When two parallel lines are intersected by a transversal the pairs of corresponding angles are formed 4 2 8 1?

Before jumping into the topic of corresponding angles, let’s first remind ourselves about angles, parallel and non-parallel lines, and transversal lines.

In Geometry, an angle is composed of three parts: vertex and two arms or sides. The vertex of an angle is where two sides or lines of the angle meet, while arms of an angle are simply the angle’s sides.

Parallel lines are two or more lines on a 2-D plane that never meet or cross. On the other hand, non-parallel lines are two or more lines that intersect. A transversal line is a line that crosses or passes through two other lines. A transverse line can pass through two parallel or non-parallel lines.

What is a Corresponding Angle?

Angles formed when a transversal line cuts across two straight lines are known as corresponding angles. Corresponding angles are located in the same relative position, an intersection of transversal and two or more straight lines.

The angle rule of corresponding angles or the corresponding angles postulates that the corresponding angles are equal if a transversal cuts two parallel lines.

Corresponding angles are equal if the transversal line crosses at least two parallel lines.

The diagram below illustrates corresponding angles formed when a transversal line crosses two parallel lines:

From the above diagram, the pair of corresponding angles are:

• < a and < e
• < b and < g
• < d and <f
• < c and < h

Proof of Corresponding Angles

In the figure above, we have two parallel lines.

We need to prove that.

We have the straight angles:

From the transitive property,

From the alternate angle’s theorem,

Using substitution, we have,

Hence,

Corresponding angles formed by non-parallel lines

Corresponding angles are formed when a transversal line intersects at least two non-parallel lines that are not equal, and in fact, they don’t have any relation with each other.

Illustration:

Corresponding interior angle

A pair of corresponding angles is composed of one interior and another exterior angle. Interior angles are angles that are positioned within the corners of the intersections.

Corresponding exterior angle

Angles that are formed outside the intersected parallel lines. An exterior angle and interior angle make a pair of corresponding angles.

Illustration:

Interior angles include; b, c, e, and f, while exterior angles include; a, d, g, and h.

Therefore, pairs of corresponding angles include:

• < a and < e.
• < b and < g
• < d and < f
• < c and < h

We can make the following conclusions about corresponding angles:

• A pair of corresponding angles lie on the same side of the transversal.
• The corresponding pair of angles comprises one exterior angle and another interior angle.
• Not all corresponding angles are equal. Corresponding angles are equal if the transversal intersects two parallel lines. If the transversal intersects non-parallel lines, the corresponding angles formed are not congruent and are not related in any way.
• Corresponding angles form are supplementary angles if the transversal perpendicularly intersects two parallel lines.
• Exterior angles on the same side of the transversal are supplementary if the lines are parallel. Similarly, interior angles are supplementary if the two lines are parallel.

How to find corresponding angles?

One technique of solving corresponding angles is to draw the letter F on the given diagram. Make the letter to face in any direction and relate the angles accordingly.

Example 1

Given ∠d = 30°, find the missing angles in the diagram below.

Solution

Given that ∠d = 30°

∠d = ∠b (Vertically opposite angles)

Therefore, ∠b = 30°

∠b = ∠ g= 30° (corresponding angles)
Now, ∠ d = ∠ f (Corresponding angles)

Therefore, ∠f = 30°
∠ b + ∠ a = 180° (supplementary angles)

∠a+ 30° = 180°

∠ a = 150°

∠ a = ∠ e = (corresponding angles)

Therefore, ∠e = 150°

∠d = ∠h = 30° (corresponding angles)

Example 2

The two corresponding angles of a figure measure 9x + 10 and 55. Find the value of x.

Solution

The two corresponding angles are always congruent.

Hence,

9x + 10 = 55

9x = 55 – 10

9x = 45

x = 5

Example 3

The two corresponding angles of a figure measure 7y – 12 and 5y + 6. Find the magnitude of a corresponding angle.

Solution

First, we need to determine the value of y.

The two corresponding angles are always congruent.

