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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 117 lessons
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Vertical angles are a pair of non-adjacent angles formed when two lines intersect. We see intersecting lines all the time in our real world. Here, we see two vapor trails that intersect. Therefore, they have created the pair of vertical angles labeled as 1 and 2.
Here is a pair of vertical angles formed in nature and that are more terrestrial.
If we draw a pair of intersecting lines, we have created two pairs of vertical angles. Here, angles AOC and BOD are a pair of vertical angles. Angles AOB and COD are also a pair of vertical angles.
Notice that vertical angles are never adjacent angles. In other words, they never share a side. For example, angles AOC and AOB are not a pair vertical angles, but they are adjacent angles. However, vertical angles always have a common vertex. Here, each pair of vertical angles share vertex O.
Let's look at some more examples of vertical angles.
Line c intersects two lines, a and b. Vertical angles are formed at each intersection. The vertical pairs of angles are as follows:
1 and 6
2 and 5
3 and 8
4 and 7
A primary property of vertical angles is that they are congruent. In other words, they have the same angle measure. Here, if we add in the angle measures, we'll see that vertical angles are congruent.
Let's do a simple proof for this. Before we begin, we should acknowledge some definitions and theorems in geometry. First of all, a linear pair of angles is a pair of adjacent angles. Their non-common sides are always opposite rays. In addition, angles that form a linear pair are also supplementary, so their sum is always 180 degrees. Here is our proof.
1. Lines m and n intersect forming angles 1, 2, 3, and 4 (given).
2. Angles 1 and 2 are a linear pair, so they are supplementary (definition of linear pair).
3. Angle 1 + angle 2 = 180 degrees (definition of supplementary angles).
4. Angles 2 and 3 are a linear pair, so they are supplementary (definition of linear pair).
5. Angle 2 + angle 3 = 180 degrees (definition of supplementary angles).
6. Angle 1 + angle 2 = angle 2 + angle 3 (substitution; see statements 3 and 5).
7. Angle 1 = angle 3 (subtract angle 2 from the equation in statement 6).
QED (our proof is complete)
Let's now complete a problem! If angle 1 is 115 degrees, what are the measures of the other angles?
The measure of angle 3 is 115 degrees because angles 1 and 3 are a pair of vertical angles. Angles 1 and 2 are a linear pair, so their sum is 180 degrees; therefore, the measure of angle 2 is 180 - 115 = 65 degrees. The measure of angle 4 is 65 degrees because angles 2 and 4 are a pair of vertical angles.
Let's finish this lesson by showing another non-example of vertical angles. Here, angles 1 and 3 are not a pair of vertical angles. Even though they share a vertex and are not adjacent, they are not formed by the same pair of intersecting lines. Angle 1 is formed by lines r and t while angle 3 is formed by lines s and t.
Whenever two lines intersect, they form two pairs of vertical angles. Vertical angles have a common vertex, but they are never adjacent angles. Finally, vertical angles are always congruent.
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NY Regents Exam - Geometry: Tutoring Solution10 chapters | 117 lessons