4.9 Parallel and also perpendicular lines through circles 4.9 Parallel and perpendicular lines through circles

In this chapter, you will discover to attract geometric figures accurately. Girlfriend will likewise explore the properties that various figures have.

You are watching: Shapes with parallel and perpendicular lines

Angles revision


When 2 lines point in various directions, we say they are at an angle to every other. If the direction are practically the same, us say the angle in between them is small. If the directions are very different, we say the angle in between them is big.

Words we use to describe angles:
Arms of the angle: the two lines that room at an edge to each other The vertex: the suggest where the two arms meet Vertices: plural of "vertex"

Arrowheads ~ above the lines median that the lines save on going. The length of one angle"s arms does not adjust the size of the angle. Even if it is the eight are long or short, the edge size continues to be the same.

There room two angles at a vertex so that is important to display which one we room talking about


There are plenty of different methods to brand angles. Look in ~ the examples below:

You have the right to name the edge on the appropriate in various ways: you have the right to say \(A\hatBC\) or \(C\hatBA\) or just \(\hatB\). The "hat" on the letter shows where the edge is.



room these lines in ~ an edge to every other?

perform the lines have actually to fulfill to be at an angle?

usage a pencil and your leader to attract the lines a little bit longer for this reason they meet. Did you readjust the angle in between the lines as soon as you expanded them?

arrange the angles from greatest to smallest. Just write the letters (a) come (f) in the correct order.


How deserve to you check that an angle is a ideal angle there is no using any type of special math equipment? (Hint: Think around where friend can discover right angles around you.)

room these 2 angles the same size? describe how you uncovered your answer. (Hint: A item of scrap paper may help!)


2 lines are attracted by holding down a ruler and also drawing currently on both sides. What have the right to you say about the 2 lines?


Look in ~ the analogue clock face on the right. The minute hand and also the hour hand do an angle. Focus on the smaller sized angle because that now.


explain why the angle in between the hands at 8 o" clock is the very same size together the angle at 4 o" clock.

to compare the edge at 2 o" clock through the angle at 4 o" clock. What carry out you notice? Why is this so?

Is the edge at 3 o" clock the exact same as the edge at a quarter previous 12? Explain.

as soon as you open up the cover of a hardcover publication you deserve to make various angles. Have the right to you think of in ~ least 5 other situations in daily life wherein objects space turned v angles? speak what the arms and the vertices are in every of her examples.

The degree: a unit because that measuring angles

Imagine if us didn"t have actually units because that measuring length.

How would tailors make garments to the right size without a ice cream measure? How could an architect style a safe and beautiful residence without a ruler? How could we lay out a skilled soccer ar without being able to measure accurately in metres?

We require units and measuring tools in plenty of situations. You understand that we usage metres, centimetres, kilometres, millimetres, etc. Because that measuring lengths.

We should also have devices for measure angles. The units we usage for measuring angle are an extremely ancient. No one now is completely sure why, yet our ancestors decided countless thousands of years earlier that a revolution should be separated into 360 equal parts. We call these components degrees. The symbol because that a level is °.

Some acquainted angles in degrees

complete the table by filling in the dimension of every angle described.

Angle (in words)

Angle (degrees)

right angle


straight angle



half a best angle

a third of a best angle

a 4 minutes 1 of a ideal angle


half a right angle

three quarters of a revolution

a 3rd of a revolution

Look at the clock shown. How countless degrees does:

the minute hand move in one hour?

the hour hand relocate in one hour?

In grade 6 girlfriend learnt the angles space classified into types. Finish the table. The very first one has actually been excellent as an example for you.


Size the the angle

Sketch the the angle

Acute angle

Between 0° and 90°


Right angle

Obtuse angle

directly angle

Reflex angle


Comparing angles making use of paper

You need a paper of A4 paper. At the corners you have 4 right angles. Number them and also tear the corners off as displayed in the diagram. Execute not do them too small.

Now use your best angles to inspection the complying with situations:

show that a directly angle is two appropriate angles.

You can sketch what you have actually done

present that a transformation is 4 right angles.

You have the right to sketch what you have done

develop a right angle using 3 of your corners.

You deserve to sketch what you have done

define how you deserve to use among your corners to inspect if an edge is acute, ideal or obtuse.

Fold corner 1 so the you can use that to measure up 45°.

Fold corner 2 so that you deserve to use that to measure up 30°.

Fold corner 3 so that you can measure 22,5°.

which is bigger: a best angle or fifty percent a ideal angle + a third of a appropriate angle + a 4 minutes 1 of a appropriate angle? have the right to you execute a calculation to display that?

Important: keep your folded piece of record for the following lesson!

