Vertical Angles

When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, the image below shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.

In the figure below, one angle measures [latex]40^\circ[/latex]. Find the measures of the remaining angles.

 

Show Solution

The 40-degree angle and [latex]∠2[/latex] are vertical angles. Therefore, [latex]m∠2=40^\circ[/latex]. Notice that [latex]∠2[/latex] and [latex]∠1[/latex] are supplementary angles, meaning that the sum of [latex]m∠2[/latex] and [latex]m∠1[/latex] equals [latex]180^\circ[/latex]. Therefore, [latex]m∠1=180^\circ−40^\circ=140^\circ[/latex]. Since [latex]∠1[/latex] and [latex]∠3[/latex] are vertical angles, then [latex]m∠3[/latex] equals [latex]140^\circ[/latex].

Transversals

When two parallel lines are crossed by a straight line or transversal, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. In the image below, angles [latex]1[/latex], [latex]2[/latex], [latex]7[/latex], and [latex]8[/latex] are called exterior angles, and angles [latex]3[/latex], [latex]4[/latex], [latex]5[/latex], and [latex]6[/latex] are called interior angles.

Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example in the image below, [latex]∠3[/latex] and [latex]∠6[/latex] are alternate interior angles and have equal measure; [latex]∠4[/latex] and [latex]∠5[/latex] are alternate interior angles and have equal measure as well.

Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example in the image below, [latex]∠2[/latex] and [latex]∠7[/latex] are alternate exterior angles and have equal measures; [latex]∠1[/latex] and [latex]∠8[/latex] are alternate exterior angles and have equal measures as well.

Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In the image below, [latex]∠1[/latex] and [latex]∠5[/latex] are corresponding angles and have equal measures; [latex]∠3[/latex] and [latex]∠7[/latex] are corresponding angles and have equal measures; [latex]∠2[/latex] and [latex]∠6[/latex] are corresponding angles and have equal measures; [latex]∠4[/latex] and [latex]∠8[/latex] are corresponding angles and have equal measures as well.

Using the figure below and given that angle [latex]3[/latex] measures [latex]40^\circ[/latex], find the measures of the remaining angles and give a reason for your solution.

 

Show Solution

[latex]m∠2=m∠3=40^\circ[/latex] by vertical angles. [latex]m∠7=m∠3=40^\circ[/latex] by corresponding angles. [latex]m∠7=m∠6=40^\circ[/latex] by vertical angles. [latex]m∠1=180−40=140^\circ[/latex] by supplementary angles. [latex]m∠4=m∠1=140^\circ[/latex] by vertical angles. [latex]m∠8=m∠1=140^\circ[/latex] by alternate exterior angles. [latex]m∠5=m∠8=140^\circ[/latex] by vertical angles.