Apply vertical angle relationships

Vertical angles are the opposite, nonadjacent angles formed when two lines intersect, and they have equal measures; adjacent angles form a linear pair and sum to 180°. Learners apply these relationships in diagrams and simple equations to determine unknown angle measures, distinguishing equality of vertical angles from supplementation of adjacent angles; formal proof and more advanced angle theorems are not included.

Detailed Explanation: Apply vertical angle relationships

When two lines intersect, they form two pairs of vertical angles.

Vertical angles are opposite each other, so they have equal measures.
Adjacent angles share a side. If they form a straight line, they make a linear pair and add to $180^\circ$ .

Suppose two lines intersect at point $O$ . The opposite angles are labeled:

$\angle AOC = 3x+10$
$\angle BOD = 5x-30$

Because $\angle AOC$ and $\angle BOD$ are vertical angles, set their measures equal:

3x+10=5x-30

Solve for $x$ :

40=2x

x=20

Now substitute $20$ into either angle expression:

\angle AOC=3(20)+10=70^\circ

So the vertical angle $\angle BOD$ is also:

\angle BOD=70^\circ

If $\angle AOD$ is next to $\angle AOC$ , they form a linear pair. Therefore, they add to $180^\circ$ :

\angle AOD+70^\circ=180^\circ

\angle AOD=110^\circ

Remember: use equal measures for vertical angles and use a sum of $180^\circ$ for adjacent angles that form a straight line.

Learn by doing: Apply vertical angle relationships

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Geometry of Lines - Crossing Lines Solve Angle

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Apply vertical angle relationships

Detailed Explanation: Apply vertical angle relationships

Learn by doing: Apply vertical angle relationships

Practice with unlimited practice problems

Practice:

Geometry of Lines - Crossing Lines Solve Angle