Using the protractor

We have actually a unique instrument for measuring angles. It is called a protractor. Look in ~ the snapshot of a common protractor v its essential parts labelled.


Protractors have the right to be large or small but they every measure angle in exactly the very same way. The dimension of the protractor provides no distinction to one angle’s size.

Measuring some acquainted angles

You require the 4 folded angle from the ahead activity. If you didn"t carry out that activity, climate go earlier now and follow the indict in concern 5.

In a team of 3 or four, use your protractor to measure up the angle that girlfriend made: 90°; 45°; 30° and 22,5°.

walk you measure up the correct sized angle? If not, climate ask you yourself the complying with questions:

go you placed the peak of the edge at the origin of the protractor? Is the bottom eight of her angle inside wall up v the basic line? Did friend fold your corners correctly?

How to use a protractor to measure an angle

Step 1: space the edge arms long enough?

The edge arms need to be a little longer 보다 the distance from the origin of the protractor come its edge. If they room too short, use a spicy pencil and a leader to do them longer. Be mindful to line the leader up through the arm.


Once her protractor is in the correct ar keep a finger on the protractor to prevent it from moving. If it moves… start again! you are currently ready to make a measurement.

Step 3: measure up the angle

A protractor provides a clockwise degree scale and also an anticlockwise level scale. You select the correct range by detect the one the starts v 0° top top the angle arm. Look at wherein the other angle eight passes under the degree scale. That is wherein your measure up is.


You can likewise place the protractor on the angle utilizing the various other arm. Then the correct position looks favor this:


The edge in the pictures above is 37°. Perform you agree? execute you view that there room two methods to measure up an angle?

Practise measuring through a protractor

measure the angles and also complete the table on the following page. Girlfriend can extend the eight if you need to; that doesn"t matter if castle go throughout text or an additional drawing.














Angle dimension in degrees

measure up all the numbered angle in the adhering to figure. Some angles can be measure up directly, others not. Her protractor cannot measure up reflex angles favor angles 7 and 8. So girlfriend will have to make a plan!















compose a brief note for yourself about measuring reflex angles here:

Some things to think about

Look at your answers in inquiry 2.

just how do angles 3 and 4 compare?

What around angles 6 and 7?

What around angles 4 and also 5?

There space some interesting concepts here. Try to carry out some further investigation and also show your teacher what girlfriend discover.

Using a protractor to construct angles

Constructing angles to a given line

Work in addition to a partner on this activity. You need your protractor, a spicy pencil and a right ruler.

your first difficulty is to build a heat at specifically right angle to the one below. Begin by picking a suggest on the line. You must note this point clearly and nicely with a little dot. Then usage your knowledge of a protractor to attract a 90° angle.


now fill in the lacking words in the steps:

Step 1: pick a suggest anywhere ~ above the line. Make a tiny mark top top the line. (You don"t constantly have a choice here. Sometimes you should use a particular point ~ above the line.)

Step 2: place the protractor v its ______ on the line and its origin specifically on height of the ______

Step 3: make a small, clear note at the ______

Step 4: usage a leader to heat up the two ______ and draw a directly line that passes exactly through them.

construct the angles utilizing the line below. The line below will be one eight of the angle you are going to construct. The vertex because that each that your angles is the point labelled O where the small vertical line cuts the lengthy horizontal one. Her angles must be measure anticlockwise native the line.












Angle direction

The heat you have been given below is dubbed a reference line.

Mathematicians typically measure angles anticlockwise indigenous the reference line.


usage the line below. In ~ each end you require to attract lines in ~ an angle of 60° to form a triangle. What sort of triangle is this?


finish the square below. The angle at P have to be 52° and also the one in ~ Q, 23°.


Parallel and perpendicular lines

Perpendicular lines fulfill each various other at one angleof 90°.

The sketch shows two perpendicular lines.

We say: ab is perpendicular come DC.

We write: ab \(\perp\) DC


Parallel lines never accomplish each other. They room an equal distance apart. They have actually the exact same direction.

The sketch shows two parallel lines.

We say: PQ is parallel to RS.

We write: PQ \(\parallel\) RS

The arrows on the center of the lines present that the lines are parallel to each other.


Constructing perpendicular and parallel lines

When creating parallel lines, remember that the lines constantly stay the exact same distance apart. Monitor the steps below to draw perpendicular and also parallel lines using a protractor and also a ruler.

We desire to attract a line the is parallel to XY and that passes through allude A.


Step 1: attract a perpendicular line between A and XY.

Use her protractor to draw a line that goes v A and is in ~ 90° to XY.Label the allude C whereby your new line touches XY.Look at the sketch below if you acquire stuck.

