To solve this problem involving parallel lines and a transversal, let's understand how the angles are formed and how their sums are calculated.
When two parallel lines are intercepted by a transversal, several types of angles are created. Among these are corresponding angles, alternate interior angles, alternate exterior angles, and consecutive (also known as same side) interior angles. The focus here is on the interior angles.
1. Consecutive (same side) Interior Angles: These angles are on the same side of the transversal and inside the parallel lines. One of the key properties of consecutive interior angles is that they are supplementary. This means that their measures add up to 180 degrees.
Let's denote the two parallel lines by [tex]\( L_1 \)[/tex] and [tex]\( L_2 \)[/tex] and the transversal by [tex]\( T \)[/tex].
2. Supplementary Nature of Consecutive Interior Angles: Suppose the consecutive interior angles formed by these lines and the transversal are [tex]\( \angle A \)[/tex] and [tex]\( \angle B \)[/tex]. Then:
[tex]\[
\angle A + \angle B = 180^\circ
\][/tex]
This is because consecutive interior angles are supplementary.
3. Calculating the Sum for Multiple Pairs:
Given the problem, we need to find the total sum of the measures of 10 pairs of interior angles formed by the transversal and the two parallel lines. Since each pair of consecutive interior angles adds up to 180 degrees, and we have 10 such pairs, we proceed as follows:
Let's denote the number of pairs as [tex]\( n \)[/tex], where [tex]\( n = 10 \)[/tex].
Therefore, the total sum can be calculated by multiplying the number of pairs by the sum of angles in each pair:
[tex]\[
\text{Total Sum} = n \times 180^\circ
\][/tex]
Substituting [tex]\( n = 10 \)[/tex]:
[tex]\[
\text{Total Sum} = 10 \times 180^\circ = 1800^\circ
\][/tex]
Thus, the sum of the measures of 10 pairs of interior angles formed by two parallel lines and a transversal is [tex]\( 1800^\circ \)[/tex].
