In this chapter, you will learn how to construct, or describe, different lines, angles and shapes. You lot will utilise drawing instruments, such as a ruler, to depict straight lines, a protractor to measure and describe angles, and a compass to draw arcs that are a sure distance from a bespeak. Through the various constructions, you will investigate some of the properties of triangles and quadrilaterals; in other words, y'all will find out more about what is always truthful almost all or certain types of triangles and quadrilaterals.

Bisecting lines

When we construct, or draw, geometric figures, we often need to bifurcate lines or angles.Bisect means to cut something into two equal parts. There are different ways to bifurcate a line segment.

Bisecting a line segment with a ruler

  1. Read through the following steps.

    Step i: Draw line segment AB and decide its midpoint.

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    Pace two: Draw any line segment through the midpoint.

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    The small-scale marks on AF and FB show that AF and FB are equal.

CD is chosen a bisector considering it bisects AB. AF = FB.

  1. Use a ruler to draw and bisect the following line segments: AB = 6 cm and XY = 7 cm.

In Grade half dozen, you lot learnt how to use a compass to draw circles, and parts of circles called arcs. We can use arcs to bisect a line segment.

Bisecting a line segment with a compass and ruler

  1. Read through the following steps.

    Step one

    Place the compass on one endpoint of the line segment (point A). Draw an arc above and below the line. (Notice that all the points on the arc aboveand beneath the line are the aforementioned altitude from signal A.)

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    Step two

    Without changing the compass width, place the compass on betoken B. Describe an arc above and beneath the line so that the arcs cross the beginning two. (The two points where the arcs cross are the same distance away from point A and from point B.)

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    Step three

    Utilise a ruler to join the points where the arcs intersect .This line segment (CD) is the bisector of AB.

    58860.png

    Intersect means to cantankerous or run into.

    A perpendicular is a line that meets another line at an angle of 90°.

Notice that CD is also perpendicular to AB. So it is also called a perpendicular bisector.

  1. Work in your exercise book. Apply a compass and a ruler to practise drawing perpendicular bisectors on line segments.

    Endeavour this!

    Work in your exercise book. Use only a protractor and ruler to describe a perpendicular bisector on a line segment. (Recollect that we use a protractor to measure angles.)

Amalgam perpendicular lines

A perpendicular line from a given betoken

  1. Read through the following steps.

    Footstep one

    Identify your compass on the given point (point P). Draw an arc beyond the line on each side of the given point. Exercise not conform the compass width when drawing the second arc.

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    Step 2

    From each arc on the line, draw another arc on the opposite side of the line from the given bespeak (P). The two new arcs will intersect.

    58905.png

    Stride 3

    Use your ruler to join the given indicate (P) to the point where the arcs intersect (Q).

    58928.png

    PQ is perpendicular to AB. We also write it like this: PQ ⊥ AB.

  2. Use your compass and ruler to draw a perpendicular line from each given signal to the line segment:

    58974.png

    58966.png

A perpendicular line at a given point on a line

  1. Read through the following steps.

    Step 1

    Place your compass on the given point (P). Draw an arc beyond the line on each side of the given point. Exercise non adjust the compass width when cartoon the second arc.

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    Pace 2

    Open your compass so that it is wider than the altitude from 1 of the arcs to the point P. Place the compass on each arc and describe an arc to a higher place or beneath the indicate P. The ii new arcs will intersect.

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    Footstep three

    Utilize your ruler to join the given point (P) and the point where the arcs intersect (Q).

    PQ ⊥ AB

    59055.png

  2. Utilize your compass and ruler to draw a perpendicular at the given point on each line:

    85742.png

Bisecting angles

Angles are formed when whatever two lines meet. We use degrees (°) to measure angles.

Measuring and classifying angles

In the figures below, each angle has a number from 1 to 9.