Step 2: measure up the perpendicular distance in between the allude and the line.

Write under the length of AC:

Step 3: attract a point that is the same distance native the line.

Draw an additional line the is perpendicular to line XY.Mark turn off the same size as AC on the line.The sketch mirrors what you should do.


Step 4: draw the parallel line.

Join A with the brand-new point that is an equal street away native XY.You now have actually a parallel line.


In your exercise book, practise creating perpendicular and also parallel lines making use of a protractor and also a ruler.

Circles are very special figures

And now for something slightly different. . . Allow us have a look at circles.

A circle through string

You might need to work with a partner here. You require two spicy pencils and also a short length that string, an A4 paper of paper and a ruler.

Tie the string to both pencils with twin knots. The knots have to be firm but not tight. The string should swing easily roughly the pencils without falling off. As soon as you have tied your string, the distance in between the pencils when the wire is tight have to not be an ext than 8 cm.

Your partner must hold one pencil vertically with its suggest near the center of the sheet of paper.

Now closely move the pointer of the other pencil around the middle one, drawing as friend go. Try to save the wire stretched and also the pencil vertical together you draw.If you have been careful, you have actually a one (well, hope something nice close come a circle). You can swop currently so her partner also has a turn drawing while you host the centre pencil.

note three clues on the one line. Measure the distance between the point and the center of the circle because that each. If you have a circle girlfriend should find that the distances are the same.

Think about it

Can girlfriend think of any other number where the distance between the centre and the sheet is continuous in every directions?

A square? A hexagon? What about an oval form (ellipse)?

Do some investigation to watch what you deserve to find.

Circles room special for numerous reasons. The most essential reason is the following:

The distance from the center of a circle to the edge is the very same in any kind of direction.

This distance is called the radius. Us pronounce this "ra-dee-us".

The many of radius is radii. We pronounce this:"ray-dee-eye".

Do friend agree that the two pencils and also string room not a good way to attract circles? The wire is stretchy. It is challenging to adjust the radius. And, the drawing pencil deserve to wander turn off course and also make a spiral or a wobbly curve. We require something better.

Using the compass

We need a special instrument for illustration circles. That must have a pointy tip, like the centre pencil. It must likewise have a drawing tip, like the pencil you moved. If girlfriend can set the distance in between these two tips, girlfriend can draw circles of any type of radius. This instrument is dubbed a pair the compasses, or frequently just a compass.


Constructing circles with a compass

at the peak of the next page friend will view a allude labelled A. Follow the actions below and also on the following page to attract a circle through a radius of 2 cm. The centre need to be at A.

action 1: ar the pointed tip on the zero heat of her ruler. Very closely widen the angle between the arms. Move the pencil reminder until it is specifically at 2 cm. Make sure that the pointed tip is tho on zero. Be careful not to readjust the space once it is set to 2 cm.

step 2: Gently push the pointed reminder into suggest A. Push just deep sufficient into the paper to store it in place. This will certainly be the centre of her circle.


Step 3: host the handle in between the forefinger and thumb of your writing hand. Save your various other hand out of the way. Use only one hand once you draw a circle with a compass.

Step 4: twisted the handle in between your thumb and forefinger. If you space right-handed that is simplest to twist the compass clockwise. If you room left-handed, rotate anticlockwise. Let the pencil reminder drag over the paper. Don"t push down too tough on the pencil. Rather, push down lightly on the pointed arm as you draw. The pencil reminder must relocate smoothly and also easily.

attract concentric one at center A above with radii of 3 cm, 4 cm, 5 cm and 6 cm. Collection the gap very closely each time. Create the radius ~ above the leaf of each circle.

Concentric circles have actually the exact same midpoint.

Learning to usage a compass is like learning to drive a bicycle. It takes co-ordination and practice. Don"t it is in embarrassed if that goes wrong. With practice you will acquire very an excellent at it. If her circles end up being all wobbly lines, just begin again!

Here are some tips for drawing circles:
If your circles are turning into spirals that is since the arms of your compass have moved. Check their width again versus a ruler. If the eight of her compass won"t remain in the place you collection them at, it is due to the fact that the nut in ~ the hinge below the handle is loose. Ask her teacher to aid you if you can"t tighten the yourself. If friend can"t do the twist, imagine you have actually a small piece the soft clay in between your thumb and forefinger and you room trying to roll it into a little strip. The twisted for transforming your compass provides the same kind of sliding movement. Allow the compass cave from her hand in the air and also twist the handle. Then try it ~ above scrap paper a couple of times until you can turn the compass easily.

