This document discusses proving that lines are parallel. It defines six postulates and theorems related to parallel lines, including the converse of corresponding angles, alternate exterior angles, consecutive interior angles, and alternate interior angles. An example problem demonstrates using the corresponding angles converse to show that two lines are parallel based on congruent corresponding angles. A second example finds the measure of an angle given that two lines are parallel based on the alternate interior angles theorem.
LL- If the two legs of a right triangle are congruent to the two legs of another triangle, then the triangles are congruent.
LA-If the leg and an acute angle of one right triangle are congruent to the corresponding leg and an acute angle of another right triangle, then the right triangles are congruent.
If two angles and a non-included side of one triangle are congruent to the corresponding two angles and a non-included side of another triangle, then the triangles are congruent.
LL- If the two legs of a right triangle are congruent to the two legs of another triangle, then the triangles are congruent.
LA-If the leg and an acute angle of one right triangle are congruent to the corresponding leg and an acute angle of another right triangle, then the right triangles are congruent.
If two angles and a non-included side of one triangle are congruent to the corresponding two angles and a non-included side of another triangle, then the triangles are congruent.
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4. Postulates & Theorems
1. Converse of Corresponding Angles Postulate
If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines are
parallel.
6. Postulates & Theorems
2. Parallel Postulate
If given a line and a point not on the line, then there
exists exactly one line through the point that is parallel
to the given line.
8. Postulates & Theorems
3. Alternate Exterior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of alternate exterior angles is congruent, then the
two lines are parallel.
10. Postulates & Theorems
4. Consecutive Interior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of consecutive interior angles is supplementary,
then the lines are parallel.
12. Postulates & Theorems
5. Alternate Interior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of alternate interior angles is congruent, then the
two lines are parallel.
14. Postulates & Theorems
6. Perpendicular Transversal Converse
In a plane, if two lines are perpendicular to the same
line, then they are parallel.
15. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
a. ∠1≅ ∠3
16. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
a. ∠1≅ ∠3
a ! b since these
congruent angles are
also corresponding,
the Corresponding
Angles Converse
holds
17. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
b. m∠1= 103° and m∠4 = 100°
18. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
b. m∠1= 103° and m∠4 = 100°
are
alternate interior
angles, but not
congruent, so a is not
parallel to c
∠1and ∠4
