If three lines all meet in a single point, how many pairs of vertical angles are formed?
There are 3 pairs of vertical angles that are formed.
(Type a whole number.)

Question

If three lines all meet in a single point, how many pairs of vertical angles are formed? There are 3 pairs of vertical angles that are formed. (Type a whole number.)

UpStudy AI · Accepted Answer

To determine how many pairs of vertical angles are formed when three lines meet at a single point, we can follow these steps:

1. **Understanding Vertical Angles**: Vertical angles are formed when two lines intersect. Each pair of intersecting lines creates two pairs of vertical angles.

2. **Counting Intersections**: With three lines intersecting at a single point, we can visualize the situation. Each pair of lines will intersect and form vertical angles.

3. **Calculating Pairs of Lines**: The number of ways to choose 2 lines from 3 lines can be calculated using the combination formula $$ C(n, k) $$, where $$ n $$ is the total number of lines and $$ k $$ is the number of lines to choose. Here, $$ n = 3 $$ and $$ k = 2 $$.

$$
   C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3
   $$

4. **Calculating Vertical Angle Pairs**: Each pair of lines creates 1 pair of vertical angles. Since there are 3 pairs of lines, there will be 3 pairs of vertical angles.

Thus, the total number of pairs of vertical angles formed is:

$$
\text{Total pairs of vertical angles} = 3
$$

Therefore, the answer is $$ 3 $$.
There are 3 pairs of vertical angles formed.

If three lines all meet in a single point, how many pairs of vertical angles are formed?
There are 3 pairs of vertical angles that are formed.
(Type a whole number.)

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