state and prove theorem of parallel line
Answers
1. THEOREM : If two parallel lines are cut by a transversal, the alternate interior angles are congruent.
CONVERSE : If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
PROOF : When the lines are parallel, the alternate exterior angles are equal in measure.
ie; In the given fig. 1, m∠1 = m∠2 and m∠3 = m∠4
2. THEOREM : If two parallel lines are cut by a transversal, the corresponding angles are congruent.
CONVERSE : If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
PROOF : When the lines are parallel, the corresponding angles are equal in measure.
ie; In the given fig. 2, m∠1 = m∠2 and m∠3 = m∠4
m∠5 = m∠6 and m∠7 = m∠8
3. THEOREM : If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.
CONVERSE : If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.
PROOF : When the lines are parallel, the interior angles on the same side of the transversal are supplementary.
ie; In the given fig. 3, m∠1 + m∠2 = 180
m∠3 + m∠4 = 180
4. THEOREM : Vertical angles are congruent.
PROOF : There are 4 sets of vertical angles in fig.4!
∠1 and ∠2
∠3 and ∠4
∠5 and ∠6
∠7 and ∠8
Remember: the lines need not be parallel to have vertical angles of equal measure.
5. THEOREM : If two angles form a linear pair, they are supplementary.
PROOF : Since a straight angle contains 180º, the two angles forming a linear pair also contain 180º when their measures are added (making them supplementary).
In fig. 5, m∠1 + m∠4 = 180
m∠1 + m∠3 = 180
m∠2 + m∠4 = 180
m∠2 + m∠3 = 180
m∠5 + m∠8 = 180
m∠5 + m∠7 = 180
m∠6 + m∠8 = 180
m∠6 + m∠7 = 180
