Skip to main content

Grade 7 Mathematics Learner’s Book

INNODEMS

Section 4.1 Angles

Subsection 4.1.1 Introduction

Definition 4.1.1.

The word angle comes from the latin word angulus, meaning “corner”. An angle is represented by the symbol \(\angle.\)
🔗
Angles are formed when two lines intersect at point (also known as vertex). In geometry, an angle is formed when two rays are joined at their endpoints. Rays are also known as sides or arms of the angle.
🔗
Observe the diagram below.
🔗
angles(Created with GeoGebra®,[Geogebra Content Team])
lIne \(OA\) and \(OB\) intersect at point \(O\) to form \(\angle\) \(x \text{,}\) shaded in red.
🔗
🔗

Activity 4.1.1.

Here is an interactive activity for you to explore the different types of angles.
🔗
Use the spy glass by dragging it over the objects in the given picture. You can click on the ’NEW PICTURE’ button to get a new image.
🔗
Figure 4.1.2. Angles
🔗
What angles have you discovered from this activity?
🔗
Now, explore the objects around you and find the different types of angles in your sorrounding. What angles did you find?
🔗
Learning Point.
🔗
Did you know one of the real life applications of angles is in Carpentary?
🔗
For example, when carpenters are making a chair, they use angles to cut the wood so that the legs of the chair slants at the desired angle thereby creating a stable and beautiful designs.
🔗
They typically use a combination of measuring tools, like a protractor and a combination square, to mark the angles on the wood before cutting, often aiming for a slight outward slant on the legs for stability and comfort
🔗
🔗

Checkpoint 4.1.3. True/False.

    Angles are formed when two rays intersect at a vertex. The opening between the two rays is called an angle, represented by a symbol \(\angle .\)
    🔗

  • True.

  • An angle is described as a ’opening’ formed by two rays/lines meeting at a vertex.

  • False.

  • An angle is described as a ’opening’ formed by two rays/lines meeting at a vertex.
Hint.
Refer to the definition in the introduction section , Definition 4.1.1.
🔗
🔗
🔗
🔗

Subsection 4.1.2 Angles in a straight line

Activity 4.1.2.

What you need: a ruler and a protractor
🔗
line
  1. Draw a straight line \(AB\) and mark point \(O\) at the centre as shown above.
    🔗
    🔗
  2. Draw two lines to make an angle with line \(AB\) at point \(O\) as shown in the diagram below.
    🔗
    🔗
    1. Measure the angles you have drawn.
      🔗
      🔗
    2. How do the angles relate to each other?
      🔗
      🔗
    🔗
🔗
Learning point
🔗
Angles on a straight line always add up to \(180^\circ\text{.}\) This is because a straight line itself is considered a "straight angle" which measures exactly \(180^\circ\text{.}\)
🔗
Any two angles that add up to \(180^\circ\) are supplementary angles.
🔗
🔗

Activity 4.1.3.

Calculate the size of the angle labeled \(k\) in the figure below.
🔗
Angles on a straight line.
Solution.
Angles on a straight line add upto \(180^\circ\)
🔗
The angle \(k\) can be solved by adding \(55^\circ+k+50^\circ\text{;}\)
🔗
Equating
\begin{equation*} 55^\circ +k + 50^\circ =1 80^\circ\text{,} \end{equation*}
we get;
🔗
\begin{equation*} k=180^\circ-105^\circ \end{equation*}
🔗
\begin{equation*} \text{Therefore}, k = 75^\circ \end{equation*}
🔗
🔗
🔗

Checkpoint 4.1.4.

Figure 4.1.5. Angles in a straight line(Created with GeoGebra®,[Geogebra Content Team])
🔗
🔗
🔗

Subsection 4.1.3 Angles at a point.

Activity 4.1.4.