  1. Utilize a protractor to measure the sizes of all the angles in each figure. Write your answers on each figure.

    1. 59104.png

    2. 59104.png

  2. Employ your answers to fill in the angle sizes below.

    \(\hat{1} = \text{_______} ^{\circ}\)

    \(\lid{ane} + \hat{2} = \text{_______} ^{\circ}\)

    \(\hat{ane} + \lid{four} = \text{_______} ^{\circ}\)

    \(\hat{2} + \hat{3} = \text{_______} ^{\circ}\)

    \(\hat{3} + \hat{iv} = \text{_______} ^{\circ}\)

    \(\lid{1} + \chapeau{2} + \hat{4} = \text{_______} ^{\circ}\)

    \(\lid{1} + \lid{2} + \chapeau{3} + \hat{4} = \text{_______} ^{\circ}\)

    \(\chapeau{6} = \text{_______} ^{\circ}\)

    \(\lid{7} + \lid{eight} = \text{_______} ^{\circ}\)

    \(\hat{half-dozen} + \hat{7} + \hat{8} = \text{_______} ^{\circ}\)

    \(\hat{5} + \lid{6} + \chapeau{7} = \text{_______} ^{\circ}\)

    \(\lid{half-dozen} + \hat{v} = \text{_______} ^{\circ}\)

    \(\hat{5} + \chapeau{half-dozen} + \hat{7} + \hat{viii} = \text{_______} ^{\circ}\)

    \(\chapeau{v} + \hat{6} + \lid{7} + \chapeau{8} + \hat{9} = \text{_______} ^{\circ}\)

  3. Side by side to each answer higher up, write downward what type of angle it is, namely acute, obtuse, correct, directly, reflex or a revolution.

Bisecting angles without a protractor

  1. Read through the following steps.

    Step ane

    Place the compass on the vertex of the angle (point B). Draw an arc across each arm of the angle.

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    Stride 2

    Place the compass on the bespeak where i arc crosses an arm and describe an arc inside the bending. Without changing the compass width, repeat for the other arm so that the two arcs cantankerous.

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    Pace 3

    Use a ruler to join the vertex to the point where the arcs intersect (D).

    DB is the bisector of \(\hat{ABC}\).

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  2. Use your compass and ruler to bisect the angles below.

    85788.png

    You could measure each of the angles with a protractor to check if you have bisected the given angle correctly.

Constructing special angles without a protractor

Constructing angles of and

  1. Read through the post-obit steps.

    Step ane

    Draw a line segment (JK). With the compass on point J, depict an arc beyond JK and up over above betoken J.

    60355.png

    Stride 2

    Without changing the compass width, motility the compass to the point where the arc crosses JK, and describe an arc that crosses the first one.

    60310.png

    Stride 3

    Join indicate J to the point where the two arcs meet (signal P). \(\lid{PJK}\) = threescore°

    60318.png

    When y'all learn more than near the properties of triangles afterwards, you will understand whythe method in a higher place creates a 60° angle. Or can you lot already work this out now? (Hint: What exercise you know near equilateral triangles?)

    1. Construct an angle of 60° at signal B below.
    2. Bisect the bending you synthetic.
    3. Exercise you observe that the bisected angle consists of two 30° angles?
    4. Extend line segment BC to A. Then measure the angle next to the 60° angle.

      Adjacent ways "next to".

      What is its size?

    5. The 60° angle and its side by side angle add together up to

    85772.png

Constructing angles of and

  1. Construct an angle of 90° at point A. Go back to section x.2 if you lot need help.
  2. Bisect the xc° angle, to create an angle of 45°. Go back to department x.3 if you need help.

    85825.png

Challenge

Work in your do volume. Try to construct the following angles without using a protractor: 150°, 210° and 135°.

Constructing triangles

In this department, you will learn how to construct triangles. Yous volition need a pencil, a protractor, a ruler and a compass.

A triangle has three sides and three angles. Nosotros can construct a triangle when we know some of its measurements, that is, its sides, its angles, or some of its sides and angles.

Constructing triangles

Amalgam triangles when iii sides are given

  1. Read through the following steps. They depict how to construct \( \triangle ABC\) with side lengths of 3 cm, v cm and vii cm.

