Measuring and Identifying Types of Angles for JSS 1

Angles

Performance Objectives

By the end of the lesson, students should be able to:

1. measure angles;
2. identify vertically opposite, adjacent, alternate and corresponding angles;
3. identify angles at a point and angles on a straight line and state their properties;
4. relate angles with real life situation.

Meaning of an Angle

An angle is the amount of rotation between two lines that meet at a common point.

The common point where the two lines meet is called the vertex, and the two lines are called the arms of the angle. Angles are measured in degrees, represented by the symbol $^\circ$. A full rotation is equal to $360^\circ$.

Measuring Angles

A protractor is a semi-circular mathematical instrument used to measure and draw angles. It has two scales running from $0^\circ$ to $180^\circ$: an outer scale running clockwise and an inner scale running counter-clockwise.

The procedure for measuring an angle using a protractor involves the following steps:

1. Place the center mark of the protractor exactly on the vertex of the angle.
2. Align the baseline of the protractor with one of the arms of the angle, ensuring the line points to the $0^\circ$ mark.
3. Follow the scale starting from $0^\circ$ along the aligned arm up to where the second arm of the angle crosses the scale.
4. Read the value on the scale to determine the size of the angle in degrees.

Angles on a Straight Line

Angles on a straight line are angles that lie next to each other along a straight line and share a common vertex.

The sum of all angles on a straight line is always equal to $180^\circ$. This property is useful for finding unknown angles when other angles on the line are known.

Worked Examples on Straight Line Angles

Example 1: Find the value of $x$ in the diagram where a straight line is divided into two angles, $x$ and $125^\circ$.

Solution:

Since the angles lie on a straight line, their sum must be $180^\circ$:

$$x + 125^\circ = 180^\circ$$

Subtract $125^\circ$ from both sides of the equation:

$$x = 180^\circ – 125^\circ$$
$$x = 55^\circ$$

Example 2: Three angles lie on a straight line. Their values are $2a$, $3a$, and $70^\circ$. Calculate the value of $a$ and the sizes of the unknown angles.

Solution:

The sum of the angles on a straight line is $180^\circ$:

$$2a + 3a + 70^\circ = 180^\circ$$

Combine the like terms:

$$5a + 70^\circ = 180^\circ$$

Subtract $70^\circ$ from both sides:

$$5a = 180^\circ – 70^\circ$$
$$5a = 110^\circ$$

Divide both sides by 5:

$$a = \frac{110^\circ}{5}$$
$$a = 22^\circ$$

Now, calculate the sizes of the two unknown angles:

$$2a = 2 \times 22^\circ = 44^\circ$$
$$3a = 3 \times 22^\circ = 66^\circ$$

Angles at a Point

Angles at a point are angles formed all the way around a single vertex where multiple lines meet.

The sum of all angles at a point is always equal to $360^\circ$. This represents a complete rotation or a full circle.

Worked Examples on Angles at a Point

Example 1: Four angles meet at a point. Three of the angles are $90^\circ$, $110^\circ$, and $85^\circ$. Find the size of the fourth angle, $y$.

Solution:

The sum of angles at a point is $360^\circ$:

$$y + 90^\circ + 110^\circ + 85^\circ = 360^\circ$$

Add the known angles together:

$$y + 285^\circ = 360^\circ$$

Subtract $285^\circ$ from both sides:

$$y = 360^\circ – 285^\circ$$
$$y = 75^\circ$$

Example 2: Find the value of $k$ if the angles at a point are $3k$, $4k$, $5k$, and $120^\circ$.

Solution:

The sum of angles at a point is $360^\circ$:

$$3k + 4k + 5k + 120^\circ = 360^\circ$$

Combine the like terms:

$$12k + 120^\circ = 360^\circ$$

Subtract $120^\circ$ from both sides:

$$12k = 360^\circ – 120^\circ$$
$$12k = 240^\circ$$

Divide both sides by 12:

$$k = \frac{240^\circ}{12}$$
$$k = 20^\circ$$

Adjacent Angles

Adjacent angles are two angles that lie next to each other, sharing a common vertex and a common arm, without overlapping.

Adjacent angles do not have a fixed sum unless they lie on a straight line or form a right angle. Their primary property is their shared boundary.

Vertically Opposite Angles

Vertically opposite angles are the angles opposite each other at the intersection of two straight lines.

Vertically opposite angles are always equal in size. When two lines cross, they form two pairs of vertically opposite angles.

Worked Examples on Vertically Opposite Angles

Example 1: Two straight lines intersect to form four angles. One of the angles is $115^\circ$. Find the sizes of the other three angles, labeled $p$, $q$, and $r$, where $p$ is vertically opposite to the $115^\circ$ angle.

