Let's draw the figure for ourselves and then we can start classifying each of the angles.

Corresponding Angles

If two parallel lines are cut by a transversal, any corresponding angles are congruent. Examining the figure, we notice that $∠ 1$ and $∠ 5$ are corresponding angles. If $m ∠ 1 = 8 0^{\circ},$ it follows that $m ∠ 5 = 8 0^{\circ} .$

Alternate Interior Angles

If two parallel lines are cut by a transversal, any alternate interior angles are congruent. Examining the figure, we notice that $∠ 5$ and $∠ 3$ are alternate interior angles. Therefore, if $m ∠ 5 = 8 0^{\circ},$ we must have $m ∠ 3 = 8 0^{\circ} .$

Alternate Exterior Angles

If two parallel lines are cut by a transversal, any alternate exterior angles are congruent. Examining the figure, we notice that $∠ 1$ and $∠ 7$ are alternate exterior angles. Therefore, if $m ∠ 1 = 8 0^{\circ},$ it is also true that $m ∠ 7 = 8 0^{\circ} .$

Consecutive Interior Angles

If two parallel lines are cut by a transversal, any consecutive interior angles are congruent. Examining the figure, we notice that

∠ 2

and

∠ 5

are consecutive interior angles and so are

∠ 3

and

∠ 8

. Therefore, if

m ∠ 3 = 8 0^{\circ}

and

m ∠ 5 = 8 0^{\circ}

we can solve for these unknown angles.

m ∠ 2 + 8 0^{\circ} m ∠ 8 + 8 0^{\circ} = 18 0^{\circ} \Leftrightarrow m ∠ 2 = 10 0^{\circ} = 18 0^{\circ} \Leftrightarrow m ∠ 8 = 10 0^{\circ}

Let's mark these angles in our figure.

Remaining Angles

The remaining angles can be determined in a number of different ways. For example, we can use the fact that

∠ 4

and

∠ 1

are a linear pair and so are

∠ 5

and

∠ 6 .

This means that they are supplementary angles and their measures add up to

180 .

m ∠ 4 + 8 0^{\circ} m ∠ 6 + 8 0^{\circ} = 18 0^{\circ} \Leftrightarrow m ∠ 4 = 10 0^{\circ} = 18 0^{\circ} \Leftrightarrow m ∠ 6 = 10 0^{\circ}

Let's finish our figure.

Exercise

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