Circles ~ above circles

It"s time to have some fun with the compass when getting much better at making use of it. Follow this instructions to draw the beautiful pattern displayed on the right in your practice book.

Make sure your pencil is sharp; then location it in the compass.

set the radius to 4 cm. Attract a circle in ~ the centre of her page. Important: your radius should stay the exact same for the totality activity.

placed your compass point anywhere on the one edge. Draw one more circle. This circle must pass v the center of your very first circle (they have the very same radius).

Your 2nd circle cut the very first circle at two points. Pick one of this points. Place your compass suggest at this point. Draw an additional circle the radius 4 cm.

Repeat step 3 through your 3rd circle, 4th circle etc. Friend should end up with six circles ~ above your first circle. The is, 7 circles in total.

Decorate it together you please. (You deserve to decorate your pattern additional by adding an ext circles or joining points with straight lines, and also so on. See what patterns and also shapes you deserve to discover among all the circles.)

Using circles to draw other figures

Geometric figures hiding in the circles

Below is a set of seven circles choose the one you drew. Sit through a companion and shot to find hidden polygons.

You will uncover these polygon by authorized the points where the circles reduced each other. The points will certainly be the vertices the the polygons. Watch carefully. There are triangles, quadrilaterals, pentagons and hexagons. Once you can see them, neatly and also carefully dominion in your sides v a pencil. If over there is no enough an are on the set of circles below, redraw the circles on a separate item of record and display the numbers there. If girlfriend wish, you have the right to measure the angles at each vertex and also the lengths that the sides.


Arcs of circles

We carry out not have to draw whole one to construct figures. We are just really interested in the points whereby the circles cross each other, so we can just attract arcs whereby they cross. Following year, friend will use arcs in her geometric constructions.

An arc is a small component of a circle. We use the hatchet circumference once we describe the distance approximately a circle or around any kind of other bent shape.

figure 1 (on the right) has actually been done as an example.



Name the figure



Regular hexagon

6-sided figure. Every the sides are equal. Every the internal angles are equal.






Construct some much more figures

review the indict carefully and also follow them exactly.

draw a heat in your practice book. The line need to be in between 3 and also 6 centimeter long. Draw it in the center of her page.

brand the end A and also B.

ar the suggest of your compass at suggest A. Carefully set the radius of your compass come the distance between A and also B.

attract a circle with the compass allude at A.

Draw another circle through the compass allude at B without transforming the radius width.

The circles cross at two points. Choose one of this points. Label it C. Check that you space on the best track by compare your sketch to the one on the right.

closely rule the currently AC and also BC.

What sort of figure is ABC? examine this by measure angles. Why execute you think this happened?

draw two currently PQ and QR in your exercise book.

The lines satisfy and kind an edge at Q. You can make her angle any kind of size. Make your line lengths different. carry out not make her lines much longer than 6 cm each.

ar your compass allude at point Q. Set the radius of her compass come the distance QP. Ar the compass suggest at R. Attract a circle.


ar the compass point earlier at Q. Collection the radius come the size QR. Ar the compass point at P. Draw a circle.

The 2 circles cross at 2 points. Decide which suggest will be the peak of a parallelogram. Speak to this point S.


sign up with the currently SP and SR. Is PQRS a parallelogram?

Something to think about

Why walk this method kind a parallelogram?

Parallel and perpendicular lines through circles

Parallel and perpendicular

Revision: complete these definitions.

as soon as one heat is parallel to another line, the present

as soon as one line is perpendicular to one more line, the lines

A seven-circle number has been drawn below. The intersection points have been marked. A line segment has been drawn in. Use a ruler and also pencil to sign up with pairs of points so the the present are:

parallel to the heat segment

perpendicular to the heat segment.


When 2 lines (or arcs) cross each various other we say they intersect.

The intersection point is the ar where they meet.

You need to have attracted 7 currently (2 parallel and also 5 perpendicular come the line segment).Compare your lines v a friend"s lines. Do you agree?

In your exercise book, draw a few circles through the exact same radius follow me a line. Begin by illustration a line. Then use your compass to attract a circle through the midpoint top top the line.


Keep your compass the very same width and draw an additional circle with the centre whereby the very first circle crossed the line. Repeat as numerous times together you wish. In the instance at the bottom the the ahead page only three circles have actually been drawn.

have the right to you uncover that example in the seven-circle figure? Look closely until you check out it.

can you watch where you can construct currently that are perpendicular come the offered line? attract them carefully with a pencil and your ruler.

can you watch the two lines that room parallel come the given line? attract them in too.

use circles to build a line the is perpendicular come the heat below.


use circles to construct a line the is parallel come the heat below.



collection your compass at a certain distance, for instance 3 cm, and also investigate point out that are the very same distance native a solved point, P.

usage your compass and investigate every the clues that space at the same distance, for instance 3 cm, native two fixed points, A and B.


