In this chapter, yous will larn how to construct, or draw, dissimilar lines, angles and shapes. You will utilize drawing instruments, such as a ruler, to draw straight lines, a protractor to measure out and describe angles, and a compass to draw arcs that are a sure distance from a point. Through the various constructions, you will investigate some of the properties of triangles and quadrilaterals; in other words, you will observe out more about what is always true about all or certain types of triangles and quadrilaterals.

Bisecting lines

When nosotros construct, or describe, geometric figures, we oft demand to bisect lines or angles.Bisect means to cut something into two equal parts. There are unlike ways to bisect a line segment.

Bisecting a line segment with a ruler

  1. Read through the following steps.

    Step 1: Draw line segment AB and decide its midpoint.

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    Step 2: Draw any line segment through the midpoint.

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

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

  1. Use a ruler to draw and bifurcate the post-obit line segments: AB = 6 cm and XY = seven cm.

In Grade 6, you lot learnt how to use a compass to draw circles, and parts of circles called arcs. Nosotros can use arcs to bisect a line segment.

Bisecting a line segment with a compass and ruler

  1. Read through the following steps.

    Pace i

    Identify the compass on i endpoint of the line segment (point A). Depict an arc higher up and beneath the line. (Detect that all the points on the arc higher upand beneath the line are the aforementioned distance from point A.)

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    Step 2

    Without changing the compass width, identify the compass on point B. Draw an arc higher up and below the line so that the arcs cross the starting time 2. (The two points where the arcs cross are the same altitude away from point A and from point B.)

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    Step 3

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

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    Intersect means to cross or meet.

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

Find that CD is also perpendicular to AB. Then 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.

    Try this!

    Work in your exercise book. Use merely a protractor and ruler to draw a perpendicular bisector on a line segment. (Think that nosotros use a protractor to measure angles.)

Amalgam perpendicular lines

A perpendicular line from a given point

  1. Read through the following steps.

    Stride 1

    Identify your compass on the given point (point P). Describe an arc across the line on each side of the given point. Practise not adjust the compass width when cartoon the second arc.

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    Step 2

    From each arc on the line, draw another arc on the contrary side of the line from the given point (P). The two new arcs volition intersect.

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    Step 3

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

    58928.png

    PQ is perpendicular to AB. Nosotros also write it similar this: PQ ⊥ AB.

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

    58974.png

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A perpendicular line at a given betoken on a line

  1. Read through the following steps.

    Step one

    Place your compass on the given bespeak (P). Draw an arc across the line on each side of the given bespeak. Exercise not adjust the compass width when drawing the second arc.

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    Stride two

    Open your compass so that it is wider than the distance from 1 of the arcs to the point P. Place the compass on each arc and depict an arc above or below the point P. The two new arcs volition intersect.

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    Step 3

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

    PQ ⊥ AB

    59055.png

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

    85742.png

Bisecting angles

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

Measuring and classifying angles

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

  1. Use 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. Utilize your answers to fill up in the bending sizes below.

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

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

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

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

    \(\hat{3} + \hat{4} = \text{_______} ^{\circ}\)

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

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

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

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

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

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

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

    \(\hat{5} + \hat{6} + \lid{7} + \hat{8} = \text{_______} ^{\circ}\)

    \(\hat{5} + \hat{vi} + \lid{7} + \lid{eight} + \chapeau{9} = \text{_______} ^{\circ}\)

  3. Next to each respond above, write downward what blazon of bending it is, namely acute, obtuse, right, straight, reflex or a revolution.

Bisecting angles without a protractor

  1. Read through the following steps.

    Step 1

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

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    Step 2

    Identify the compass on the point where i arc crosses an arm and draw an arc inside the angle. Without changing the compass width, repeat for the other arm so that the two arcs cross.

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    Step 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 beneath.

    85788.png

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

Constructing special angles without a protractor

Constructing angles of and

  1. Read through the following steps.

    Pace ane

    Draw a line segment (JK). With the compass on point J, describe an arc beyond JK and upwards over above indicate J.

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    Step 2

    Without changing the compass width, move the compass to the point where the arc crosses JK, and draw an arc that crosses the offset ane.

    60310.png

    Step 3

    Join betoken J to the point where the ii arcs come across (indicate P). \(\hat{PJK}\) = lx°

    60318.png

    When you lot learn more than most the backdrop of triangles afterward, you lot volition understand whythe method above creates a threescore° angle. Or can you already piece of work this out now? (Hint: What do you know about equilateral triangles?)

    1. Construct an angle of lx° at point B below.
    2. Bisect the angle you lot constructed.
    3. Do you find that the bisected angle consists of two thirty° angles?
    4. Extend line segment BC to A. Then measure the angle adjacent to the 60° angle.

      Adjacent means "side by side to".

      What is its size?

    5. The 60° angle and its next angle add together upward to

    85772.png

Constructing angles of and

  1. Construct an bending of 90° at betoken A. Go back to section 10.ii if yous need help.
  2. Bisect the xc° angle, to create an bending of 45°. Get back to section x.three if you demand assist.

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Challenge

Work in your exercise book. Endeavor to construct the post-obit angles without using a protractor: 150°, 210° and 135°.

Constructing triangles

In this section, you lot will acquire how to construct triangles. You will need a pencil, a protractor, a ruler and a compass.

A triangle has three sides and 3 angles. We tin 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

Constructing triangles when three sides are given

  1. Read through the post-obit steps. They depict how to construct \( \triangle ABC\) with side lengths of iii cm, 5 cm and seven cm.

    Step 1

    Draw one side of the triangle using a ruler. It is often easier to get-go with the longest side.