Angles at a point.
🔗
1.Trace and cut out the fugure below.
🔗
Angles at a point
2.Use a protractor to measure the following angles:
🔗
(a). Angle \(a\)
🔗
(b). Angle \(b\)
🔗
(c). Angle \(c\)
🔗
(d). Angle \(d\)
🔗
3.Which angles are equal?
🔗
4.Find the sum of angles \(a\) , \(b\) , \(c\) and \(d\)
🔗
5. What is their sum?
🔗
6. Share your work with other learners in class.
🔗
Learning point
🔗
Angle \(a\) is vertically opposite to angle \(c\) .
🔗
Angle \(b\) is vertically opposite to angle \(d\text{.}\)
🔗
Vertically opposite angles are equal.
🔗
Angle \(a\) = Angle \(c\)
🔗
Angle \(b\) = Angle \(d\)
🔗
🔗

Example 4.1.6.

Work out the value of the angle marked \(x\) in the figure
🔗
Solution.
Angles \(x,45^\circ, 70^\circ, 60^\circ \text{and } 120^\circ\) meet at a point.
🔗
Therefore:
🔗
\begin{equation*} x +45^\circ + 70^\circ + 60^\circ + 120^\circ = 360^\circ \end{equation*}
🔗
\begin{equation*} x + 300^\circ = 360^\circ \end{equation*}
🔗
\begin{equation*} x + 300^\circ - 300^\circ = 360^\circ - 300^\circ \end{equation*}
🔗
\begin{equation*} x = 60^\circ \end{equation*}
🔗
🔗
🔗

Checkpoint 4.1.7.

Figure 4.1.8. Angles at a point
🔗
🔗

Checkpoint 4.1.9.

🔗
🔗

Subsection 4.1.4 Complementary and supplimentary angles

Angles that add together to make a straight line are called supplementary angles. If you know one supplementary angle you can subtract it from 180° to find a missing angle.
🔗

Activity 4.1.5.

Here is an activity for you to explore on supplementary and complementary angles.
🔗
Figure 4.1.10. Supplimentary and complementary angles(Created with GeoGebra®,[Geogebra Content Team])
🔗
Hint.
Add the two angles
🔗
🔗
🔗

Activity 4.1.6.

Here is an activity for you to explore on supplementary angles formed by a parallel lines and a transversal.
🔗
Figure 4.1.11. Supplimentary angles(Created with GeoGebra®,[Geogebra Content Team])
🔗
🔗

Checkpoint 4.1.12.

Here is another interactive activity to help you discover the properties of complementary and supplimentary angles.
🔗
Figure 4.1.13. Complementary and supplimentary angles(Created with GeoGebra®,[Geogebra Content Team])
🔗
What do you notice from this activity?
🔗
🔗
🔗

Subsection 4.1.5 Angles on a transversal

A transversal is any line that intersects two straight lines at distinct points.
🔗
When a transversal cuts two parallel lines, several angles are formed by these two intersections. Those are called transversal angles. Those types of angles on a transversal are given below:
🔗
🔗

Subsubsection 4.1.5.1 Corresponding angles

Definition 4.1.14.
Corresponding angles are the angles which are formed in matching corners or corresponding corners with the transversal when two parallel lines are intersected by any other line (i.e. the transversal).
🔗
Here ia an activity for you to explore on corresponding angles
🔗
Figure 4.1.15. Corresponding angles(Created with GeoGebra®,[Geogebra Content Team])
🔗
🔗
Example 4.1.16.
Find the value of angle \(k\) in the figure below.
🔗
Corresponding angles
Solution.
Corresponding angles are equal.
🔗
Angle \(k\) corresponds with \(110^\circ.\)
🔗
Angle \(k = 110^\circ.\)
🔗
🔗
🔗
🔗

