Question

13 explain how you can use the measures of alternate exterior angles to prove that two lines are parallel. 14. if two parallel lines are cut by a transversal, what is the relationship between the measures of consecutive interior angles? 15. are consecutive interior angles located on the same side or opposite sides of the transversal? are they located on the interior or exterior of the parallel lines? 16 two parallel lines are intersected by a transversal. one of the consecutive interior angles is represented by the expression (3x + 10) and the other by (2x + 40). find the value of x.

Explanation:

Step1: Recall alternate - exterior angles property

If the alternate - exterior angles formed when two lines are cut by a transversal are congruent, then the two lines are parallel. This is a postulate in geometry.

Step2: Recall consecutive - interior angles property for parallel lines

When two parallel lines are cut by a transversal, consecutive - interior angles are supplementary. That is, the sum of their measures is 180°.

Step3: Determine location of consecutive - interior angles

Consecutive - interior angles are located on the same side of the transversal and on the interior of the parallel lines.

Step4: Solve for x in problem 16

Since consecutive - interior angles of parallel lines are supplementary, we set up the equation:
$(3x + 10)+(2x + 40)=180$.
First, combine like terms:
$3x+2x+10 + 40=180$,
$5x+50 = 180$.
Then, subtract 50 from both sides:
$5x=180 - 50$,
$5x=130$.
Finally, divide both sides by 5:
$x=\frac{130}{5}=26$.

Answer:

1. If the alternate - exterior angles formed by two lines and a transversal are congruent, the two lines are parallel.
2. Consecutive - interior angles are supplementary (sum to 180°).
3. Consecutive - interior angles are on the same side of the transversal and on the interior of the parallel lines.
4. $x = 26$