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    Footstep 2

    Prepare the compass width to 5 cm. Draw an arc v cm away from indicate A. The third vertex of the triangle will be somewhere along this arc.

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    Step 3

    Set the compass width to 3 cm. Draw an arc from point B. Annotation where this arc crosses the offset arc. This will be the third vertex of the triangle.

    60588.png

    Footstep 4

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

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  2. Piece of work in your exercise volume. Follow the steps above to construct the following triangles:
    1. \( \triangle ABC\) with sides 6 cm, 7 cm and 4 cm
    2. \(\triangle KLM\) with sides 10 cm, 5 cm and viii cm
    3. \(\triangle PQR\) with sides 5 cm, ix cm and 11 cm

Constructing triangles when certain angles and sides are given

  1. Use the rough sketches in (a) to (c) beneath to construct accurate triangles, using a ruler, compass and protractor. Exercise the structure side by side to each rough sketch.
    • The dotted lines show where you take to use a compass to measure out the length of a side.
    • Use a protractor to measure out the size of the given angles.
    1. Construct \( \triangle ABC\), with two angles and one side given.

      60766.png

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

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

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  2. Measure 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 iii(a) to (c) with a classmate's triangles. Are the triangles exactly the aforementioned?

If triangles are exactly the aforementioned, we say they are congruent.

Challenge

  1. Construct these triangles:
    1. \( {\triangle}\text{STU}\), with iii angles given: \(Due south = 45^{\circ}\), \(T = lxx^{\circ}\) and \(U = 65^{\circ}\) .
    2. \( {\triangle}\text{XYZ}\), with two sides and the angle opposite ane of the sides given: \(X = fifty^{\circ}\) , \(XY = 8 \text{ cm}\) and \(XZ = 7 \text{ cm}\).
  2. Can you find more than ane solution for each triangle higher up? 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 exist the aforementioned length or different lengths.

Properties of equilateral triangles

    1. Construct \( \triangle ABC\) adjacent to its rough sketch below.
    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 2 are called equilateral triangles. Talk over with a classmate if the following is true for an equilateral triangle:
    • All the sides are equal.
    • All the angles are equal to sixty°.

Properties of isosceles triangles

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

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

    2. Measure and characterization all the sides and angles of each triangle.
  2. Both triangles to a higher place are chosen isosceles triangles. Discuss with a classmate whether the following is true for an isosceles triangle:
    • Only two sides are equal.
    • Only 2 angles are equal.
    • The ii 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 \({\triangle}\text{JKL}\) higher up and on the previous page. What is the sum of the three angles each fourth dimension?
  2. Did yous find that the sum of the interior angles of each triangle is 180°? Do the following to check if this is true for other triangles.
    1. On a clean sheet of paper, construct any triangle. Characterization the angles A, B and C and cut out the triangle.

      61190.png

    2. Neatly tear the angles off the triangle and fit them next to one some other.
    3. Detect that \(\hat{A} + \lid{B} + \hat{C} = \text{______}^{\circ}\)

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

Backdrop of quadrilaterals

A quadrilateral is whatever closed shape with four direct sides. We allocate quadrilaterals according to their sides and angles. We notation which sides are parallel, perpendicular or equal. We likewise note which angles are equal.

Properties of quadrilaterals

  1. Measure and write downwards the sizes of all the angles and the lengths of all the sides of each quadrilateral below.

    Foursquare

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    Rectangle

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    Parallelogram

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    Rhomb

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    Trapezium

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    Kite

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  2. Use your answers in question 1. Identify a âœÂ" in the correct box beneath to prove which property is correct for each shape.

    Properties

    Parallelogram

    Rectangle

    Rhombus

    Foursquare

    Kite

    Trapezium

    Simply i pair of sides are parallel

    Reverse sides are parallel

    Reverse sides are equal

    All sides are equal

    Two pairs of adjacent sides are equal

    Opposite angles are equal

    All angles are equal

Sum of the angles in a quadrilateral

  1. Add up the 4 angles of each quadrilateral on the previous page. What exercise you notice about 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 bank check if this is true for other quadrilaterals.
    1. On a make clean sheet of paper, utilize a ruler to construct any quadrilateral.
    2. Label the angles A, B, C and D. Cutting out the quadrilateral.
    3. Neatly tear the angles off the quadrilateral and fit them adjacent to one some other.
    4. What practise you notice?

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

Amalgam quadrilaterals

You learnt how to construct perpendicular lines in section x.2. If y'all know how to construct parallel lines, you should be able to construct any quadrilateral accurately.

Amalgam parallel lines to draw quadrilaterals

  1. Read through the following steps.

    Step one

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

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    Step 2

    Draw an arc from A that crosses Advertising and AB. Keep the same compass width and draw an arc from point D as shown.

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    Step 3

    Set up the compass width to the altitude between the 2 points where the starting time arc crosses Advertisement and AB. From the point where the second arc crosses AD, draw a third arc to cantankerous the second arc.

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    Step 4

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

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  2. Practise drawing a parallelogram, square and rhombus in your exercise volume.
  3. Use a protractor to try to draw quadrilaterals with at least 1 set of parallel lines.
  1. Do the post-obit construction in your practise book.
    1. Use a compass and ruler to construct equilateral \( \triangle ABC\) with sides 9 cm.
    2. Without using a protractor, bisect \(\hat{B}\). Let the bisector intersect Air-conditioning at point D.
    3. Utilise a protractor to measure \(\hat{ADB}\). Write the measurement on the drawing.
  2. Proper noun the post-obit 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 post-obit quadrilaterals matches each description below? (There may be more than one reply 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. Reverse sides are parallel and all angles are equal.
    6. All sides are equal.