Elements of the plane

On a hiking map, a straight trail connects point $A$ (a shelter) to point $B$ (a waterfall). If we extend the trail beyond the waterfall towards the mountain peak (point $C$), identify which geometric concept each of the following sections represents:

1. The path that starts at the shelter and ends at the waterfall.

2. The path that starts at the shelter and continues indefinitely passing through the waterfall and the peak.

3. The theoretical line containing the shelter, the waterfall and the peak, extending endlessly in both directions.

Look at the following image and identify which of the drawn elements corresponds to a line, which to a ray, and which to a line segment. Briefly explain the difference between them by looking at their endpoints.

The following segments have these lengths:

• Segment $AB = 40$ $mm$

• Segment $CD = 2.5$ $cm$

• Segment $EF = 0.5$ $dm$

Calculate the total sum of their lengths in centimeters ($cm$).

Remember to convert all measurements to the same unit before adding.

On the map of your neighbourhood, two streets can cross or be parallel. Look at the diagram of lines $a$, $b$, and $c$. Indicate which pair of lines are parallel and which pair are intersecting. What is the orange point where two of them intersect called?

An architect is designing a new residential area. They have drawn three streets represented by the lines $r$, $s$ and $t$. Look at the plan and answer with reasoning:

• If street $r$ is perpendicular to street $s$, and street $t$ is parallel to street $r$, how are streets $s$ and $t$ related to each other?

• If a car drives along street $s$ and wants to turn into street $t$, through what angle (in degrees) will it have to turn?

Two parallel lines, $r$ and $s$, are cut by a transversal $t$. It is known that two alternate interior angles are given by the measures $2x + 15^\circ$ and $3x - 10^\circ$.

1. a) Find the value of $x$.
2. b) Calculate the measure of all eight angles formed at the intersection.

Classify the following angles according to their size as acute, right, or obtuse. To help you, remember that a right angle measures exactly $90^\circ$.

• Angle $A = 45^\circ$

• Angle $B = 120^\circ$

• Angle $C = 90^\circ$

• Angle $D = 10^\circ$

Draw a three-sided polygon (a triangle) and label its main elements. You must indicate where the vertices, the sides and the interior angles are. How many of each does a triangle have?

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

The hands of a clock form different angles throughout the day.

1. a) What type of angle do the hands form when it is 3:00 o'clock?
2. b) What about when it is 6:00 o'clock?
3. c) What angle (in degrees) does the minute hand turn through in 15 minutes?

Lucas has to calculate the complementary angle of $\hat{A} = 42^\circ\ 35'$. In his notebook, he wrote the following solution:

"Since two complementary angles add up to $90^\circ$, I subtract: $90^\circ - 42^\circ\ 35'$. I convert $90^\circ$ into $89^\circ\ 100'$, so I do $89^\circ\ 100' - 42^\circ\ 35' = 47^\circ\ 65'$. Since $65'$ exceeds one degree, I transfer $60'$ to the degrees and get a final result of $48^\circ\ 5'$".

1. a) Identify the mistake made by Lucas and explain in detail why his reasoning about units fails and how it affects the final result, recalling the base of the sexagesimal system.
2. b) Perform the operation correctly and find the exact measure of the complementary angle of $\hat{A}$.

A traffic radar rotates on its axis sweeping a total angle of $145^\circ$ $30'$. If the radar has already rotated $82^\circ$ $45' $ $20''$, how much more does it need to rotate to complete its full sweep? Express the result in degrees, minutes and seconds.

Observe the hands of an analog clock when it shows exactly $15:00$ (three o'clock). The figure shows that clock: the short hand is the hour hand and the long one is the minute hand.

Answer:

1. a) What type of angle do the two hands form at $15:00$?

1. b) If the clock moves forward to $15:10$, the minute hand has turned through a certain angle. Knowing that each minute corresponds to a $6^\circ$ turn of the minute hand, what angle has the minute hand covered?

1. c) At $15:10$, is the angle formed by the two hands greater or smaller than the one they formed at $15:00$? Keep in mind that the hour hand also moves a little: it turns $0.5^\circ$ every minute.

