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Proving Lines Parallel Proving Lines Parallel •Determine the relationships between angle pairs formed by a transversal. •Use these relationships to prove two lines are parallel. •Language Objectives: • Explain the supplementary and congruent natures of the angle pairs iff •If two things equal the same thing, then they are equal to each other. iff •If two things equal the same thing, then they are equal to each other. •Two things equal the same thing ⇒ they are equal iff •If two things equal the same thing, then they are equal to each other. •Two things equal the same thing ⇒ they are equal •If two things equal each other, then they equal the same thing. iff •If two things equal the same thing, then they are equal to each other. •Two things equal the same thing ⇒ they are equal •If two things equal each other, then they equal the same thing. •Two things are equal ⇒ they equal the same thing Iff •Since that sentence works in both directions, we say: •Iff two things equal the same thing, then they equal each other. •Two things equal the same thing ⇔ they equal each other. Iff •If angles are vertical, then they are congruent. Iff •If angles are vertical, then they are congruent. •Angle are vertical ⇒ they are congruent Iff •If angles are vertical, then they are congruent. •Angle are vertical ⇒ they are congruent •If angles are congruent, then they are vertical. Iff •If angles are vertical, then they are congruent. •Angle are vertical ⇒ they are congruent •If angles are congruent, then they are vertical. •Angles are congruent ⇏ they are vertical. Notecards •Vertical Angles are always congruent. •If an angle or segment is bisected, both sides are congruent. Relationships Notecards •Corresponding Angles: Iff two lines are parallel, the corresponding angles of the transversal are congruent. Notecards •Alternate Interior Angles: Iff two lines are parallel, the alternate interior angles of the transversal are congruent. Notecards •Alternate Exterior Angles: Iff two lines are parallel, the alternate exterior angles of the transversal are congruent. Notecards •Consecutive Interior Angles: Iff two lines are parallel, the consecutive interior angles of the transversal are supplementary. Example 1 •Solve for 𝑥 and 𝑦: Examples 2 and 3 •Solve for 𝑥 and 𝑦. Examples 3 and 4 •Solve for 𝑥. Example 5 Example 6 Example 7 Example 8 Example 9 •Are these lines parallel? Prove why or why not. Example 10 •Are these lines parallel? Prove why or why not.