Chapter 3 Guided Notes Parallel and Perpendicular ?· Chapter 3 Guided Notes Parallel and Perpendicular…

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<ul><li><p>Geometry Winter Semester (2012-13) </p><p>Name: ______________ </p><p>Chapter 3 Guided Notes </p><p>Parallel and Perpendicular Lines </p><p> Chapter Start Date:_____ </p><p>Chapter End Date:_____ Chapter Test Date:_____ </p></li><li><p>CH.3 Guided Notes, page 2 </p><p>3.1 Identify Pairs of Lines and Angles </p><p>Term Definition Example </p><p> parallel lines (// or ||) </p><p> parallel to (// or ||) </p><p> not parallel to </p><p>(// or ||) </p><p> skew lines </p><p> parallel planes </p><p>What is a Named Theorem ? </p><p>Postulate 13 Parallel Postulate </p><p>If there is a line and a point not on the line, </p><p>then there is exactly one line through the </p><p>point parallel to the given line. </p><p>Postulate 14 Perpendicular </p><p>Postulate </p><p>If there is a line and a point not on the line, </p><p>then there is exactly one line through the </p><p>point perpendicular to the given line. </p><p>transversal </p><p> The lines the transversal intersects do not </p><p>need to be parallel; the transversal can also </p><p>be a ray or line segment. </p></li><li><p>CH.3 Guided Notes, page 3 </p><p>Special Angles formed by Transversals </p><p>exterior angles </p><p> interior angles </p><p>corresponding angles </p><p>alternate interior angles </p><p>alternate exterior angles </p><p>consecutive (same-side) </p><p>interior angles </p><p>consecutive (same-side) </p><p>exterior angles </p></li><li><p>CH.3 Guided Notes, page 4 </p><p>3.2 Use Parallel Lines and Transversals </p><p>Term Definition Example </p><p>Postulate 15 Corresponding </p><p>Angles Postulate (not named) </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of corresponding angles are congruent. </p><p>Proof Abbrieviation: </p><p>Theorem 3.1 Alternate </p><p>Interior Angles Theorem </p><p>(not named) </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of alternate interior angles are </p><p>congruent. </p><p>Proof Abbrieviation: </p><p>Theorem 3.2 Alternate </p><p>Exterior Angles Theorem </p><p>(not named) </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of alternate exterior angles are </p><p>congruent. </p><p>Proof Abbrieviation: </p><p>Theorem 3.3 Same-Side </p><p>Interior Angles Theorem </p><p>(not named) </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of Same-Side Interior (AKA Consecutive </p><p>Interior) angles are supplementary. </p><p>Proof Abbrieviation: </p><p>BONUS Theorem Same-Side </p><p>Exterior Angles Theorem </p><p>(not named) </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of Same-Side Exterior angles are </p><p>supplementary. </p><p>Proof Abbrieviation: </p></li><li><p>CH.3 Guided Notes, page 5 </p><p>3.3 Prove Lines are Parallel </p><p>Term Definition Example </p><p>Postulate 16 Corresponding </p><p>Angles Converse (not named) </p><p>If two lines are cut by a transversal so the </p><p>corresponding angles are congruent, then the lines </p><p>are parallel. </p><p>Proof Abbrieviation: </p><p>Theorem 3.4 Alternate </p><p>Interior Angles Converse </p><p>(not named) </p><p>If two lines are cut by a transversal so the </p><p>alternate interior angles are congruent, then the </p><p>lines are parallel. </p><p>Proof Abbrieviation: </p><p>Theorem 3.5 Alternate </p><p>Exterior Angles Converse </p><p>(not named) </p><p>If two lines are cut by a transversal so the </p><p>alternate exterior angles are congruent, then the </p><p>lines are parallel. </p><p>Proof Abbrieviation: </p><p>Theorem 3.6 Same-Side </p><p> Interior Angles Converse </p><p>(not named) </p><p>If two lines are cut by a transversal so the Same-</p><p>Side (Consecutive) Interior angles are </p><p>supplementary, then the lines are parallel. </p><p>Proof Abbrieviation: </p><p>BONUS Theorem Same-Side </p><p>Exterior Angles Converse </p><p>If two parallel lines are cut by a transversal, then </p><p>the pairs of Same-Side Exterior angles are </p><p>supplementary. </p><p>Proof Abbrieviation: </p><p> paragraph proof </p></li><li><p>CH.3 Guided Notes, page 6 </p><p>Theorem 3.7 Transitive Property of Parallel Lines </p><p>If two lines are parallel to the same line, then they </p><p>are parallel to each other. </p></li><li><p>CH.3 Guided Notes, page 7 </p><p>3.4 Find and Use Slopes of Lines </p><p>Term Definition Example </p><p>slope </p><p> positive slope </p><p> negative slope </p><p>zero slope (slope of zero) </p><p>(no slope) </p><p>A horizontal line. </p><p> undefined slope </p><p>A vertical line. </p><p> Postulate 