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Converse of Parallel Lines Theorem - Problem 1
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By applying the converse of the alternating interior angles parallel lines theorem, you can determine the angle measure of two angles so that the lines are parallel.

If the measures of the angles are given with expressions of variables, set these expressions equal to one another: recall that the parallel lines theorem says that alternating interior angles of parallel lines are congruent. So, if these angles are congruent, the lines are parallel. Solve for y. Then, plug this value back into one of the expressions to get the measure of the angle needed for these lines to be parallel.

You can apply the Converse of the Parallel Lines Theorem to a problem like this which asks for what values of x are these lines parallel? Well first you have to ask what type of angles are these? Well they’re in between the lines and they’re on opposite sides of the transversal, so these are alternate interior angles.

If alternate interior angles are congruent or equal, then these 2 lines must be parallel. So to solve this all we have to do is set those 2 equal to each other. We can say that 4x is equal to x+10, so to solve this we’re going to subtract x, 3x is equal to 10 and if you divide by 3, you get x equals 10/3, so for what value of x are these lines parallel? 10/3 and what do we do? We said that if alternate interior angles are congruent, then these 2 lines must be parallel.

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