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Warm Up x Given: b ∥ c, ∠1 and ∠2 are supplementary 2 1 Find x. Explain your reasoning. (No proof required.) 70 a b c Practice Given: a ∥ b, ∠1 and ∠2 are supplementary Prove: ∠3 ≅ ∠4 3 a b 1 2 4 c Quick Review Parallel Postulate: If there is a line and a point not on that line, there there is exactly one line through the point and parallel to the original line. Perpendicular Postulate: If there is a line and a point not on that line, there is exactly one line through the point and perpendicular to the original line. Theorem 3.1: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Theorem 3.2: If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Theorem 3.3: If two lines are perpendicular, then they intersect to form four right angles. Quick Review Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Alternative Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternative interior angles are congruent. Alternative Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternative exterior angles are congruent. Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Quick Review Corresponding Angles Converse: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Converse: If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse: If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Consecutive Interior Angles Converse: If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. Quick Review Theorem 3.11: If two lines are parallel to the same line, then they are parallel to each other. (Transitivity of parallelism) Theorem 3.12: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Pseudo-Quiz Please write a 2-column proof of the following. t 1 a Given: ∠1 and ∠2 are a linear pair of congruent angles. ∠3 is a right angle. a∥c b Prove: b ∥ c c 2 3