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.

Classify the following angles according to their size. State whether they are acute, right or obtuse:

• $\alpha = 45^\circ$

• $\beta = 120^\circ$

• $\gamma = 90^\circ$

A group of friends is designing a new skatepark circuit on wasteland. To do this, they have drawn several lines on the ground representing the edges of the ramps. Look at the plan they have drawn and answer the questions about the relative position of the lines:

1. a) Name a pair of lines that are parallel to each other.
2. b) Find two lines that are intersecting but not perpendicular.
3. c) Identify a line that is perpendicular to line $s$.
4. d) If line $t$ and line $u$ never cross no matter how far they are extended, what geometric relationship do they have?

Four intersecting lines have been drawn on a plane in such a way that each line intersects all the others, but no three lines pass through the same point.

1. a) What is the maximum number of intersection points that can be formed?
2. b) Into how many regions is the plane divided?

Draw a diagram representing this and justify your answer by counting the regions.

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

When two lines intersect, vertically opposite angles are equal. Look at the figure and, without using a protractor, state the size of angle $\hat{A}$ and angle $\hat{B}$.

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.

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 $n$.

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.

An architect is designing a square in the shape of a scalene triangle whose vertices are three fountains located at points $A$, $B$ and $C$ (the figure shows a sketch of the square). He wants to place a lamppost at a point $P$ that meets two conditions:

1. It must be at the same distance from fountain $A$ as from fountain $B$.

2. It must be at the same distance from the street joining $A$ with $C$ as from the street joining $B$ with $C$.

Which geometric constructions (perpendicular bisector, angle bisector, etc.) must the architect carry out to find the exact point for the lamppost? Explain the property of the points on those lines that justifies your answer.

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)?

Two angles are complementary if their sum is $90^\circ$. If angle $\alpha$ measures $35^\circ$, what must angle $\beta$ measure for them to be complementary?

A navigation radar turns a total of $5^\circ$ every second. Express this movement in arcminutes ($'$). Remember that $1^\circ = 60'$.

Find the value of angle $\hat{X}$ in the following figure, knowing that the two angles are supplementary (they add up to $180^\circ$).

A security camera in a shopping centre has a total field of view of $110^\circ$. The maintenance technician decides to divide this angle into two zones: Zone A, which covers the shops, and Zone B, which covers the corridors.

If Zone A measures $64^\circ 35' 50''$, calculate the exact size (in degrees, minutes and seconds) that Zone B must have to cover the rest of the field of view.

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.

A wall clock shows 15:40 in the afternoon.

1. a) Calculate the angle formed by the hour hand and the minute hand. Take into account that the hour hand moves a little as the minutes advance.
2. b) How would you classify that angle according to its size (acute, right, obtuse or straight)?

A communications satellite orbits the Earth and takes three angular rotation measurements to correct its trajectory. The first rotation is $42^\circ \ 35' \ 48''$. The second rotation is double the first. If by the end of the day the satellite must have rotated through a total angle equal to a full straight angle to align with the ground station, what must the size of the third rotation be? Express the result in sexagesimal form (degrees, minutes and seconds).

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?

Calculate the exact angle formed by the hands of an analog clock when it shows 03:40.

Hint: Keep in mind that the hour hand does not stay fixed at 3, but moves proportionally as the minute hand moves.

1. a) How many degrees does the hour hand move for each minute that passes?

1. b) Calculate the position in degrees of both hands with respect to 12:00 and find the difference.

Consider two adjacent angles, $\widehat{AOB}$ and $\widehat{BOC}$, which share side $OB$ and whose other sides $OA$ and $OC$ form a straight line (that is, $A$, $O$ and $C$ are collinear and the two angles add up to $180°$). The bisectors $OX$ of angle $\widehat{AOB}$ and $OY$ of angle $\widehat{BOC}$ are drawn.

Prove, by performing the calculation step by step, how much the angle $\widehat{XOY}$ formed by both bisectors measures. Does this result depend on the individual measure of the initial angles? Justify your answer.