Solution:

Since vertically opposite angles are equal:

$$p = 115^\circ$$

The angle $q$ lies on a straight line with the $115^\circ$ angle. Therefore, their sum is $180^\circ$:

$$q + 115^\circ = 180^\circ$$
$$q = 180^\circ – 115^\circ$$
$$q = 65^\circ$$

The angle $r$ is vertically opposite to $q$, so they are equal:

$$r = q = 65^\circ$$

The three unknown angles are $p = 115^\circ$, $q = 65^\circ$, and $r = 65^\circ$.

Example 2: In an intersection of two straight lines, one angle is $(3x – 10)^\circ$ and its vertically opposite angle is $(2x + 15)^\circ$. Find the value of $x$.

Solution:

Since vertically opposite angles are equal, set the two expressions equal to each other:

$$3x – 10 = 2x + 15$$

Subtract $2x$ from both sides:

$$3x – 2x – 10 = 15$$
$$x – 10 = 15$$

Add 10 to both sides:

$$x = 15 + 10$$
$$x = 25$$

Parallel Lines and Transversals

Parallel lines are lines on a flat surface that never meet, no matter how far they are extended. They are always the same distance apart.

A transversal is a straight line that cuts across two or more parallel lines. When a transversal intersects parallel lines, several special pairs of angles are formed, including alternate and corresponding angles.

Alternate Angles

Alternate angles are pairs of angles on opposite sides of the transversal line and between the parallel lines, forming a “Z” shape.

Alternate angles are always equal in size. They can be identified by looking for the “Z” or backward “Z” pattern formed by the parallel lines and the transversal.

Corresponding Angles

Corresponding angles are pairs of angles in the same relative position at each intersection where a transversal crosses parallel lines, forming an “F” shape.

Corresponding angles are always equal in size. They can be identified by looking for the “F” pattern, which can be facing forward, backward, upside down, or tilted.

Worked Examples on Alternate and Corresponding Angles

Example 1: Two parallel lines are cut by a transversal. If one of the corresponding angles is $72^\circ$, what is the size of the other corresponding angle?

Solution:

Corresponding angles are equal. Therefore, the size of the other corresponding angle is also $72^\circ$.

Example 2: In a diagram of parallel lines cut by a transversal, an alternate interior angle is labeled $(4x + 20)^\circ$ and its partner alternate interior angle is labeled $100^\circ$. Find the value of $x$.

Solution:

Since alternate angles are equal, set the two values equal to each other:

$$4x + 20 = 100$$

Subtract 20 from both sides:

$$4x = 100 – 20$$
$$4x = 80$$

Divide both sides by 4:

$$x = \frac{80}{4}$$
$$x = 20$$

Real-Life Applications of Angles

Angles are not just mathematical drawings; they exist in many everyday objects, structures, and activities. Recognizing angles helps in construction, design, and navigation.

Examples of angles in real-life situations include:

1. Building Roofs: The slope of a house roof is designed at a specific angle to allow rainwater to run off easily without damaging the structure.
2. Scissors and Tools: When using a pair of scissors, the blades open at different angles depending on the thickness of the material being cut.
3. Road Intersections: Roads meet at various angles, which traffic engineers must measure carefully to ensure safe turns for vehicles.
4. Open Doors: A door swings open at different angles. A fully closed door represents $0^\circ$, while a door opened halfway forms a right angle of $90^\circ$.
5. Ladders: When leaning a ladder against a wall, it must be placed at a safe angle (usually around $75^\circ$) to prevent it from slipping or falling backward.
6. Bending Joints: Human elbows and knees form different angles when bending during activities like running, jumping, or lifting objects.

Practice Exercises

The following exercises are designed to test understanding of measuring, identifying, and calculating angles:

1. Use a protractor to draw and measure the following angles: $45^\circ$, $90^\circ$, $120^\circ$, and $155^\circ$.
2. Two lines intersect. If one angle is $58^\circ$, find the sizes of the other three angles.
3. Three angles on a straight line are $x$, $2x$, and $60^\circ$. Find the value of $x$ and the sizes of the unknown angles.
4. Find the value of $y$ if the angles at a point are $110^\circ$, $130^\circ$, and $2y$.
5. State three real-life situations where the measurement of angles is highly important.
6. Identify the relationship (alternate, corresponding, adjacent, or vertically opposite) between the angles described below:
   1. Angles that form a ‘Z’ shape on parallel lines cut by a transversal.
   2. Angles opposite each other when two straight lines cross.
   3. Angles that share a common vertex and a common arm but do not overlap.
   4. Angles that form an ‘F’ shape on parallel lines cut by a transversal.