A pizzeria wants to divide a circular pizza of $30 \text{ cm}$ diameter into 7 equal slices.

1. a) What is the exact measure of the central angle of each slice in degrees, minutes and seconds?
2. b) If by mistake the first cut is made at $52^\circ$, how many extra minutes and seconds does that slice have compared to the correct measure?

Calculate exactly the smaller angle formed by the hands of an analogue clock when it shows $10:20$ in the morning. Bear in mind that the hour hand moves proportionally to the movement of the minute hand. (Hint: every minute, the hour hand moves $0.5^\circ$).

Two angles are complementary if their sum is $90^\circ$ and supplementary if their sum is $180^\circ$. Calculate the missing angle in each case:

1. a) The complement of an angle of $35^\circ$.
2. b) The supplement of an angle of $110^\circ$.

In a geometric design workshop, you are asked to find an unknown angle $\hat{A}$ that satisfies a very specific mathematical property: the measure of its supplementary angle is exactly three times the measure of its complementary angle.

1. a) Reason, without needing to calculate angle $\hat{A}$ yet, how many degrees of difference there always are between the supplementary angle and the complementary angle of any angle in the plane.
2. b) Use that difference and the condition that one is three times the other to deduce what the measure of the complementary angle of $\hat{A}$ must necessarily be.
3. c) Determine from the above the exact measure of angle $\hat{A}$ and verify that the initial condition is met by calculating its complement and its supplement.

In the design of a skateboard ramp, three interior angles have been calculated that must add up to a straight angle so that the structure is stable. If the first two angles measure $\hat{A} = 48^\circ$ $25'$ and $\hat{B} = 72^\circ$ $50'$, calculate the measure of the third angle $\hat{C}$. What type of angle is it according to its turn (acute, right or obtuse)?

Two angles are supplementary. If one of them measures three times as much as the other, what is the measure of each of the angles? Draw a diagram to help you visualise the situation.

The rotating arm of a sorting machine in a logistics center in Limerick starts from a reference position and rotates through an angle of $58^\circ\ 48'$. To complete a calibration that requires reaching exactly a right angle ($90^\circ$), the operator must program a second complementary turn following the first one.

1. a) Before working with pencil and paper, estimate with reasoning between which two whole tens of degrees (for example, between $20^\circ$ and $30^\circ$, or between $30^\circ$ and $40^\circ$) the value of the remaining turn will lie. Explain what you look at to make the estimate without calculating the exact value.
2. b) Calculate with sexagesimal precision (in degrees and minutes) the measure of the turn the machine still needs to make to reach $90^\circ$.
3. c) Compare the exact value obtained with your previous estimate and evaluate with reasoning whether it makes geometric sense.

In the design of a metal part, two angles are supplementary. If one of them is three times the other minus $20^\circ$, how much does each angle measure? Set out a logical or mathematical strategy to find their measures in degrees, minutes and seconds.

When two lines intersect, vertically opposite angles are equal. In the following drawing, if we know that the orange angle $\hat{\alpha}$ measures $40^\circ$, how much will angles $\hat{\beta}$, $\hat{\gamma}$, and $\hat{\delta}$ measure?

Two friends, Ana and Borja, want to install a Wi-Fi repeater antenna in the school playground that is exactly at the same distance from both of them. If Ana is at point $A$ and Borja is at point $B$:

1. What is the name of the imaginary line where they could place the antenna?

2. If the distance between Ana and Borja is $12$ metres, what is the minimum distance from each of them that the antenna will be?

3. Draw a diagram showing points $A$ and $B$, the line segment joining them, and the line where they must place the antenna.

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

A gardener wants to divide a flowerbed shaped like a circular sector into two equal sectors. To do this, he needs to draw the angle bisector of the central angle formed by the original flowerbed, which measures $135^\circ$.

• Calculate the measure of each of the new angles after drawing the bisector.

• If the gardener makes a mistake and draws a line that leaves an angle of $60^\circ$ $40'$, how many degrees, minutes, and seconds of difference are there compared to the correct bisector?

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

In the urban redevelopment of an area in Vitoria-Gasteiz, two straight avenues, Avenida del Norte and Avenida Central, intersect at a point $P$. One of the four angles generated at that intersection measures $68^\circ\ 40'$.

