Elements of the Plane

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.

Use a ruler to measure the following line segments and calculate the total sum of their lengths in centimeters ($cm$).

• Segment $AB = 40$ $mm$

• Segment $CD = 2.5$ $cm$

• Segment $EF = 0.5$ $dm$

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

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.

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?

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.

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 $D = 10^\circ$

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.

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)?

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?

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 $110^\circ$.

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.

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?

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.

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.

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?

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.

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 friends, Ana and Carlos, want to install a Wi-Fi repeater antenna in the schoolyard that is exactly at the same distance from both of them. If Ana is at point $A$ and Carlos 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 Carlos is $12$ meters, 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.

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

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.