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Geometry: Chapter 3
Ch. 3. 4: Prove Lines are Parallel
Ch. 3.5 Using Properties of
Parallel Lines
Postulate 16: Corresponding Angles
Converse
If two lines are cut
by a transversal so
the corresponding
angles are congruent,
then the lines are parallel.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 161.
Ex. 1. Find the value of y that makes a || b.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 162.
Ex. 1 (cont.)
Solution: Lines a and b are parallel if the
marked alternate exterior angles are
congruent.
(5y +6)o =121o
5y=121-6
5y=115
y = 23
Theorem 3.8: Alternate Interior Angles
Converse
If two lines are cut
by a transversal so the
alternate interior angles
are congruent, then the
lines are parallel.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 162.
Theorem 3.9: Consecutive Interior Angles
Converse
If two lines are cut
by a transversal so the
consecutive interior
angles are supplementary,
then the lines are parallel.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 162.
Theorem 3.10: Alternate Exterior Angles
Converse
If two lines are cut
by a transversal so the
alternate exterior angles
are congruent, then the
lines are parallel.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 162.
Example 2: A woman was stenciling this
design on her kitchen walls. How can she
tell if the top and bottom are parallel?
She can measure alternate interior angles or
corresponding angles and see if they are
congruent.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 162.
Ex. 3: Prove that if 1 and 4 are
supplementary, then a||b.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 163.
Ex. 4: In the figure, a || b and 1 is congruent
to 3. Prove c || d. Use a paragraph proof.
Theorem 3.11: Transitive Property of
Parallel Lines.
If two lines are parallel to the same line,
then they are parallel to each other.
Theorem 3.12: Lines Perpendicular to a
Transversal Theorem
In a plane, if two lines are perpendicular to the
same line, then they are parallel to one
another.
If m ┴ p and n┴ p, then m || n.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 192.