1. a) Model the situation by classifying the relative position of both avenues in the plane and determine with reasoning the measure of the other three angles created at intersection $P$, justifying which ones are vertically opposite (vertical angles) and which ones are supplementary.
2. b) To make it easier for city buses to turn, a platform is planned whose central axis coincides exactly with the bisector of the $68^\circ\ 40'$ angle. Calculate the exact measure of the angle (in degrees and minutes) that this platform forms with Avenida Central.
3. c) If later a new street is opened completely parallel to Avenida del Norte that also intersects Avenida Central at another different point, justify what angles that new street will form with Avenida Central without needing to measure them on the ground.

Three villages, Ballina ($A$), Bantry ($B$) and Carrick ($C$), are situated such that the angle with vertex at $A$ formed by the roads going to $B$ and $C$ measures $74^\circ \ 12'$. A mobile phone mast is to be built that is exactly the same distance from road $AB$ as from road $AC$. In addition, the mast must be situated at the same distance from villages $B$ and $C$. Describe with reasoning which geometric construction lines (special lines) you must draw to find the exact location of the mast and state the name of their point of intersection.

Draw an obtuse angle $\widehat{AOB} = 150^\circ$. Draw its bisector $OC$. Afterwards, draw the bisector $OD$ of angle $\widehat{AOC}$. Without using a protractor for the final measurement, calculate arithmetically the measure of the supplementary angle of angle $\widehat{DOB}$. Show all the steps of the calculation in the sexagesimal system.

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

Two 1st Year students, Aoife and Seán, want to find out the measure of the angle formed by the bisectors of any two adjacent supplementary angles, $\hat{A}$ and $\hat{B}$ (which add up to $180^\circ$ and share the vertex and a common side).

• Aoife's method (numerical verification): She chooses a specific case, for example $\hat{A} = 120^\circ$ and $\hat{B} = 60^\circ$. She divides each by two to find their bisectors ($60^\circ$ and $30^\circ$), adds them ($60^\circ + 30^\circ = 90^\circ$), and concludes that they will always form $90^\circ$.

• Seán's method (general deduction): He states that if $\hat{A} + \hat{B} = 180^\circ$, the angle formed by the two bisectors is the sum of their halves: $\frac{\hat{A}}{2} + \frac{\hat{B}}{2} = \frac{\hat{A} + \hat{B}}{2} = \frac{180^\circ}{2} = 90^\circ$.

1. a) Critically compare both approaches: why does Aoife's method serve as a clue but not formally prove that the property holds for all possible pairs of supplementary angles? What advantage does Seán's method provide?
2. b) Apply Seán's general deductive method to calculate and prove what angle the bisectors of two adjacent complementary angles will always form.
3. c) Based on your deduction, reason whether it is possible for the bisectors of two adjacent complementary angles to form an obtuse angle in any case.

At the Maritime Rescue center in Howth (Dublin), controllers track the navigation of several vessels on a flat screen:

• The vessel Alfa follows a straight-line path represented by the line $r$.

• The vessel Bravo follows another straight-line route represented by the line $s$. Both routes are intersecting and cross at a point $P$, forming an acute angle of $41^\circ\ 35'$.

• A rescue boat departs from the intersection $P$ sailing in a straight line following exactly the bisector of the obtuse angle formed by the routes of Alfa and Bravo at that point.

• A mothership Charlie moves along a straight-line path $t$ that is perpendicular to the path of vessel Alfa ($r \perp t$) at a point located several miles from $P$.

1. a) Determine with reasoning the exact measure of the obtuse angle formed by the paths of Alfa and Bravo at point $P$.
2. b) Calculate the exact measure (in sexagesimal degrees, minutes, and seconds) of the angle that the heading of the rescue boat forms with the path of vessel Bravo.
3. c) Prove in a purely geometric way whether the paths of vessel Charlie ($t$) and vessel Bravo ($s$) will intersect at some point in the plane or if there is a possibility that they are parallel. To justify this, deduce what angular condition $r$ and $s$ would have to satisfy for $t$ to be parallel to $s$.