Hence,

7y – 12 = 5y + 6

7y – 5y = 12 + 6

2y = 18

y = 9

The magnitude of a corresponding angle,

5y + 6 = 5 (9) + 6 = 51

Applications of Corresponding Angles

There exist many applications of corresponding angles which we ignore. Observe them if you ever get a chance.

• Usually, windows have horizontal and vertical grills, which make multiple squares. Each vertex of the square makes the corresponding angles.
• The bridge stands on the pillars. All pillars are connected in such a way that corresponding angles are equal.
• The railway tracks are designed so that all the corresponding angles are equal on the track.

In this chapter, you will explore the relationships between pairs of angles that are created when straight lines intersect (meet or cross). You will examine the pairs of angles that are formed by perpendicular lines, by any two intersecting lines, and by a third line that cuts two parallel lines. You will come to understand what is meant by vertically opposite angles, corresponding angles, alternate angles and co-interior angles. You will be able to identify different angle pairs, and then use your knowledge to help you work out unknown angles in geometric figures.

Angles on a straight line

In the figures below, each angle is given a label from 1 to 5.

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

   A

   B

2. Use your answers to fill in the angle sizes below.
   1. \( \hat{1} + \hat{2} = \text{______}^{\circ} \)
   2. \( \hat{3} + \hat{4} + \hat{5}= \text{______}^{\circ} \)

The sum of angles that are formed on a straight line is equal to 180°. (We can shorten this property as: \(\angle\)s on a straight line.)

Two angles whose sizes add up to 180° are also called supplementary angles, for example \( \hat{1} + \hat{2}\).

Angles that share a vertex and a common side are said to be adjacent. So \( \hat{1} + \hat{2}\) are therefore also called supplementary adjacent angles.

When two lines are perpendicular, their adjacent supplementary angles are each equal to 90°.

In the drawing below, DC A and DC B are adjacent supplementary angles because they are next to each other (adjacent) and they add up to 180° (supplementary).

Work out the sizes of the unknown angles below. Build an equation each time as you solve these geometric problems. Always give a reason for every statement you make.

1. Calculate the size of \(a\).

   \( \begin{align} a + 63^{\circ} &= \text{______} [\angle\text{s on a straight line}] \\ a &= \text{______} - 63^{\circ} \\ &= \text{______} \end{align}\)

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

3. Calculate the size of \(y\).

1. Calculate the size of:

2. Calculate the size of:

3. Calculate the size of:

4. Calculate the size of:

5. Calculate the size of:

Vertically opposite angles

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

2. Notice which angles are equal and how these equal angles are formed.

Vertically opposite angles (vert. opp. \(\angle\)s) are the angles opposite each other when two lines intersect.

Vertically opposite angles are always equal.

Calculate the sizes of the unknown angles in the following figures. Always give a reason for every statement you make.

1. Calculate \(x,~ y\) and \(z\).

   \( \begin{align} x &= \text{______}^{\circ} &&[\text{vert. opp.}\angle\text{s}] \\ \\ y + 105^{\circ} &= \text{______}^{\circ} &&[\angle\text{s on a straight line}] \\ y &= \text{______} - 105^{\circ} && \\ & = \text{______} \\ \\ z &= \text{______} &&[\text{vert. opp.}\angle\text{s}] \end{align}\)

2. Calculate \(j,~ k\) and \(l\).

3. Calculate \(a,~ b,~ c\) and \(d\).

Vertically opposite angles are always equal. We can use this property to build an equation. Then we solve the equation to find the value of the unknown variable.

1. Calculate the value of \(m\).
   \( \begin{align} m + 20^{\circ} &= 100^{\circ} [\text{vert. opp.}\angle\text{s}] \\ m &= 100^{\circ} - 20^{\circ} \\ &= \text{______} \end{align}\)
2. Calculate the value of \(t\).

3. Calculate the value of \(p\).

4. Calculate the value of \(z\).

5. Calculate the value of \(y\).

6. Calculate the value of \(r\).

Lines intersected by a transversal

A transversal is a line that crosses at least two other lines.

