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Euclid's Fifth Postulate

Definition (Parallel Lines) Two distinct lines and are said to be parallel when they lie in the same plane and they do not meet.

Definition (Transversal) If lines and all lie in the same plane and intersects and then is called a transversal of lines and

Theorem (Parallelism) If two lines in the same plane are cut by a transversal so that a pair of alternating interior angles are congruent, then the lines are parallel.

    Proof.  We are given two lines and meeting line at points and respectively. We also are given a pair of alternating interior angles, say and with Assume for a contradiction, that is not parallel to By definition of parallel lines, and meet at some point By definition of  exterior angles, is an exterior angle of By the Exterior Angle Inequality Theorem, since We have reached the contradiction:, and because of the trichotomy property of the real numbers. Therefore, in fact and must be parallel.

Axiom (Euclid's Fifth Postulate) If is any line and is any point not on there exists in the plane of and one and only one line that passes through and is parallel to

Theorem (Transversal Properties) Here are some properties of transversals.

    (i) If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.
    
    (ii) If two lines in the same plane are cut by a transversal, then the two lines are parallel if and only if a pair of interior angles on the same side of the transversal are supplementary.
    
    (iii) If two lines in the same plane are cut by a transversal, then the two lines are parallel if and only if a pair of corresponding angles are congruent.
    
    (iv) If two lines in the same plane are cut by a transversal, then the lines are parallel if and only if a pair of alternating interior angles are congruent.
    
    (v) If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also.

Theorem (Euclidean Exterior Angle)

    (i) The measure of an exterior angle of any triangle equals the sum of the measure of the two opposite interior angles.    

    (ii) The sum of the measures of the angles of any triangle is 180.
    
    Proof. Say we are given triangle with exterior angle we can construct segment so that the midpoint of is also the midpoint of . Since and we know that by Therefore, Then using the above properties,

Theorem (Midpoint Connector)

    (i) The segment joining the midpoints of two sides of a triangle is parallel to the third side and has length one-half that of the third side.

    (ii) If a line bisects one side of a triangle and is parallel to the second, it also bisects the third side.

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