Subsubsection 4.1.5.2 Co-interior angle

Activity 4.1.7.
Work in groups
🔗
\(1.\) Draw a pair of parallel lines and a transversal
🔗
\(2. \) Mark the angles as shown below.
🔗
Co-interior angles
\(3. \) Cut the angles \(n\) and \(p\) using a pair of scissors
🔗
\(4.\) Place the angles on a straight line.
🔗
What do you notice?
🔗
\(5\) Share your findings with other learners in class.
🔗
Learning point
🔗
Angles \(n\) and \(p\) are co-interior. Co-interior angles add upto \(180^\circ\text{.}\)
🔗
🔗
Example 4.1.17.
What is the value of angle \(n\) in the figure below?
🔗
Co-interior angles
Solution.
Co-interior angles add up to \(180^\circ\)
🔗
\begin{equation*} n + 50^\circ = 180^\circ \end{equation*}
🔗
\begin{equation*} n + 50^\circ - 50^\circ = 180^\circ - 50^\circ \end{equation*}
🔗
\begin{equation*} n = 130^\circ \end{equation*}
🔗
🔗
🔗
🔗

Subsubsection 4.1.5.3 Alternate interior and alternate exterior angles

Alternate angles in parallel lines are angles that occur on opposite sides of the transversal line and have the same size.
🔗
Activity 4.1.8.
Here is an exercise for you to explore on alternate angles
🔗
Figure 4.1.18. Alternate angles
🔗
From the given figure, we have the following categories of alternate angles.
🔗
Alternate interior angles
🔗
  • \(\displaystyle \angle u \text{ and } \angle v\)
    🔗
    🔗
  • \(\displaystyle \angle r \text{ and } \angle h\)
    🔗
    🔗
🔗
Alternate exterior angles
🔗
  • \(\displaystyle \angle p \text{ and } \angle t\)
    🔗
    🔗
  • \(\displaystyle \angle q \text{ and } \angle s\)
    🔗
    🔗
🔗
Alternate angles are also called the Z angles and can be identified by drawing a Z on the figure as shown in the figure above.
🔗
🔗
Checkpoint 4.1.19.
🔗
Checkpoint 4.1.20.
🔗
🔗
🔗

Subsection 4.1.6 Angles in a parallelogram

A parallelogram is a geometrical shape whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length. The Sum of adjacent angles of a parallelogram is equal to 180 degrees
🔗
Properties of a parallelogram
🔗
  • The opposite sides of a parallelogram are equal in length.
    🔗
    🔗
  • The opposite angles are equal in measure.
    \begin{equation*} \angle p = \angle r \end{equation*}
    \begin{equation*} \angle q = \angle s \end{equation*}
    🔗
    🔗
  • Also, the interior angles on the same side of the transversal are supplementary.
    \begin{equation*} \angle p + \angle q = 180^\circ \end{equation*}
    \begin{equation*} \angle q + \angle r = 180^\circ \end{equation*}
    \begin{equation*} \angle r + \angle s = 180^\circ \end{equation*}
    \begin{equation*} \angle s + \angle p = 180^\circ \end{equation*}
    🔗
    🔗
  • The Sum of all the interior angles equals \(360^\circ\)
    🔗
    🔗
🔗

Example 4.1.21.

In the figure below, work out the sizes of angles marked \(x\text{,}\) \(y\) and \(z\text{.}\)
🔗
Parallelogram
Solution.
Angles marked \(z\) and \(180^\circ\) are co-interior angles.
🔗
Therefore:
🔗
Angle marked \(z = 180^\circ \)
🔗
\begin{equation*} x=72^\circ \end{equation*}
🔗
Angles marked \(x\) and \(180^\circ\) are co-interior angles.Therefore:
🔗
\begin{equation*} \text{Angle marked } x = 180^\circ-108^\circ \end{equation*}
🔗
\begin{equation*} x=72^\circ \end{equation*}
🔗
Angle marked \(x\) and \(y\) are co-interior angles.Therefore:
🔗
\begin{equation*} x + y = 180^\circ \end{equation*}
🔗
\begin{equation*} y = 180^\circ - x \end{equation*}
🔗
\begin{equation*} y = 180^\circ - 72^\circ \end{equation*}
🔗
\begin{equation*} y = 108^\circ \end{equation*}
🔗
🔗
🔗

Checkpoint 4.1.22.