When a transversal intersects two lines, we can compare the sets of angles on the two lines by looking at their positions.

The angles that lie on the same side of the transversal and are in matching positions are called corresponding angles (corr.\(\angle\)s). In the figure, these are corresponding angles:

• \(a\) and \(e\)
• \(b\) and \(f\)
• \( d\) and \(h\)
• \(c\) and \(g\).

1. In the figure, \(a\) and \(e\) are both left of the transversal and above a line.

   Write down the location of the following corresponding angles. The first one is done for you.

   \(b\) and \(f\): Right of the transversal and above lines

   \(d\) and \(h\):

   \(c\) and \(g\):

Alternate angles (alt.\(\angle\)s) lie on opposite sides of the transversal, but are not adjacent or vertically opposite. When the alternate angles lie between the two lines, they are called alternate interior angles. In the figure, these are alternate interior angles:

• \(d\) and \(f\)
• \(c\) and \(e\)

When the alternate angles lie outside of the two lines, they are called alternate exterior angles. In the figure, these are alternate exterior angles:

• \(a\) and \(g\)
• \(b\) and \(h\)

1. Write down the location of the following alternate angles:

   \(d\) and \(f\):

   \(c\) and \(e\):

   \(a\) and \(g\):

   \(b\) and \(h\):

Co-interior angles (co-int.\(\angle\)s) lie on the same side of the transversal and between the two lines. In the figure, these are co-interior angles:

• \(c\) and \(f\)
• \(d\) and \(e\)

1. Write down the location of the following co-interior angles:

   \(d\) and \(e\):

   \(c\) and \(f\):

Two lines are intersected by a transversal as shown below.

Write down the following pairs of angles:

1. two pairs of corresponding angles:
2. two pairs of alternate interior angles:
3. two pairs of alternate exterior angles:
4. two pairs of co-interior angles:
5. two pairs of vertically opposite angles:

Parallel lines intersected by a transversal

In the figure below left, EF is a transversal to AB and CD. In the figure below right, PQ is a transversal to parallel lines JK and LM.

1. Use a protractor to measure the sizes of all the angles in each figure. Write the measurements on the figures.
2. Use your measurements to complete the following table.

   Corr.\(\angle\)s	\( \hat{1} = \text{_______};~\hat{5} = \text{_______}\) \( \hat{4} = \text{_______};~\hat{8} = \text{_______}\) \( \hat{2} = \text{_______};~\hat{4} = \text{_______}\) \( \hat{3} = \text{_______};~\hat{7} = \text{_______}\)	\( \hat{9} = \text{_______};~\hat{13} = \text{_______}\) \( \hat{12} = \text{_______};~\hat{16} = \text{_______}\) \( \hat{10} = \text{_______};~\hat{14} = \text{_______}\) \( \hat{11} = \text{_______};~\hat{15} = \text{_______}\)
   Alt.int.\(\angle\)s	\( \hat{4} = \text{_______};~\hat{6} = \text{_______}\) \( \hat{3} = \text{_______};~\hat{5} = \text{_______}\)	\( \hat{12} = \text{_______};~\hat{14} = \text{_______}\) \( \hat{11} = \text{_______};~\hat{13} = \text{_______}\)
   Alt.ext.\(\angle\)s	\( \hat{1} = \text{_______};~\hat{7} = \text{_______}\) \( \hat{2} = \text{_______};~\hat{8} = \text{_______}\)	\( \hat{9} = \text{_______};~\hat{15} = \text{_______}\) \( \hat{10} = \text{_______};~\hat{16} = \text{_______}\)
   Co-int.\(\angle\)s	\( \hat{4} + \hat{5} = \text{_______}\) \( \hat{3} + \hat{6} = \text{_______}\)	\( \hat{12} + \hat{13} = \text{_______}\) \( \hat{11} + \hat{14} = \text{_______}\)

3. Look at your completed table in question 2. What do you notice about the angles formed when a transversal intersects parallel lines?