    Pace i

    Describe one side of the triangle using a ruler. It is often easier to start with the longest side.

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    Pace ii

    Set the compass width to 5 cm. Describe an arc 5 cm away from point A. The 3rd vertex of the triangle will be somewhere along this arc.

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    Pace 3

    Set the compass width to 3 cm. Depict an arc from point B. Annotation where this arc crosses the first arc. This will be the tertiary vertex of the triangle.

    60588.png

    Step 4

    Apply your ruler to join points A and B to the point where the arcs intersect (C).

    60598.png

  2. Work in your exercise book. Follow the steps above to construct the following triangles:
    1. \( \triangle ABC\) with sides six cm, 7 cm and iv cm
    2. \(\triangle KLM\) with sides 10 cm, 5 cm and eight cm
    3. \(\triangle PQR\) with sides 5 cm, 9 cm and 11 cm

Constructing triangles when certain angles and sides are given

  1. Use the rough sketches in (a) to (c) below to construct accurate triangles, using a ruler, compass and protractor. Do the construction side by side to each crude sketch.
    • The dotted lines show where yous accept to use a compass to measure the length of a side.
    • Use a protractor to measure the size of the given angles.
    1. Construct \( \triangle ABC\), with two angles and ane side given.

      60766.png

    2. Construct a \(\triangle KLM\), with 2 sides and an bending given.

      60789.png

    3. Construct correct-angled \(\triangle PQR\), with thehypotenuse and ane other side given.

      60804.png

  2. Measure out the missing angles and sides of each triangle in 3(a) to (c) on the previous page. Write the measurements at your completed constructions.
  3. Compare each of your synthetic triangles in 3(a) to (c) with a classmate'due south triangles. Are the triangles exactly the same?

If triangles are exactly the same, we say they are coinciding.

Challenge

  1. Construct these triangles:
    1. \( {\triangle}\text{STU}\), with three angles given: \(S = 45^{\circ}\), \(T = 70^{\circ}\) and \(U = 65^{\circ}\) .
    2. \( {\triangle}\text{XYZ}\), with 2 sides and the bending opposite 1 of the sides given: \(10 = 50^{\circ}\) , \(XY = 8 \text{ cm}\) and \(XZ = seven \text{ cm}\).
  2. Tin you find more than 1 solution for each triangle above? Explain your findings to a classmate.

Properties of triangles

The angles of a triangle can be the same size or different sizes. The sides of a triangle can be the same length or different lengths.

Properties of equilateral triangles

    1. Construct \( \triangle ABC\) side by side to its rough sketch beneath.
    2. Measure and label the sizes of all its sides and angles.

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  2. Measure and write downward the sizes of the sides and angles of \({\triangle}DEF\) below.

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  3. Both triangles in questions 1 and ii are called equilateral triangles. Discuss with a classmate if the following is true for an equilateral triangle:
    • All the sides are equal.
    • All the angles are equal to 60°.

Properties of isosceles triangles

    1. Construct \({\triangle}\text{DEF}\) with \(EF = 7 \text{cm}, ~\hat{Eastward} = 50^{\circ} \) and \(\hat{F} = 50^{\circ}\).

      Also construct \({\triangle}\text{JKL}\) with \(JK = 6 \text{cm},~KL = 6 \text{cm}\) and \(\hat{J}=70^{\circ}\).

    2. Measure and label all the sides and angles of each triangle.
  2. Both triangles above are called isosceles triangles. Discuss with a classmate whether the following is true for an isosceles triangle:
    • Only 2 sides are equal.
    • Only 2 angles are equal.
    • The 2 equal angles are opposite the two equal sides.

The sum of the angles in a triangle

  1. Look at your constructed triangles \({\triangle}\text{ABC},~{\triangle}\text{DEF} \) and \({\triangle}\text{JKL}\) above and on the previous page. What is the sum of the 3 angles each time?
  2. Did you find that the sum of the interior angles of each triangle is 180°? Practice the post-obit to check if this is true for other triangles.
    1. On a clean sheet of paper, construct whatsoever triangle. Label the angles A, B and C and cutting out the triangle.