🔗

Checkpoint 4.1.23.

🔗
🔗

Subsection 4.1.7 Angle properties of polygons.

What do you understand by the term ’polygon’?
🔗
A polygon is a two dimensional shape with three or more sides, where the sides are all straight lines.
🔗
’Poly’ is a greek word for ’many’ and ’gon’ means ’angles’.
🔗
Examples of polygons include: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon e.t.c.
🔗

Activity 4.1.10.

Explore the connection between the name and number of sides of a polygon in this activity.
🔗
Figure 4.1.24. Angle properties of polygons(Created with GeoGebra®,[Geogebra Content Team])
🔗
🔗
In order to explore the different properties of polygons in a fun and interactive way, you will be using GeoGebra.
🔗
GeoGebra is a free Mathematical tool for interactive geometry, algebra, statistics.
🔗

Activity 4.1.11.

For your activity, you will be using GeoGebra to construct and analyse polygons.
🔗
Click on the link below to access the GeoGebra platform, Beginner Tutorial, where you will learn and practice on how to construct and analyse polygons.
🔗
What did you learn? Share your findings with other learners.
🔗
🔗

Activity 4.1.12.

Use GeoGebra.
🔗
  1. Construct the following polygons; triangle, quadrilateral, pentagon and hexagon.
    🔗
    🔗
  2. Measure the sizes of angles in each polygons.
    🔗
    🔗
  3. Determine the sum of all angles in each polygon.
    🔗
    🔗
  4. Copy and fill in the table below.
    🔗
    🔗
🔗
Polygon Number of sides Sum of angles
Triangle
Quadrilateral
Pentagon
Hexagon
Identify a formula that relates the angles in each polygon to the number of sides.
🔗
Learning Point
🔗
The angles you have measured in activity Activity 4.1.11 are called interior angles. Polygons have different names according to the number of sides.
🔗
If a polygon has \(n\) sides, the number of right angles is equal to \((2n-4)\text{.}\)
🔗
Reflect
🔗
How do we get the sum of interior angles in a polygon?
🔗
🔗
🔗

Subsection 4.1.8 Interior angles of a polygon

Interior angles of a polygon are the angles within a polygon made by two sides.
🔗
We can calculate the sum of the interior angles of a polygon by subtracting 2 from the number of sides and then multiplying by \(180^\circ\text{.}\)
🔗

Subsubsection 4.1.8.1 Triangle

Activity 4.1.13.
Sum of the interior angles in a triangle
🔗
We know that the angles in any triangle add up to \(180^\circ.\)
🔗
Therefore, the sum of interior angles for a triangle is \(180^\circ.\)
🔗
🔗
🔗

Subsubsection 4.1.8.2 Rectangle

Activity 4.1.14.
Sum of the interior angles in a rectangle
🔗
To find the sum of interior angles of the rectangle below;
🔗
We can split the rectangle into two triangles by drawing a line from one corner to an opposite one.
🔗
Rectangle
Rectangle2
The sum of interior angles one triangle is \(180^\circ.\) Therefore, the sum of the interior angles of the two triangles in the rectangle is \(180^\circ \times 2 = 360^\circ\)
🔗
Properties of a rectangle.
🔗
  • A rectangle has four sides.
    🔗
    🔗
  • It has four interior angles.
    🔗
    🔗
  • Each interior angle is \(90^\circ\) (right angle).
    🔗
    🔗
  • The sum of interior angles in a rectangle is \(360^\circ\)
    🔗
    🔗
  • The sum of interior angles is equal to four right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.3 Square

