How to Solve the Unknown Variable of Triangles With Parallel Lines & Theorems

A transversal is a line that crosses two parallel lines.
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There are several theorems in geometry that describe the relationship of angles formed by a line that transverses two parallel lines. If you know the measures of some of the angles formed by the transversal of two parallel lines, you can use these theorems to solve for the measure of other angles in the diagram. Use the Triangle Angle Sum theorem to solve for additional angles in the triangle.

    Determine the two lines you need to prove are parallel. These will usually be lines that form angles with known measures as well an unknown angle in the triangle with the variable you need to solve.

    Identify a transverse line to the two lines you need to prove are parallel. This is a line that intersects both of the two lines.

    Prove that the lines are parallel using one of the parallel line transversal theorems and postulates. The Corresponding Angles postulate states that if corresponding angles in a transversal are congruent, the lines are parallel. The Alternate Interior Angles theorem and Alternate Interior Angles theorem state that if alternate interior or angles are congruent, the two lines are parallel. The Same-Side Interior theorem states that if same-side interior angles are supplementary, then the lines are parallel.

    Use the converses of the parallel line transversal theorems to solve for the values of other angles in the triangle. For example, the converse of the Corresponding Angles postulate states that if two lines are parallel, then corresponding angles are congruent. Therefore, if one angle in the diagram measures 45 degrees, its corresponding angle on the other line also measures 45 degrees.

    If necessary, use the the Triangle Angle Sum theorem to find the measures of other angles in the triangle. The Triangle Angle Sum theorem states that the sum of the three angles of a triangle is always 180 degrees. If you know the measures of two angles in a triangle, subtract the sum of the two angles from 180 to find the measure of the third angle.

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