Elements of the plane
The lines on a football pitch, the streets of your neighbourhood on a map, the corner of a window... all of these are drawings made from the most basic pieces of geometry: points, lines and angles. Before classifying triangles or calculating areas, you need to know these pieces well.
Point, line, ray and line segment
Smaller pieces can be cut from a line:
The school playground, seen from above, is a piece of plane: a surface that extends in all directions and contains all the points, lines and line segments you draw on it.
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…
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…
The following segments have these lengths:
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Segment $AB = 40$ $mm$
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Segment $CD = 2,5$ $cm$
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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.
Positions of two lines
Two streets in your neighbourhood either cross at some point or never cross at all.
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…
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?
The angle: elements and measurement
An angle is named with the vertex in the middle: $\widehat{AOB}$, or with a Greek letter ($\alpha$, $\beta$...). To measure how wide it opens, we use a protractor, placing its centre on the vertex and its $0°$ mark on one of the sides.
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…
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.
Degrees, minutes and seconds
A complete turn is divided into $360$ equal parts called degrees ($°$). But sometimes a degree is not precise enough, so it is divided in turn, just as an hour is divided into minutes and seconds:
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…
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.
Complementary and supplementary angles
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: a) The complement of an…
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:
- a) The complement of an angle of $35^\circ$.
- 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…
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…
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.
Vertically opposite angles and the bisector
When two lines cross, four angles are formed. The ones facing each other, without touching, are vertically opposite and are equal in size.
The bisector of an angle is the ray that starts at the vertex and divides the angle into two equal angles.
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…
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…
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 $135^\circ$.
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Calculate the measure of each of the new angles after drawing the bisector.
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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…
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.