In this chapter, you will learn more about different kinds of triangles and quadrilaterals, and their properties. You will explore shapes that are congruent and shapes that are similar. You will also use your knowledge of the properties of 2D shapes in order to solve geometric problems.

Types of triangles

By now, you know that a triangle is a closed 2D shape with three straight sides. We can classify or name different types of triangles according to the lengths of their sides and according to the sizes of their angles.

Naming triangles according to their sides

  1. Match the name of each type of triangle with its correct description.

    Name of triangle

    Description of triangle

    Isosceles triangle

    All the sides of a triangle are equal.

    Scalene triangle

    None of the sides of a triangle are equal.

    Equilateral triangle

    Two sides of a triangle are equal.
  2. Name each type of triangle by looking at its sides.

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Naming triangles according to their angles

Remember the following types of angles:

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Study the following triangles; then answer the questions:

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  1. Are all the angles of a triangle always equal?
  2. When a triangle has an obtuse angle, it is called an ____________ triangle.
  3. When a triangle has only acute angles, it is called an ____________ triangle.
  4. When a triangle has an angle equal to ______, it is called a right-angled triangle.

Investigating the angles and sides of triangles

    1. What is the sum of the interior angles of a triangle?
    2. Can a triangle have two right angles? Explain your answer.
  2. Can a triangle have more than one obtuse angle? Explain your answer.

    If you cannot work out the answers in 1(b) and (c), try to construct the triangles to find the answers.

  3. Look at the triangles below. The arcs show which angles are equal.
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    1. \( \triangle ABC\) is an equilateral triangle. What do you notice about its angles?
    2. \({\triangle}\text{FEM}\) is an isosceles triangle. What do you notice about its angles?
    3. \({\triangle}\text{JKL}\) is a right-angled triangle. Is its longest side opposite the 90° angle?
    4. Construct any three right-angled triangles on a sheet of paper. Is the longest side always opposite the 90° angle?

Properties of triangles:

  • The sum of the interior angles of a triangle is 180°.
  • An equilateral triangle has all sides equal and each interior angle is equal to 60°.
  • An isosceles triangle has two equal sides and the angles opposite the equal sides are equal.
  • A scalene triangle has no sides equal.
  • A right-angled triangle has a right angle (90°).
  • An obtuse triangle has one obtuse angle (between 90° and 180°).
  • An acute triangle has three acute angles (<90°).

Interior angles are the angles inside a closed shape, not the angles outside of it.

Unknown angles and sides of triangles

You can use what you know about triangles to obtain other information. When you work out new information, you must always give reasons for the statements you make.

Look at the examples below of working out unknown angles and sides when certain information is given. The reason for each statement is written in square brackets.

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\( \hat{A} = \hat{B} + \hat{C} = 60^{\circ}\) [Angles in an equilateral \({\triangle}\) = 60°]

DE = DF [Given]

\( \hat{E} = \hat{F}\) [Angles opposite the equal sides of an isosceles \triangle} are equal]

\(\hat{J} = 55^{\circ}\) [The sum of the interior angles of a \({\triangle} = 180^{\circ}\); so \( \hat{J} = 180^{\circ} - 40^{\circ} - 85^{\circ} \)]

You can shorten the following reasons in the ways shown:

  • Sum of interior angles (∠s) of a triangle (\triangle}) = 180°: Interior ∠s of \({\triangle}\)
  • Isosceles triangle has 2 sides and 2 angles equal: Isosceles \({\triangle}\)
  • Equilateral triangle has 3 sides and 3 angles equal: Equilateral \({\triangle}\)
  • Angles forming a straight line = 180°: Straight line

Working out unknown angles and sides

Find the sizes of unknown angles and sides in the following triangles. Always give reasons for every statement.

  1. What is the size of \(\hat{C}\)?

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    \( \begin{align} \hat{A} + \hat{B} + \hat{C} &= \text{______} \text{[Interior } \angle\text{s of a }{\triangle}] \\ \ 50 ° + \text{______} + \hat{C} &= \text{______} \\ 145 ° + \hat{C} &= \text{______} \\ \hat{C} &= \text{______} -145 ° \\ \hat{C} &= \text{______} \end{align} \)
  2. Determine the size of \(\hat{P}\).

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    1. What is the length of KM
    2. Find the size of \(\hat{K}\).

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  4. What is the size of \(\hat{S}\)?

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    1. Find CB.
    2. Find \(\hat{C}\) if \(\hat{A} = 50^{\circ}\).

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    1. Find DF.
    2. Find \(\hat{E}\) if \(\hat{D} = 50^{\circ}\).

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Working out more unknown angles and sides

  1. Calculate the size of \(\hat{X}\) and \(\hat{Z}\).

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  2. Calculate the size of \(x\).

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  3. KLM is a straight line. Calculate the size of \(x\) and \(y\).

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  4. Angle \(b\) and an angle with size 130° form a straight angle. Calculate the size of \(a\) and \(b\).

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  5. \(m\) and \(n\) form a straight angle. Calculate the size of \(m\) and \(n\).

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  6. BCD is a straight line segment. Calculate the size of \(x\).

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  7. Calculate the size of \(x\) and then the size of \( \hat{H}\).

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  8. Calculate the size of \(\hat{N}\).

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  9. DNP is a straight line. Calculate the size of \(x\) and of \(y\).

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Types of quadrilaterals and their properties

A quadrilateral is a figure with four straight sides which meet at four vertices. In many quadrilaterals all the sides are of different lengths and all the angles are of different sizes.

You have previously worked with these types of quadrilaterals, in which some sides have the same lengths, and some angles may be of the same size.

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parallelograms

rectangles

kites

rhombuses

squares

trapeziums

The properties of different types of quadrilaterals

  1. In each question below, different examples of a certain type of quadrilateral are given. In each case identify which kind of quadrilateral it is. Describe the properties of each type by making statements about the lengths and directions of the sides and the sizes of the angles of each type. You may have to take some measurements to be able to do this.
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  2. Use your completed lists and the drawings in question 1 to determine if the following statements are true (T) or false (F).
    1. A rectangle is a parallelogram.
    2. A square is a parallelogram.
    3. A rhombus is a parallelogram.
    4. A kite is a parallelogram.
    5. A trapezium is a parallelogram.
    6. A square is a rhombus.
    7. A square is a rectangle.
    8. A square is a kite.
    9. A rhombus is a kite.
    10. A rectangle is a rhombus.
    11. A rectangle is a square.

A convention is something (such as a definition or method) that most people agree on, accept and follow.

If a quadrilateral has all the properties of another quadrilateral, you can define it in terms of the other quadrilateral, as you have found above.

  1. Here are some conventional definitions of quadrilaterals:
    • A parallelogram is a quadrilateral with two opposite sides parallel.
    • A rectangle is a parallelogram that has all four angles equal to 90°.
    • A rhombus is a parallelogram with all four sides equal.
    • A square is a rectangle with all four sides equal.
    • Atrapezium is a quadrilateral with one pair of opposite sides parallel.
    • A kite is a quadrilateral with two pairs of adjacent sides equal.

    Write down other definitions that work for these quadrilaterals.

    1. Rectangle:
    2. Square:
    3. Rhombus:
    4. Kite:
    5. Trapezium:

Unknown angles and sides of quadrilaterals

Finding unknown angles and sides

Find the length of all the unknown sides and angles in the following quadrilaterals. Give reasons to justify your statements. (Also recall that the sum of the angles of a quadrilateral is 360°.)

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  3. ABCD is a kite.

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  4. The perimeter of RSTU is 23 cm.

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  5. PQRS is a rectangle and has a perimeter of 40 cm.

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Congruency

What is congruency?

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  1. \( \triangle ABC\) is reflected in the vertical line (mirror) to give \(\triangle KLM\).

    Are the sizes and shapes of the two triangles exactly the same?

  2. \({\triangle}\text{MON}\) is rotated 90° around point F to give you \({\triangle}\text{TUE}\).

    Are the sizes and shapes of \({\triangle}\text{MON}\) and \({\triangle}\text{TUE}\) exactly the same?

  3. Quadrilateral ABCD is translated 6 units to the right and 1 unit down to give quadrilateral XRZY.

    Are ABCD and XRZY exactly the same?

In the previous activity, each of the figures was transformed (reflected, rotated or translated) to produce a second figure. The second figure in each pair has the same angles, side lengths, size and area as the first figure. The second figure is thus an accurate copy of the first figure.

When one figure is an image of another figure, we say that the two figures are congruent.

The word congruent comes from the Latin word congruere, which means "to agree". Figures are congruent if they match up perfectly when laid on top of each other.

The symbol for congruent is: \(\equiv\)

Notation of congruent figures

When we name shapes that are congruent, we name them so that the matching, or corresponding, angles are in the same order. For example, in \({\bf{\triangle}\text{ABC}}\) and \({\bf{\triangle}\text{KLM}}\) on the previous page:

\(\hat{A}\) is congruent to (matches and is equal to) \(\hat{K}\).

\(\hat{B}\) is congruent to \(\hat{M}\).

\(\hat{C}\) is congruent to \(\hat{L}\).

We therefore use this notation: \({\bf{\triangle}\text{ABC}} \equiv {\bf{\triangle}\text{KML}}\).

We cannot assume that, when the angles of polygons are equal, the polygons are congruent. You will learn about the conditions of congruence in Grade 9.

Similarly for the other pairs of figures on the previous page: \({\bf{\triangle}\text{MON}} \equiv {\bf{\triangle}\text{ETU}}\) and \({\bf ABCD} \equiv {\bf XRZY}\).