Activity 4.1.15.
Sum of the interior angles in a square
🔗
1.Trace and cut out the square below.
🔗
Split the sqaure into two triangles. Sum of the interior angles of the two triangles is the sum of the interior angles of the square.
🔗
Square
Square2
Properties of a square
🔗
  • A square has four sides
    🔗
    🔗
  • It has four interior angles.
    🔗
    🔗
  • Each interior angle is \(90^\circ\) (right angle).
    🔗
    🔗
  • The sum of interior angles is \(360^\circ\)
    🔗
    🔗
  • The sum of interior angles is equal to four right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.4 Rhombus

Activity 4.1.16.
Properties of a rhombus.
🔗
  • A rhombus has four equal sides.
    🔗
    🔗
  • It has four interior angles.
    🔗
    🔗
  • Its opposite angles are equal.
    🔗
    🔗
  • The sum of interior angles is \(360^\circ\)
    🔗
    🔗
  • The sum of interior angles is equal to four right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.5 Parallelogram

Activity 4.1.17.
Properties of a parallelogram
🔗
  • A parallelogram has four sides.
    🔗
    🔗
  • Opposite sides are equal and parallel.
    🔗
    🔗
  • It has four interior angles.
    🔗
    🔗
  • Opposite angles are equal
    🔗
    🔗
  • The sum of interior angles is \(360^\circ\)
    🔗
    🔗
  • The sum of interior angles is equal to four right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.6 Trapezium

Activity 4.1.18.
Properties of a trapezium
🔗
  • Trapezium has four sides.
    🔗
    🔗
  • It has one pair of parallel lines.
    🔗
    🔗
  • It has four interior angles.
    🔗
    🔗
  • The sum of interior angles is \(360^\circ\)
    🔗
    🔗
  • The sum of interior angles is equal to four right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.7 Pentagon

Activity 4.1.19.
Sum of angles in a pentagon
🔗
Given the regular pentagon below;
🔗
Using the knowledge of triangles you can find the sum of the interior angles of the pentagon by splitting it into triangles as shown.
🔗
Pentagon
Square2
Properties of a pentagon.
🔗
  • A pentagon has five sides.
    🔗
    🔗
  • It has five interior angles.
    🔗
    🔗
  • Regular pentagons have equal sides and equal angles. Each interior angle is \(108^\circ\)
    🔗
    🔗
  • Irregular pentagons do not have equal lines and equal angles.
    🔗
    🔗
  • The sum of interior angles of any pentagon is \(540^\circ\text{.}\)
    🔗
    🔗
  • The sum of interior angles is equal to six right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.8 Hexagon

Activity 4.1.20.
Sum of angles in a hexagon
🔗
Given the hexagon below;
🔗
Sum of interior angles of the hexagon is:
🔗
Hexagon
Properties of a hexagon.
🔗
  • Hexagon has six sides.
    🔗
    🔗
  • It has six interior angles.
    🔗
    🔗
  • Regular hexagons have equal sides and equal angles. Each interior angle is \(120^\circ\)
    🔗
    🔗
  • Irregular hexagon do not have equal lines and equal angles.
    🔗
    🔗
  • The sum of interior angles of any hexagon is \(720^\circ\text{.}\)
    🔗
    🔗
  • The sum of interior angles is equal to eight right angles.
    🔗
    🔗
🔗
🔗
🔗

Subsubsection 4.1.8.9 Summary

In order to find the sum of interior angles of any polygon:
🔗
  1. Identify the number of sides of the polygon. Check if the polygon is regular or irregular.
    🔗
    🔗
  2. Work out how many triangles could be created within the polygon by drawing lines from a single vertex to all the other vertices
    🔗
    🔗
  3. Multiply the number of triangles in the polygon by \(180^\circ\) to find the sum of the interior angles.
    🔗
    🔗
🔗
🔗
🔗

