Straight lines

In the following drawing of a straight road, identify the name given to each of the marked elements.

State which is a line segment, which is a ray, and which represents a full line.

Look at the plan of these streets. State the relative position (parallel, intersecting or perpendicular) of the following pairs of streets:

• Street A and Street B.

• Street B and Street C.

• Street A and Street C.

We are analysing the structure of a modern bookcase. The horizontal shelves are parallel to each other and are intersected by a sloping support.

Knowing that angle $\hat{1}$ measures $124^\circ$, use the properties of angles formed by parallel lines cut by a transversal to calculate, without using a protractor, the size of angles $\hat{2}, \hat{3}$ and $\hat{4}$. Justify each answer (being vertically opposite, corresponding, alternate interior, etc.).

Look at the following figure showing several related angles. We know that lines $m$ and $n$ are parallel. If the angle $\hat{A} = 48^\circ$, calculate with clear reasoning the size of angles $\hat{B}$ and $\hat{C}$.

In the design of a metallic roof structure, there are two main parallel beams, $r$ and $s$. A third transversal beam $t$ intersects them forming an obtuse angle of $135^\circ \ 20'$. It is decided to install a brace that coincides exactly with the angle bisector of the acute angle formed by the intersection of beams $s$ and $t$. What angle will this brace form with beam $r$ on the inner part of the structure? Reason whether the angles will be alternate interior or corresponding angles.

Draw in your notebook a line $r$ and a point $P$ that does not lie on it. Using set squares, draw a line $s$ that passes through $P$ and is parallel to $r$. How many lines parallel to $r$ can pass through that point $P$?

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

Given a line $r$ and an external point $P$, we know that there is only one line parallel to $r$ that passes through $P$.

Now imagine that we have three lines parallel to each other ($r, s, t$) and two transversal lines ($m, n$) that are not parallel to each other.

1. a) What is the total number of angles formed at the intersection points?
2. b) If all the acute angles formed by line $m$ with the parallel lines measure $70^\circ$, and the acute angles formed by line $n$ with the parallel lines measure $55^\circ$, calculate the angle formed between lines $m$ and $n$.

On the plan of a children's playground, you want to place a lamppost that is exactly the same distance from the slide ($T$) and the swings ($C$). The figure shows the position of both and the distance between them.

1. a) What is the name of the imaginary line where all the points that meet this condition should lie?

1. b) If the distance between the slide and the swings is $12$ meters and you place the lamppost at the midpoint of segment $TC$, how far will the lamppost be from each of them?

1. c) Draw in your notebook a sketch of segment $TC$ and its perpendicular bisector.

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

Two main avenues in a city intersect forming an angle of $72^\circ 15'$. They want to build a pedestrian street that divides this angle exactly in half.

1. a) What is the technical name of the line representing that pedestrian street?
2. b) Calculate the size of the angle that the new street will form with each of the avenues.
3. c) What type of angle is the result (acute, right or obtuse)?

In a strategy video game, a laser beam reflects off a metallic surface. It is known that the angle of incidence $\alpha$ and the angle of reflection $\beta$ are equal.

If the total angle formed by the incoming beam and the outgoing beam is $115^\circ 20'$, calculate the size of angle $\alpha$. Then state whether the supplementary angle of $\alpha$ is acute, right or obtuse.

The angle $\alpha$ is such that its supplementary angle measures three times its complementary angle.

1. a) Set up an equation using the concepts of complementary angle ($90^\circ - \alpha$) and supplementary angle ($180^\circ - \alpha$).
2. b) Calculate the value of angle $\alpha$.
3. c) How would you classify angle $\alpha$ according to its size?