Which shapes on the opposite web page are circles?

Which forms on the opposite page are triangles?

Which forms on the opposite page are quadrilaterals?

Use your leader to do the following:

do a illustration of one triangle with 3 acute angles, and also another triangle v one obtuse angle.

attract a quadrilateral v two obtuse angles.

deserve to you attract a triangle through two obtuse angles?

attract a triangle v one ideal angle, and a triangle without any kind of right angles.

deserve to you attract a triangle v two right angles?

deserve to you draw a quadrilateral with four right angles?

These four lines kind quadrilateral ABCD.


The 2 red sides, BC and also AD, are dubbed opposite sides of quadrilateral ABCD.

Which various other two political parties of ABCD are also opposite sides?

The lines DA and abdominal muscle in the number in question 7 are called adjacent sides. They satisfy at a point that is just one of the vertices (corner points) that the quadrilateral.

Name another two adjacent sides in ABCD.

abdominal muscle is nearby to DA in the square ABCD. Which other side the ABCD is also nearby to DA?

wilhelm says:

"Each next of a quadrilateral has actually two adjacent sides.

every side of a quadrilateral likewise has two opposite sides."

Is wilhelm correct? give reasons for her answer.

William additionally says:

"In a triangle, every side is surrounding to all the other sides."

Is this true? give a factor for her answer.

In every case, say even if it is the two sides space opposite sides or surrounding sides that the square PQRS.

QP and also PS

QP and also SR

PQ and also RQ

PS and QR

SR and QR

Different species of triangles

Equilateral, isosceles, scalene and right-angled triangles

A triangle v two equal sides is referred to as an isosceles triangle.

A triangle with three equal political parties is dubbed an equilateral triangle.

A triangle v a best angle is called a right-angled triangle.

A triangle with 3 sides with different lengths and also no best angle is referred to as a scalene triangle.


measure every edge in every of the isosceles triangles provided above. Do you notice anything special? If you are not sure, draw much more isosceles triangles in your exercise book.

measure the angles and sides of the adhering to triangles. What is special about these triangles? In other words, what renders these triangles different to various other triangles?


These triangles are referred to as equilateral triangles.

measure each edge in every of the complying with triangles. Perform you notice anything special about these angles?


Identify the longest next in every of the triangles. If you are not sure which one is the longest side, measure up the sides. What carry out you notice about the longest side in each of this triangles?

These triangle are dubbed right-angled triangles.

Comparing and describing triangles

When 2 or an ext sides the a form are same in length, we show this using short lines on the same sides.

usage the following triangles come answer the questions that follow:



i beg your pardon triangle has actually only two sides that space equal?

What is this kind of triangle called?

i beg your pardon triangle has actually all three sides equal?

What is this kind of triangle called?

which triangle has actually an angle same to 90°?

What is this type of triangle called?

compose down the type of each of the complying with triangles in the an are provided:

Is this triangle scalene, isosceles, or equilateral?

name the 2 sides the the triangle that space equal.

What is the size of JK?

Name 2 equal angles in this triangle.

What is the size of \(\hatJ\) and also \(\hatL\) ?

Different species of quadrilaterals

Investigating quadrilaterals

The 2 pages that follow show various groups the quadrilaterals.

In which teams are both bag of opposite sides parallel?

In which teams are just some surrounding sides equal?

In which groups are all 4 angles equal?

In which teams are every the sides in each quadrilateral equal?

In which groups are all 4 sides equal?

In which teams is each side perpendicular to the sides nearby to it?

In which teams are opposite political parties equal?

In which teams is at the very least one pair of nearby sides equal?

In which groups is at the very least one pair the opposite political parties parallel?

In which groups are every the angles right angles?

The figures in group 1 are referred to as parallelograms.

What execute you observe about the opposite political parties of parallelograms?

What carry out you observe around the angles of parallelograms?

The numbers in team 2 are called kites.

What perform you observe around the sides of kites?

What else execute you observe around the kites?

Group 1


Group 2


Group 3


Group 4


Group 5


Group 6


The numbers in group 3 are called rhombi.

What carry out you observe around the sides of rhombi?

What else perform you observe about the rhombi?

Note: One rhombus; two or an ext rhombi.

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The numbers in team 4 are dubbed rectangles.

What perform you observe about the opposite sides of rectangles?

What do you observe about the angles of rectangles?

What execute you observe around the surrounding sides that rectangles?

The figures in team 5 are called trapeziums. What do you observe around the opposite political parties of trapeziums?