Section 5: Proving Lines Parallel
Converse of Corresponding Angles Postulate:
If two lines are cut by a transversal so corresponding angles are congruent, the lines are identified as parallel.
Parallel Postulate:
Given a line and a point not on that line, there is one and only one line through the given point parallel to the given line.
Alternate Exterior Angles Converse:
When two parallel lines are cut by a transversal, the alternate exterior angles are congruent.
Consecutive Interior Angles Converse:
If two lines are cut by a transversal so that consecutive interior angles are supplementary, the lines are parallel
Alternate Interior Angles Converse:
If two lines are cut by a transversal, such that the pair of alternate interior angles is congruent, then the lines are parallel.
Perpendicular Transversal Converse:
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also
Given the following information, is it possible to prove that the lines shown in the image below are parallel? If they are, identify the theorem or postulate you used to justify your answer.
1. <3 ≅ <8
<3 and <8 are alternate exterior angles of line l and n.
Since <3 ≅ <8, l is || to n by the Converse of the Alternate Exterior Angles Theorem.
2. < 6 ≅ <7
<6 and <7 are Consecutive Interior angles of lines m and n.
Since <6 and <7, m is || to n by the Converse of the
consecutive Interior angles.
<3 and <8 are alternate exterior angles of line l and n.
Since <3 ≅ <8, l is || to n by the Converse of the Alternate Exterior Angles Theorem.
2. < 6 ≅ <7
<6 and <7 are Consecutive Interior angles of lines m and n.
Since <6 and <7, m is || to n by the Converse of the
consecutive Interior angles.
Real-World Problem:
Based on the image shown here, The angles are consecutive interior angles. For lines m and n to be parallel, consecutive interior angles must be supplementary , according to the Conserve of Consecutive Interior Angles Theorem.
(5x-6) + (9x-5) = 180 Definition of supplementary
14x -11 = 180 Simplify
14x= 191 Subtract 11 from each side.
x= 13.642 Divide 14 each side.
(5x-6) + (9x-5) = 180 Definition of supplementary
14x -11 = 180 Simplify
14x= 191 Subtract 11 from each side.
x= 13.642 Divide 14 each side.