Lines Angles and Transversals: Definition with Examples

Key Concepts

• Identify angles created by parallel lines cut by a transversal
• Find unknown angle measures
• Use algebra to find unknown angle measures

Lines Angles and Transversals

1. What is meant by similar figures? 

2. Which symbols is used to indicate similarity? 

3. What is the sequence of transformations for the image given?  

Answers: 

1. Similar figures are two figures having the same shape. The objects which are of exactly the same shape and size are known as congruent objects. 

2. This symbol is used for similarity. 

3. Reflect over the x-axis, then translate (x+6, y) 

Angles, Lines, and Transversals 

Angle: 

Angles are formed when two lines intersect at a point. 

Line: 

A line is a one-dimensional figure, which has length but no width. A line is made of a set of points which is extended in opposite directions infinitely. 

A transversal is a line that passes through two lines in the same plane at two distinct points.  

Transversal: 

Identify Angles Created by Parallel Lines Cut by a Transversal 

Parallel lines: 

Parallel lines are the lines that do not intersect or meet each other at any point in a plane. 

Transversal: 

When any two parallel lines are cut by a transversal, many pairs of angles are formed.  

There is a relationship that exists between these pairs of angles.  

While some of them are congruent, the others are supplementary. 

Example: 

Identify angles created by parallel lines cut by a transversal. 

Sol: 

Parallel Lines Cut by a Transversal 

From the figure corresponding angles formed by the intersection of the transversal are: 

∠1 and ∠5 

∠2 and ∠6 

∠3 and ∠7 

∠4 and ∠8 

The pair of corresponding angles are equal in measure, that is, 

 ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8 

Alternate interior angles: 

Alternate interior angles are formed on the inside of two parallel lines which are intersected by a transversal. In the figure given above, there are two pairs of alternate interior angles. 

∠3 and ∠6 

∠4 and ∠5 

The pair of alternate interior angles are equal in measure, that is, ∠3 = ∠6, and ∠4 = ∠5 

Alternate exterior angles: 

When two parallel lines are cut by a transversal, the pairs of angles formed on either side of the transversal are named as alternate exterior angles. From the figure given above, there are two pairs of alternate exterior angles. 

∠1 and ∠8 

∠2 and ∠7 

The pair of alternate exterior angles are equal in measure, that is, ∠1 = ∠8, and ∠2 = ∠7 

Consecutive interior angles: 

When two parallel lines are cut by a transversal, the pairs of angles formed on the inside of one side of the transversal are called consecutive interior angles or co-interior angles. From the figure, there are two pairs of consecutive interior angles. 

∠4 and ∠6 

∠3 and ∠5 

Unlike the other pairs given above, the pair of consecutive interior angles are supplementary, that is, ∠4 + ∠6 = 180°, and ∠3 + ∠5 = 180°. 

Find Unknown Angle Measures  

Example: 

 What is the measures of m ∠6? Explain. 

Sol: 

Use what you know about the angles created when parallel lines are cut by transversal. 

m ∠6 + 59° = 180° 

m ∠6 = 180° – 59° 

m ∠6 = 121° 

Use Algebra to Find Unknown Angle Measures 

Algebra:  

An algebraic expression in algebra is formed using integer constants, variables, and basic arithmetic operations of addition (+), subtraction (-), multiplication (×), and division (/). 

Below image is the example for algebraic expression. 

Example: 

In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find the value of x. 

Sol: 

From the given figure,  

∠ (2x + 20) ° and ∠ (3x – 10) ° are corresponding angles.  

So, they are equal.  

Then, we have 

(2x + 20) ° = (3x – 10)° 

2x + 20 = 3x – 10 

Subtract 2x from each side.  

20 = x – 10 

Add 10 to each side.  

30 = x 

Exercise

1. Given the following two parallel lines that have been cut by a transversal. ∠1 and which angle make up alternate interior angles?

2. Find the missing angle measures.

3. Find the unknown angle.

4. Find the value of x in the following figure.

5. What type of angle pair is ∠1 and ∠3?

6. Solve for x.

7. Find x.

8. Given the following two parallel lines that have been cut by a transversal.

     Which two angles would be alternate exterior?

9. For the given figure, can you conclude that r∥s?  Explain.

10.   Solve for x.

What we have learned:

• Understand Angles, Lines and Transversals
• Identify Angles Created by Parallel Lines Cut by a Transversal
• Find Unknown Angle Measures
• Use Algebra to Find Unknown Angle Measures

Concept Map: