Parallel and Perpendicular Lines

Transcription

1 Name: Chapter 3 Guided Notes arallel and erpendicular Lines Chapter Start Date: Chapter End Date: Chapter Test Date: Geometry Winter Semester ( )

2 3.1 Identify airs of Lines and Angles CH.3 Guided Notes, page 2 Term Definition Example parallel lines (// or ) parallel to (// or ) not parallel to (// or ) sew lines parallel planes What is a Named Theorem? ostulate 13 arallel ostulate ostulate 14 erpendicular ostulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. transversal The lines the transversal intersects do not need to be parallel; the transversal can also be a ray or line segment.

3 CH.3 Guided Notes, page 3 Special Angles formed by Transversals exterior angles interior angles corresponding angles alternate interior angles alternate exterior angles consecutive (same-side) interior angles consecutive (same-side) exterior angles

4 3.2 Use arallel Lines and Transversals CH.3 Guided Notes, page 4 Term Definition Example ostulate 15 Corresponding Angles ostulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. roof Abbrieviation: Theorem 3.1 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. roof Abbrieviation: Theorem 3.2 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. roof Abbrieviation: Theorem 3.3 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of Same-Side Interior (AKA Consecutive Interior) angles are supplementary. roof Abbrieviation: BONUS Theorem Same-Side Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of Same-Side Exterior angles are supplementary. roof Abbrieviation:

5 3.3 rove Lines are arallel CH.3 Guided Notes, page 5 Term Definition Example ostulate 16 Corresponding Angles Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. roof Abbrieviation: Theorem 3.4 Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. roof Abbrieviation: Theorem 3.5 Alternate Exterior Angles Converse If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel. roof Abbrieviation: Theorem 3.6 Same-Side Interior Angles Converse If two lines are cut by a transversal so the Same- Side (Consecutive) Interior angles are supplementary, then the lines are parallel. roof Abbrieviation: BONUS Theorem Same-Side Exterior Angles Converse If two parallel lines are cut by a transversal, then the pairs of Same-Side Exterior angles are supplementary. roof Abbrieviation: paragraph proof

6 CH.3 Guided Notes, page 6 Theorem 3.7 Transitive roperty of arallel Lines If two lines are parallel to the same line, then they are parallel to each other.

7 3.4 Find and Use Slopes of Lines CH.3 Guided Notes, page 7 Term Definition Example slope positive slope negative slope zero slope (slope of zero) (no slope) A horizontal line. undefined slope A vertical line. ostulate 17 Slopes of arallel Lines In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel! In a coordinate plane, two nonvertical lines ostulate 18 Slopes of erpendicular Lines if and only if form (iff) are perpendicular if and only if the product of their slopes is -1. The slopes of the two lines that are perpendicular are negative reciprocals of each other. Horizontal lines are perpendicular to vertical lines. The form used when both a conditional and its converse are true.

8 3.5 Write and Graph Equations of Lines CH.3 Guided Notes, page 8 Term Definition Example slope-intercept form standard form x-intercept y-intercept

9 CH.3 Guided Notes, page 9 Chap3 Constructing arallel & erpendicular Lines emember that the complete construction guide (all 7 Basic constructions) has been posted online at 4. Construct the perpendicular bisector of a line segment. Or, construct/find the midpoint of a line segment. 1. Begin with line segment. 2. lace the compass at point. Adjust the compass radius so that it is more than (½). Draw two arcs as shown here. 3. Without changing the compass radius, place the compass on point. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B. A B 4. Using the straightedge, draw line AB. Label the intersection point M. oint M is the midpoint of line segment, and line AB is perpendicular to line segment. A M B

10 CH.3 Guided Notes, page Given a point () ON a line (), construct a line through, perpendicular to. 1. Begin with line, containing point. 2. lace the compass on point. Using an arbitrary radius, draw arcs intersecting line at two points. Label the intersection points and. 3. lace the compass at point. Adjust the compass radius so that it is more than (½). Draw an arc as shown here. 4. Without changing the compass radius, place the compass on point. Draw an arc intersecting the previously drawn arc. Label the intersection point A. A 5. Use the straightedge to draw line A. Line A is perpendicular to line. A

11 CH.3 Guided Notes, page Given a point (), NOT ON a line (), construct a line through, perpendicular to. 1. Begin with point line and point, not on the line. 2. lace the compass on point. Using an arbitrary radius, draw arcs intersecting line at two points. Label the intersection points and. 3. lace the compass at point. Adjust the compass radius so that it is more than (½). Draw an arc as shown here. 4. Without changing the compass radius, place the compass on point. Draw an arc intersecting the previously drawn arc. Label the intersection point B. B 5. Use the straightedge to draw line B. Line B is perpendicular to line. B

12 CH.3 Guided Notes, page Given a line & a point not on the line, construct a line through the point, parallel to the given line. 1. Begin with point and line. 2. Draw an arbitrary line through point, intersecting line. Call the intersection point Q. Now the tas is to construct an angle with vertex, congruent to the angle of intersection. Q 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 and draw another arc. Q 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 Q. Mar the arc intersection point. Q 5. Line is parallel to line. Q

13 CH.3 Guided Notes, page rove Theorems about erpendicular Lines Term Definition Example Theorem 3.8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles. Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Theorem 3.11 erpendicular Transversal Theorem Theorem 3.12 Lines erpendicular to a Transversal Theorem distance from a point to a line If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. distance between two parallel lines