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 of any triangle is always $180^\circ$.

To decorate her room, Lucy has cut out a fabric pennant in the shape of a triangle. When measuring its sides, she saw 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?

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

Observe the "Yield" traffic sign. It has the shape of an equilateral triangle. Indicate how much each of its interior angles must measure without needing to use a protractor. Justify your answer based on the properties of this type of polygon.

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

• 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 graphic design workshop, they are creating a new logo in the shape of 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 according to its angles (acute, right, or obtuse).
3. c) Classify the triangle according to its sides (equilateral, isosceles, or scalene).

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

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

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

2. Classify the triangle according to its sides and according to its angles.

Marcus has three wooden rods to build a triangular structure. The lengths of the rods are $12\text{ cm}$, $5\text{ cm}$, and $4\text{ cm}$. Explain with reasoning whether Marcus will be able to form a triangle with them or if it will be impossible for him.

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

1. a) Explain, using the property of the sum of the sides of a triangle, whether it is possible to build the frame with those measurements.
2. b) If he decides to change the $24\text{ cm}$ slat for a $20\text{ cm}$ one, could he build it now? Reason your answer.

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

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

2. If $x = 3$, is it possible to build that triangle? Justify your answer using the property of the sum of the sides of a triangle.

A garden has the shape of an equilateral triangle. If one of its sides measures $8\text{ meters}$, how many feet of fence will we need to surround it completely? Calculate its perimeter.

In a right 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 triangles? And what is the longest side opposite the right angle called?

A playground has a sandbox in the shape of 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.

Observe 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 $\overline{CD} = 7\text{ cm}$.

A tent has a front in the shape of 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.

An explorer has placed a $24$-foot rope surrounding a triangular plot of land to set up his camp.

1. If the land 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 right-angled with sides of $6 \text{ ft}$, $8 \text{ ft}$, and $10 \text{ ft}$, would the same rope work? Also check if the Pythagorean theorem holds in this last case.

Observe the attached figure, which represents a metal part. It is formed by a large right triangle and a smaller one inside it. We know that the horizontal leg of the large one measures $15 \text{ cm}$ 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 a third, what will happen to the height?).

Three towns, Ashland ($A$), Belmar ($B$), and Canton ($C$), are situated so that they form a scalene triangle. They want to install a cell tower that is exactly the same distance from all three towns.

1. a) Which notable point of the triangle represents the location of the tower?
2. b) Which notable lines did you have to draw to find this point? Draw a rough sketch of the situation.

Three mountain towns, Alton, Bristol, and Concord, want to build a cell tower that is exactly the same distance from all three so that everyone has the same coverage.

1. What is the name of the notable point of the triangle where they should place the tower?

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

3. Draw a rough sketch of the situation.

A triangular traffic sign has been designed which is an equilateral triangle with a side of $60\text{ in}$.

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

A garden has the shape of a rectangle of $12 \times 8$ feet. To install an irrigation system, the garden has been divided by drawing a diagonal. In one of the two resulting triangles, they want to place a circular sprinkler at the incenter.

1. Explain what angle bisectors are and why the sprinkler at the incenter is the best option so that it is at the same distance from the three sides of the triangle (the two edges of the garden and the diagonal).

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

In a garden, they want to place a fence that delimits 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 bed divided? Classify them according to their angles.

In an advanced geometry exam, a triangle is presented where the three medians intersect at a point $G$ (Centroid). If the median coming from vertex $A$ measures a total of $12 \text{ cm}$, calculate the measure 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 other).