When lines are parallel:

• corresponding angles are equal
• alternate interior angles are equal
• alternate exterior angles are equal
• co-interior angles add up to 180°

1. Fill in the corresponding angles to those given.

2. Fill in the alternate exterior angles.

3. 1. Fill in the alternate interior angles.
   2. Circle the two pairs of co-interior angles in each figure.

4. 1. Without measuring, fill in all the angles in the following figures that are equal to \(x\) and \(y\).
   2. Explain your reasons for each \(x\) and \(y\) that you filled in to your partner.

5. Give the value of \(x\) and \(y\) below.

Finding unknown angles on parallel lines

Work out the sizes of the unknown angles. Give reasons for your answers. (The first one has been done as an example.)

1. Find the sizes of \(x,~y\) and \(z\).

   \( \begin{align} x &= 74^{\circ} &&[\text{alt.}\angle\text{ with given }74^{\circ}; AB \parallel CD] \\ \\ y &= 74^{\circ} &&[\text{corr.}\angle\text{ with }x; AB \parallel CD] \\ \text{or } y &= 74^{\circ} &&[\text{vert. opp.}\angle\text{ with given }74^{\circ}] \\ \\ z &= 106^{\circ} &&[\text{co-int.}\angle\text{ with }x; AB \parallel CD] \\ \text{or } z &= 106^{\circ} &&[\angle\text{s on a straight line}] \end{align}\)

2. Work out the sizes of \(p,~ q\) and \(r\).

3. Find the sizes of \(a,~b,~c\) and \(d\).

4. Find the sizes of all the angles in this figure.

5. Find the sizes of all the angles. (Can you see two transversals and two sets of parallel lines?)

Two angles in the following diagram are given as \(x\) and \(y\). Fill in all the angles that are equal to \(x\) and \(y\).

The diagram below is a section of the previous diagram.

1. What kind of quadrilateral is in the diagram? Give a reason for your answer.
2. Look at the top left intersection. Complete the following equation:

   Angles around a point\( = 360^{\circ}\)

   \(\therefore x + y+ \text{______} + \text{______} = 360^{\circ}\)

3. Look at the interior angles of the quadrilateral. Complete the following equations:

   Sum of angles in the quadrilateral \(= x + y + + \text{______} + \text{______}\)

   From question 2: \(x + y+ \text{______} + \text{______} = 360^{\circ}\)

   \(\therefore\) Sum of angles in a quadrilateral = \(\text{______}^{\circ}\)

   Can you think of another way to use the diagram above to work out the sum of the angles in a quadrilateral?

Solving more geometric problems

1. Calculate the sizes of \(\hat{1}\) to \(\hat{7}\).

2. Calculate the sizes of \(x,~y\) and \(z\).

3. Calculate the sizes of \(a, ~b, ~c\) and \(d\).

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

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

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

7. Calculate the sizes of \(a\) and \(\hat{CEP}\).

1. Calculate the sizes of \(\hat{1}\) to \(\hat{6}\).

2. RSTU is a trapezium. Calculate the sizes of \(\hat{T}\) and \(\hat{R}\).

3. JKLM is a rhombus. Calculate the sizes of \(\hat{JML}, \hat{M_2}\) and \(\hat{K_1}\).

4. ABCD is a parallelogram. Calculate the sizes of \(\hat{ADB}, \hat{ABD}, \hat{C}\) and \(\hat{DBC}\)

1. Look at the drawing below. Name the items listed alongside.
   1. a pair of vertically opposite angles
   2. a pair of corresponding angles
   3. a pair of alternate interior angles
   4. a pair of co-interior angles
2. In the diagram, AB \(\parallel\) CD. Calculate the sizes of \(\hat{FHG}, \hat{F}, \hat{C}\) and \(\hat{D}\). Give reasons for your answers.

3. In the diagram, OK = ON, KN \(\parallel\) LM, KL\(\parallel\) MN and \(\hat{LKO} = 160^{\circ}\).

   Calculate the value of \(x\). Give reasons for your answers.