17 Slopes of </p><p>Parallel Lines </p><p>In a coordinate plane, two nonvertical lines </p><p>are parallel if and only if they have the </p><p>same slope. </p><p>Any two vertical lines are parallel! </p><p>Postulate 18 Slopes of </p><p>Perpendicular Lines </p><p>In a coordinate plane, two nonvertical lines </p><p>are perpendicular if and only if the product </p><p>of their slopes is -1. </p><p>The slopes of the two lines that are </p><p>perpendicular are negative reciprocals of </p><p>each other. Horizontal lines are </p><p>perpendicular to vertical lines. </p><p>if and only if form (iff) </p><p>The form used when both a conditional and </p><p>its converse are true. </p></li><li><p>CH.3 Guided Notes, page 8 </p><p>3.5 Write and Graph Equations of Lines </p><p>Term Definition Example </p><p> slope-intercept </p><p>form </p><p> standard form </p><p> x-intercept </p><p> y-intercept </p></li><li><p>CH.3 Guided Notes, page 9 </p><p>Chap3 Constructing Parallel &amp; Perpendicular Lines Remember that the complete construction guide (all 7 Basic constructions) has been posted online at www.behmermath.weebly.com 4. Construct the perpendicular bisector of a line segment. </p><p>Or, construct/find the midpoint of a line segment. </p><p>1. Begin with line segment XY. YX </p><p>2. Place the compass at point X. Adjust the compass radius so that it is more than ()XY. Draw two arcs as shown here. </p><p>YX</p><p>3. Without changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B. </p><p>A</p><p>B</p><p>YX</p><p>4. Using the straightedge, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY. </p><p>X Y</p><p>A</p><p>B</p><p>M</p></li><li><p>CH.3 Guided Notes, page 10 5. Given a point (P) ON a line (k), construct a line through P, perpendicular to k. </p><p>1. Begin with line k, containing point P. kP </p><p>2. Place the compass on point P. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y. </p><p>k</p><p>P YX </p><p>3. Place the compass at point X. Adjust the compass radius so that it is more than ()XY. Draw an arc as shown here. k</p><p>P YX </p><p>4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point A. k</p><p>A</p><p>P YX </p><p>5. Use the straightedge to draw line AP. Line AP is perpendicular to line k. </p><p>k</p><p>A</p><p>X YP</p></li><li><p>CH.3 Guided Notes, page 11 6. Given a point (R), NOT ON a line (k), construct a line through R, perpendicular to k. </p><p>1. Begin with point line k and point R, not on the line. k</p><p>R</p><p>2. Place the compass on point R. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y. </p><p>kR</p><p>YX </p><p>3. Place the compass at point X. Adjust the compass radius so that it is more than ()XY. Draw an arc as shown here. </p><p>kYX</p><p>R</p><p>4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point B. </p><p>k</p><p>B</p><p>YX</p><p>R</p><p>5. Use the straightedge to draw line RB. Line RB is perpendicular to line k. </p><p>k</p><p>B</p><p>YX</p><p>R</p></li><li><p>CH.3 Guided Notes, page 12 7. Given a line &amp; a point not on the line, construct a line through the point, parallel to the given line. </p><p>1. Begin with point P and line k. </p><p>k</p><p>P</p><p>2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection. </p><p>kQ</p><p>P</p><p>3. Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc. </p><p>kQ</p><p>P</p><p>4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R. k</p><p>R</p><p>Q</p><p>P</p><p>5. Line PR is parallel to line k. </p><p>k</p><p>R</p><p>Q</p><p>P</p></li><li><p>CH.3 Guided Notes, page 13 </p><p>3.6 Prove Theorems about Perpendicular Lines </p><p>Term Definition Example </p><p> Theorem 3.8 (not named) </p><p>If two lines intersect to form a linear pair </p><p>of congruent angles, then the lines are </p><p>perpendicular. </p><p> Theorem 3.9 (not named) </p><p>If two lines are perpendicular, then they </p><p>intersect to form four right angles. </p><p> Theorem 3.10 (not named) </p><p>If two sides of two adjacent acute angles </p><p>are perpendicular, then the angles are </p><p>complementary. </p><p>Theorem 3.11 Perpendicular Transversal Theorem </p><p>(not named) </p><p>If a transversal is perpendicular to one of </p><p>two parallel lines, then it is perpendicular to </p><p>the other. </p><p>Theorem 3.12 Lines </p><p>Perpendicular to a Transversal </p><p>Theorem (not named) </p><p>In a plane, if two lines are perpendicular to </p><p>the same line, then they are parallel to each </p><p>other. </p><p>distance from a point to a line </p><p>distance between two parallel lines </p></li></ul>

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