      61190.png

    2. Neatly tear the angles off the triangle and fit them next to ane another.
    3. Observe that \(\hat{A} + \hat{B} + \hat{C} = \text{______}^{\circ}\)

We can conclude that the interior angles of a triangle always add up to 180°.

Properties of quadrilaterals

A quadrilateral is any airtight shape with four straight sides. We classify quadrilaterals according to their sides and angles. We annotation which sides are parallel, perpendicular or equal. We as well note which angles are equal.

Properties of quadrilaterals

  1. Measure and write down the sizes of all the angles and the lengths of all the sides of each quadrilateral beneath.

    Foursquare

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    Rectangle

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    Parallelogram

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    Rhombus

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    Trapezium

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    Kite

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  2. Utilize your answers in question i. Place a âœÂ" in the correct box below to show which belongings is correct for each shape.

    Properties

    Parallelogram

    Rectangle

    Rhombus

    Foursquare

    Kite

    Trapezium

    But one pair of sides are parallel

    Contrary sides are parallel

    Contrary sides are equal

    All sides are equal

    Ii pairs of side by side sides are equal

    Reverse angles are equal

    All angles are equal

Sum of the angles in a quadrilateral

  1. Add upward the four angles of each quadrilateral on the previous page. What do you lot notice almost the sum of the angles of each quadrilateral?
  2. Did you notice that the sum of the interior angles of each quadrilateral equals 360°? Do the following to check if this is true for other quadrilaterals.
    1. On a clean canvas of paper, use a ruler to construct any quadrilateral.
    2. Label the angles A, B, C and D. Cut out the quadrilateral.
    3. Neatly tear the angles off the quadrilateral and fit them side by side to one another.
    4. What do you lot notice?

We can conclude that the interior angles of a quadrilateral always add up to 360°.

Amalgam quadrilaterals

Yous learnt how to construct perpendicular lines in department x.2. If you know how to construct parallel lines, you lot should be able to construct any quadrilateral accurately.

Constructing parallel lines to draw quadrilaterals

  1. Read through the following steps.

    Step 1

    From line segment AB, marking a point D. This point D will be on the line that will exist parallel to AB. Draw a line from A through D.

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    Pace 2

    Draw an arc from A that crosses AD and AB. Go along the same compass width and draw an arc from bespeak D as shown.

    62185.png

    Step 3

    Ready the compass width to the distance between the ii points where the first arc crosses Ad and AB. From the point where the second arc crosses Advertizing, draw a third arc to cantankerous the second arc.

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    Footstep iv

    Draw a line from D through the point where the two arcs meet. DC is parallel to AB.

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  2. Practise drawing a parallelogram, square and rhombus in your practise volume.
  3. Use a protractor to effort to depict quadrilaterals with at least one set of parallel lines.
  1. Do the following construction in your exercise volume.
    1. Apply a compass and ruler to construct equilateral \( \triangle ABC\) with sides ix cm.
    2. Without using a protractor, bisect \(\hat{B}\). Permit the bisector intersect AC at point D.
    3. Use a protractor to measure out \(\chapeau{ADB}\). Write the measurement on the drawing.
  2. Name the following types of triangles and quadrilaterals.

    1. 67943.png

    2. 67928.png

    3. 67936.png

    4. 68048.png

    5. 68027.png

    6. 68041.png

  3. Which of the following quadrilaterals matches each description below? (There may be more i respond for each.)

    parallelogram; rectangle; rhombus; foursquare; kite; trapezium

    1. All sides are equal and all angles are equal.
    2. Two pairs of adjacent sides are equal.
    3. One pair of sides is parallel.
    4. Opposite sides are parallel.
    5. Opposite sides are parallel and all angles are equal.
    6. All sides are equal.