# Corresponding Angles

Corresponding angles are two angles that lie in **similar relative positions** on the same side of a transversal or at each intersection. They are usually formed when two parallel or non-parallel lines are cut by a transversal.

Remember that a **transversal** is a line that intersects two or more lines.

In our illustration above, parallel lines [latex]a[/latex] and [latex]b[/latex] are cut by a transversal which as a result, formed 4 corresponding angles. For example, [latex]\angle 2[/latex] and [latex]\angle 6[/latex] are corresponding angles. Why? Because both angles are located in matching corners or corresponding positions on the right-hand side of the transversal. In other words, each angle is located **above the line** and **to the right** of the transversal.

Here are our corresponding angles (must be in pairs) from the diagram and their location.

- [latex]\angle \textbf{1}[/latex] and [latex]\angle \textbf{5}[/latex] – above the line, left of the transversal
- [latex]\angle \textbf{3}[/latex] and [latex]\angle \textbf{7}[/latex] – below the line, left of the transversal
- [latex]\angle \textbf{2}[/latex] and [latex]\angle \textbf{6}[/latex] – above the line, right of the transversal
- [latex]\angle \textbf{4}[/latex] and [latex]\angle \textbf{8}[/latex] – below the line, right of the transversal

There are a few things to remember when dealing with corresponding angles.

## Corresponding Angles Postulate

When two

parallel linesare cut by a transversal, then the pairs of corresponding angles arecongruentor have thesame measure.

Take for example in our diagram above, since [latex]\angle 1[/latex] and [latex]\angle 5[/latex] are corresponding angles, they are congruent. This also means that if [latex]\angle 1[/latex] measures [latex]70^\circ [/latex] then [latex]\angle 5[/latex] also measures [latex]70^\circ [/latex]. Therefore, [latex]\angle 1 \cong \angle 5[/latex].

On the other hand, if the transversal intersects with two** non-parallel lines**, the corresponding angles formed are **not congruent** and do not have a specific relationship to each other.

Hence, [latex]\angle a[/latex] and [latex]\angle e[/latex] are corresponding angles but are NOT congruent.

## Example Problems Involving Corresponding Angles

**Example 1:** Identify the corresponding angles.

Here we have two parallel lines, lines [latex]k[/latex] and [latex]g[/latex], that are cut by the transversal, [latex]t[/latex]. Remember that corresponding angles are angles that are in similar positions on the same side of the transversal.

So the corresponding angles are:

- [latex]\angle 2[/latex] and [latex]\angle 1[/latex]
- [latex]\angle 4[/latex] and [latex]\angle 3[/latex]
- [latex]\angle 6[/latex] and [latex]\angle 5[/latex]
- [latex]\angle 8[/latex] and [latex]\angle 7[/latex]

**Example 2:** Name the pairs of corresponding angles and their location.

As you can see, the transversal cuts across two non-parallel lines forming 4 corresponding angles. Always remember that in this case, though the angles are located in corresponding positions relative to the two lines, they are not congruent.

The corresponding angles are:

- [latex]\angle 3[/latex] and [latex]\angle 5[/latex] – above the line, left of the transversal
- [latex]\angle 4[/latex] and [latex]\angle 6[/latex] – below the line, left of the transversal
- [latex]\angle 2[/latex] and [latex]\angle 8[/latex] – above the line, right of the transversal
- [latex]\angle 1[/latex] and [latex]\angle 7[/latex] – below the line, right of the transversal

**Example 3:** Find the measure of [latex]\angle 8[/latex].

Since line [latex]h[/latex] and line [latex]w[/latex] are parallel lines, the measure of [latex]\angle 8[/latex] should be the same as its corresponding angle. Hence, the [latex]m\angle 8[/latex] is [latex]120^\circ[/latex].

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