Subsection 4.1.9 Relating interrior angles to the number of sides of a polygon

  • If a polygon has n sides, the number of triangles is \(n-2\text{.}\)
    🔗
    🔗
  • The sum of the interior angles in a polygon = number of triangles \(\times 180^\circ\)
    🔗
    \begin{equation*} 180^\circ\times (n-2) \end{equation*}
    🔗
    🔗
  • To get the size of one interior angle in a regular polygon, divide the sum of the interior angles by the number of sides.
    🔗
    \begin{equation*} \frac{90^\circ\times (2n-4)}{n}=\frac{180^\circ\times (n-2)}{n} \end{equation*}
    🔗
    🔗
🔗

Checkpoint 4.1.25.

🔗

Checkpoint 4.1.26.

🔗
🔗

Subsection 4.1.10 Relating exterior angles and the numer of sides of a polygon

Exterior angles are the angles between a polygon and the extended line from the next side.
🔗
Given the parallelogram below, the exterior angles are;
🔗
\(\angle p, \angle l, \angle s \text{ and } \angle u\)
🔗
Figure 4.1.27. Angles in a straight line
🔗
Sum of exterior angles of a polygon is always equal to \(360^\circ.\)
🔗
Exterior angle + its corresponding interior angle \(=180^\circ.\) i.e.
🔗
\begin{equation*} p + q = 180^\circ \end{equation*}
\begin{equation*} r + l = 180^\circ \end{equation*}
\begin{equation*} u + w = 180^\circ \end{equation*}
\begin{equation*} s + t = 180^\circ \end{equation*}
🔗
Learning point
🔗
To get the size of exterior angle of a regular polygon, divide \(360^\circ\) by the number of sides of the polygon.
🔗
Size of exterior angle \(= \frac{360^\circ}{n}\text{.}\)
🔗
🔗

Subsection 4.1.11 Solving angles of polygons

Example 4.1.28.

Find the sum of interior angles of this polygon
🔗
Complex polygon
Solution.
The polygon has \(6\) sides. Since all the sides are not equal, this is an irregular hexagon.
🔗
To work out the number of triangles in the hexagon, we subtract \(2\) from the number of sides of the hexagon
🔗
\begin{equation*} 6 - 2 = 4 \end{equation*}
🔗
Therefore, the sum of its interior angles can be found by \(4 \times 180^\circ = 720^\circ\)
🔗
The sum of interior angles in a hexagon is \(720^\circ\)
🔗
🔗
🔗

Example 4.1.29.

A regular polygon has six sides.determine:
🔗
(a) The size of each interior angle (b) The size of each exterior angle.
Solution.
(a) sum of interior angles is
\begin{equation*} (2n-4)\times 90^\circ=(2 \times 6-4)\times 90^\circ \end{equation*}
🔗
\begin{equation*} =(12-4)\times 90^\circ \end{equation*}
🔗
\begin{equation*} = 8 \times 90^\circ \end{equation*}
🔗
\begin{equation*} =720^\circ \end{equation*}
🔗
size of each interior angle
\begin{equation*} = \frac{720^\circ}{6} \end{equation*}
🔗
\begin{equation*} = 120^\circ \end{equation*}
🔗
(b).Size of each exterior angle is
\begin{equation*} = \frac{360^\circ}{n}=\frac{360}{6} \end{equation*}
🔗
\begin{equation*} = 60^\circ \end{equation*}
🔗
🔗
🔗

Example 4.1.30.

The sum of the interior angles of a regular polygon is \(540^\circ.\) Deterimine the number of sides of the polygon.
🔗
Solution.
\((2n-4) \times 90^\circ = 540^\circ\)
🔗
\(180n-360^\circ=540^\circ\)
🔗
\(180n-360^\circ+360^\circ=540^\circ+360^\circ\)
🔗
\begin{equation*} 180n=900^\circ \end{equation*}
🔗
\begin{equation*} n=5 \end{equation*}
🔗
🔗
🔗
🔗
🔗
PrevTopNext