The notation of congruent figures also shows which sides of the two figures correspond and are equal. For example, \({\triangle}\text{ABC} \equiv {\triangle}\text{KML}\) shows that:

\(\text{AB = KM, BC = ML and AC = KL}\)

The incorrect notation \({\triangle}\text{ABC} \equiv {\triangle}\text{KLM}\) will show the following incorrect information:

\( \hat{B} = \hat{L}, ~\hat{C} = \hat{M},~\text{AB = KL, AC = KM}\)

Identifying congruent angles and sides

Write down which angles and sides are equal between each pair of congruent figures.

  1. \({\triangle}\text{PQR} \equiv {\triangle}\text{UCT}\)
  2. \({\triangle}\text{KLM} \equiv {\triangle}\text{UWC}\)
  3. \( {\triangle}\text{GHI}\equiv {\triangle}\text{QRT}\)
  4. \({\triangle}\text{KJL}\equiv {\triangle}\text{POQ}\)

Similarity

In Grade 7, you learnt that two figures are similar when they have the same shape (their angles are equal) but they may be different sizes. The sides of one figure are proportionally longer or shorter than the sides of the other figure; that is, the length of each side is multiplied or divided by the same number. We say that one figure is an enlargement or a reduction of the other figure.

Checking for similarity

  1. Look at the rectangles below and answer the questions that follow.

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    1. Look at rectangle 1 and ABCD:

      How many times is FH longer than BC?

      How many times is GF longer than AB?

    2. Look at rectangle 2 and ABCD:

      How many times is IL longer than BC?

      How many times is LM longer than CD?

    3. Is rectangle 1 or rectangle 2 an enlargement of rectangle ABCD? Explain your answer.
  2. Look at the triangles below and answer the questions that follow.

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    1. How many times is:
      • FG longer than BC?
      • HF longer than AB?
      • HG longer than AC?
      • IK shorter than BC?
      • JI shorter than AB?
      • JK shorter than AC?
    2. Is \({\triangle}\text{HFG}\) an enlargement of \( \triangle ABC\)? Explain your answer.
    3. Is \({\triangle}\text{JIK}\) a reduction of \( \triangle ABC\)? Explain your answer.

In the previous activity, rectangle KILM is an enlargement of rectangle ABCD. Therefore, ABCD is similar to KILM. The symbol for 'is similar to' is: |||. So we write: ABCD ||| KILM.

The triangles on the previous page are also similar. \({\triangle}\text{HFG}\) is an enlargement of \( \triangle ABC\) and \({\triangle}\text{JIK}\) is a reduction of \( \triangle ABC\).

In \( \triangle ABC\) and \({\triangle}\text{HFG}, ~\hat{A} = \hat{H},~\hat{B} = \hat{F}\) and \(\hat{C} = \hat{G}\). We therefore write it like this: \({\triangle}\text{ABC} \text{|||} {\triangle}\text{HFG}\).

In the same way, \({\triangle}\text{ABC} ||| {\triangle}\text{JIK}\).

When you enlarge or reduce a polygon, you need to enlarge or reduce all its sides proportionally, or by the same ratio. This means that you multiply or divide each length by the same number.

Similar figures are figures that have the same angles (same shape) but are not necessarily the same size.

Using properties of similar and congruent figures

  1. Are the triangles in each pair similar or congruent? Give a reason for each answer.

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  2. Is \({\triangle}\text{RTU} ||| {\triangle}\text{EFG}\)? Give a reason for your answer.
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  3. \({\triangle}\text{PQR} ||| {\triangle}\text{XYZ}\). Determine the length of XZ and XY.
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  4. Are the following statements true or false? Explain your answers.
    1. Figures that are congruent are similar.
    2. Figures that are similar are congruent.
    3. All rectangles are similar.
    4. All squares are similar.
  1. Study the triangles below and answer the following questions:
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    1. Tick the correct answer. \( \triangle ABC\) is:

      ☐ acute and equilateral

      ☐ obtuse and scalene

      ☐ acute and isosceles

      ☐ right-angled and isosceles.

    2. If AB = 40 mm, what is the length of AC?
    3. If \( \hat{B}= 80^{\circ}\), what is the size of \( \hat{C}\) and of \( \hat{A}\)?
    4. \({\triangle}\text{ABC} \equiv {\triangle}\text{FDE}\). Name all the sides in the two triangles that are equal to AB.
    5. Name the side that is equal to DE.
    6. If \( \hat{F}\) is 40°, what is the size of \( \hat{B}\)?
  2. Look at figures JKLM and PQRS. (Give reasons for your answers below.)

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    1. What type of quadrilateral is JKLM?
    2. Is JKLM ||| PQRS?
    3. What is the size of \(\hat{L}\)?
    4. What is the size of \(\hat{S}\)?
    5. What is the length of KL?