On measuring its sides, she found that they measure $15\text{ cm}$, $15\text{ cm}$ and $20\text{ cm}$. How is this triangle classified according to the length of its sides?

Look at the "Give way" road sign. It is shaped like an equilateral triangle. State the measure of each of its interior angles without using a protractor. Justify your answer based on the properties of this type of polygon.

An isosceles triangle has an unequal angle measuring $40^\circ$. If the other two angles are equal to each other, what is the measure of each of those equal angles?

In a video game design workshop, a programmer needs to create a character shaped like an isosceles triangle. The top vertex angle measures $20^\circ$ less than three times one of the base angles. Calculate the measure of the three angles of the character so that the design is geometrically correct.

Classify the following triangles according to the measure of their angles (acute-angled, right-angled or obtuse-angled):

• Triangle 1: Its angles measure $90^\circ$, $40^\circ$ and $50^\circ$.

• Triangle 2: Its angles measure $110^\circ$, $30^\circ$ and $40^\circ$.

• Triangle 3: Its angles measure $60^\circ$, $60^\circ$ and $60^\circ$.

In a right-angled triangle, the two sides that form the $90^\circ$ angle measure $6\text{ cm}$ and $8\text{ cm}$. What are these two sides called in the mathematical terminology of right-angled triangles? And what is the longest side opposite the right angle called?

In a graphic design studio, they are creating a new logo shaped like a triangle. We know that two of its interior angles measure $42^\circ$ and $68^\circ$.

1. a) Calculate the measure of the third angle of the logo.
2. b) Classify the triangle by its angles (acute-angled, right-angled or obtuse-angled).
3. c) Classify the triangle by its sides (equilateral, isosceles or scalene).

Marcos has three wooden rods to build a triangular structure. The lengths of the rods are $12\text{ cm}$, $5\text{ cm}$ and $4\text{ cm}$. The figure shows the three rods drawn to the same scale.

Explain with reasoning whether Marcos will be able to form a triangle with them or whether it will be impossible.

A carpenter has three wooden battens to make a triangular frame for a mirror. The battens measure $15\text{ cm}$, $8\text{ cm}$ and $24\text{ cm}$.

1. a) Explain, using the triangle inequality property, whether it is possible to construct the frame with those measurements.
2. b) If he decides to replace the batten of $24\text{ cm}$ with one of $20\text{ cm}$, could he construct it now? Explain your answer.

An architect is designing a ramp for a skatepark. He has calculated that the main support must form a triangle with sides $x + 2$ meters, $2x - 1$ meters and $10$ meters.

1. If the triangle must be equilateral, what value must $x$ have?

2. If $x = 3$, is it possible to construct this triangle? Justify your answer using the triangle inequality theorem (the sum of the lengths of any two sides of a triangle).

A triangular road sign has been designed in the shape of an equilateral triangle with a side length of $60\text{ cm}$.

1. a) If we want to put a reflective border around the sign, how many meters of reflective material will we need?
2. b) If we draw the three angle bisectors of the triangle, what is the point where they cross called? What special property does this point have with respect to the sides of the sign?

In a garden, a fence is to be put up to enclose a flower bed in the shape of an isosceles triangle. The unequal angle measures $100^\circ$.

1. a) What is the measure of each of the other two angles of the flower bed?
2. b) If we draw the median from the vertex of the $100^\circ$ angle to the opposite side, into what two types of triangles is the region divided? Classify them by their angles.

In an advanced geometry exam, a triangle is shown where the three medians intersect at a point $G$ (Centroid). If the median coming from vertex $A$ has a total length of $12 \text{ cm}$, calculate the length of the two segments into which point $G$ divides this median. (Remember the property: the centroid divides the median into two parts, one being twice the length of the other).

A garden is shaped like a rectangle of $12 \times 8$ meters. To install an irrigation system, the garden has been divided by drawing a diagonal. In one of the two resulting triangles, a circular sprinkler is to be placed at the incenter.

1. Explain what angle bisectors are and why placing the sprinkler at the incenter is the best option for it to be equidistant from the three sides of the triangle (the two edges of the garden and the diagonal).

2. Draw a sketch of the rectangle, the diagonal, and the incenter of one of the triangles.

Three villages, Altea ($A$), Belmonte ($B$) and Calar ($C$), are located so that they form a scalene triangle, as shown in the figure. A telecommunications tower is to be installed exactly the same distance from all three villages.

1. a) Which notable point of the triangle represents the location of the tower?

1. b) Which notable lines did you have to draw to find that point? Draw a rough sketch of the situation.

You can do it on paper and send a photo with your answer.

Three mountain towns, Arteaga, Bernal, and Creel, want to build a cell phone tower that is at exactly the same distance from all three so that everyone has the same coverage.

1. What is the name of the triangle's key point where the tower should be placed?

2. If the triangle formed by the towns is obtuse, explain whether the tower will be inside or outside the area bounded by the three towns.

3. Draw a rough sketch of the situation.

A children's playground has a sandbox shaped like a right triangle. We know that the longest side (the hypotenuse) measures $13\text{ meters}$ and one of the sides forming the right angle measures $5\text{ meters}$.

1. a) Calculate the length of the other side of the sandbox.
2. b) Find the total perimeter of the sandbox.
3. c) Calculate the surface area occupied by the sand.

Look at the following figure composed of two triangles. Triangle $ABC$ is right-angled at $B$ and triangle $ACD$ is isosceles with $\overline{AC} = \overline{AD}$.

If $\overline{AB} = 8\text{ cm}$ and $\overline{BC} = 6\text{ cm}$, calculate:

1. a) The length of side $\overline{AC}$.
2. b) The total perimeter of the shaded figure $ABCD$ knowing that $\overline{CD} = 7\text{ cm}$.

An explorer has placed a rope of $24$ meters around a triangular plot of land to set up his camp.

1. If the plot is an isosceles triangle and one of the equal sides measures $9 \text{ meters}$, how long is the remaining side?

2. If the explorer wanted the triangle to be a right triangle with sides of $6 \text{ m}$, $8 \text{ m}$ and $10 \text{ m}$, would the same rope work? Check also if the Pythagorean theorem holds in this latter case.

A garden is shaped like an equilateral triangle. If one of its sides measures $8\text{ meters}$, how many meters of fencing will we need to enclose it completely? Calculate its perimeter.

A tent has a front shaped like an isosceles triangle. The base of the tent measures $2.4\text{ meters}$ and the two equal sides measure $2\text{ meters}$ each.

1. a) Use the Pythagorean theorem to calculate the height of the tent.
2. b) Calculate the area of canvas needed to cover only the front of the tent.

Observe the attached figure, which represents a metal piece. It consists of a large right-angled triangle and a smaller one inside it. We know that the horizontal leg of the large triangle is $15 \text{ cm}$ and its area is $60 \text{ cm}^2$. The small triangle is similar and has a base of $5 \text{ cm}$.

1. Calculate the height of the large triangle.

2. Using the logic of proportions, calculate the area of the small triangle. (Hint: if the base is one-third, what happens to the height?).