Exercises: 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.

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

Classify the following angles according to their opening into 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 $D = 10^\circ$

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

1. a) The complementary angle of a $35^\circ$ angle.
2. b) The supplementary angle of a $110^\circ$ angle.

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

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

1. a) What type of angle do the hands form when it is exactly 3:00?
2. b) And when it is exactly 6:00?
3. c) What angle (in degrees) does the minute hand cover in 15 minutes?

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

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

• Line segment $AB = 40$ $mm$

• Line segment $CD = 2,5$ $cm$

• Line segment $EF = 0,5$ $dm$

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

On a hiking map, a straight route joins point $A$ (a shelter) with point $B$ (a waterfall). If we extend the route beyond the waterfall towards the mountain peak (point $C$), identify which geometric element the following sections represent:

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 that contains the shelter, the waterfall and the peak, extending endlessly in both directions.

An architect is designing a new residential area. He has 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 is driving along street $s$ and wants to turn to enter street $t$, what angle (in degrees) will it have to turn?

A speed 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 does it have left to rotate to complete its total sweep? Express the result in degrees, minutes and seconds.

In the design of a skateboard ramp, three internal angles have been calculated which must add up to a straight angle for the structure to be 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 opening (acute, right or obtuse)?

Two angles are supplementary. If one of them measures three times the other, what is the measure of each angle? Use a diagram to help you visualize the situation.

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

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

2. If the distance between Ashley and Brandon is $12$ meters, what is the minimum distance the antenna will be from each of them?

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

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 how much each of the new angles will measure 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 with respect to the correct bisector?

Look at the hands of an analog clock when it shows exactly $15:00$ (three o'clock) and answer:

1. What type of angle do the two hands form?

2. If the clock advances to $15:10$, the minute hand has rotated a certain angle. Knowing that each minute is equivalent to a rotation of $6^\circ$, what angle has the minute hand covered?

3. At the time of $15:10$, 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).

Three towns, Oakridge ($A$), Pineville ($B$) and Cedarville ($C$), are situated so 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 exactly at the same distance from road $AB$ as from road $AC$. In addition, the tower must be situated at the same distance from the towns $B$ and $C$. Describe with reasoning which geometric elements of the plane (notable lines) you must draw to find the exact location of the tower and what name is given to the point of intersection.

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

Calculate exactly the smaller angle formed by the hands of an analog clock when it shows $10:20$ in the morning. Keep in mind that the hour hand moves proportionally to the movement of the minute hand. (Hint: every minute, the hour hand advances $0,5^\circ$).

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

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

Draw an obtuse angle $\widehat{AOB} = 150^\circ$. Draw its angle bisector $OC$. Then, draw the angle bisector $OD$ of the 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 operation in the sexagesimal system.

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?