Parallel Lines and Planes Chapter 3 Parallel Lines and Planes By: Jessica House and Yana Feldman

3-1 Definitions Parallel lines A B C D l n l and n are parallel lines Parallel lines (║): coplanar lines that do not intersect Skew lines: noncoplanar lines Neither ll nor intersecting Look perpendicular but not because in 2 different planes Segments and rays contained in ║ lines are also ║ Skew lines k j j and k are skew lines P Q R S PQ and RS do not intersect, but they are parts of lines PQ and RS that do intersect. Thus PQ is NOT parallel to RS Parallel planes: do not intersect A line and a plane are ║ if they do not intersect

X Y Z n l Theorem 3-1 3-1 Theorem 3-1: If two parallel planes are cut by a third plane, then the lines of intersection are parallel. Given: Plane X ║ plane Y; plane Z intersects X in line l; plane Z intersects Y in line n. Prove: l ║ n Statements l is in Z; n is in Z l and n are coplanar l is in X; n is in Y; X ║ Y l and n do not intersect. l ║ n Reasons Given Definition of coplanar Parallel planes do not intersect. (Definition of ║ planes) Definition of ║lines (2,4)

3-1 Transversal: a line that intersects two or more coplanar lines in different points. Interior angles: angles 3, 4, 5, 6 Exterior angles: angles 1, 2, 7 ,8 Alternate interior angles: two nonadjacent interior angles on opposite sides of the transversal 3 and 6 4 and 5 Same-side interior angles: two interior angles on the same side of transversal 3 and 5 4 and 6 Corresponding angles: two angles in corresponding positions relative to the two lines 1 and 5 2 and 6 3 and 7 4 and 8 h t k 1 2 3 4 5 6 7 8

3-2 Properties of Parallel Lines Postulate 10: If two ║ lines are cut by a transversal, then corresponding angles are congruent. Theorem 3-2: If two ║ lines are cut by a transversal, then alternate interior angles are congruent. Theorem 3-2 k t n 1 2 3

3-2 Theorem 3-3 k t n 1 2 4 Theorem 3-3: If two ║ lines are cut by a transversal, then same-side interior angles are supplementary. Theorem 3-4: If a transversal is perpendicular to one of two ║lines, then it is perpendicular to the other one also. Theorem 3-4 l t n 1 2

3-2 This type of arrow usage shows that the lines are parallel (2 arrows show its talking about a different line)

3-3 Proving Lines Parallel Postulate 11: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are ║. (the following theorems of this section 3-5,3-6, and 3-7 can be deducted from this postulate) The following theorems in section 3-3 are the converses of the theorems in section 3-2 Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are ║. Theorem 3-5 k t n 1 2 3

3-3 Theorem 3-6: If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are ║. Theorem 3-7: In a plane two lines perpendicular to the same line are ║. k n 1 2 3 Theorem 3-6 t Theorem 3-7 k t n 1 2

3-3 Theorem 3-8: Through a point outside a line, there is exactly one line ║ to the given line. Theorem 3-9: Through a point outside a line, there is exactly one line perpendicular to the given line. Theorem 3-10: Two lines ║ to a third line are ║ to each other. Theorem 3-10 k l n

3-3 Ways to Prove Two Lines Parallel show that a pair of corresponding angles are congruent show that a pair of alternate interior angles are congruent show that a pair of same-side interior are supplementary in a plane show that both lines are perpendicular to a 3rd line show that both lines are parallel to a 3rd line

3-4 Angles of a Triangle Triangle: the figure formed by three segments joining three noncollinear points Each point is a vertex (plural form is vertices) Segments are sides of triangle Triangles can be classified by number of congruent sides No sides congruent At least two sides congruent All sides congruent Scalene Triangle Isosceles Triangle Equilateral Triangle Acute∆ Obtuse∆ Right∆ Equiangular∆ Three acute s One obtuse One right All s congruent

3-4 Auxiliary line: a line (or ray or segment) added to a diagram to help in a proof Auxiliary lines allow you to add additional things to your pictures in order to help in proofs-this becomes very important in proofs (in the following diagram the auxiliary line is shown as a dashed line) Theorem 3-11: The sum of the measures of the angles of a triangle is 180. A B 1 2 3 4 5 D Theorem 3-11 and auxiliary line usage C

3-4 Corollary: a statement that can be proved easily by applying a theorem. Like theorems, corollaries can be used as reasons for proofs The following 4 corollaries are based off of theorem 3-11: the sum of the measures of the angles of a triangle is 180 Corollary 1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Corollary 2: Each angle of an equiangular triangle has measure 60. Corollary 3: In a triangle there can be at most one right angle or obtuse angle. Corollary 4: The acute angles of a right triangle are complementary.

3-4 Theorem 3-12: The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. A B C 120 150 30 ° Theorem 3-12 Exterior angle Remote interior angles

3-5 Angles of a Polygon Polygon: “many angles” Formed by coplanar segments such that Each segment intersects exactly two other segments, one at each endpoint. No two segments with a common endpoint are collinear. Convex polygon: a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon

3-5 Number of Sides Name 3 4 5 6 7 8 9 10 11 12 13 14 15 n triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon undecagon dodecagon tridecagon tetracagon pentadecagon n-gon

3-5 When referring to polygons, list consecutive vertices in order Diagonal: a segment joining two nonconsecutive vertices (indicated by dashes) Finding sum of measures of angles of a polygon: draw all diagonals from one vertex to divide polygon into triangles Theorem 3-13: The sum of the measures of the angles of a convex polygon with n sides is (n - 2)180 Diagonals: 4 sides 2 triangles Angle sum= 2(180) 5 sides 3 triangles Angle sum= 3(180) 6 sides 4 triangles Angle sum= 4(180)

3-5 Theorem 3-14: The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360 Regular Polygon: must be both equiangular and equilateral 120 ° This is a hexagon that is neither equiangular nor equilateral. Not a regular polygon Equilateral triangle Not a regular polygon 120 ° Equiangular hexagon Not a regular polygon Regular hexagon

So far we have only used deductive reasoning 3-6 Inductive Reasoning Deductive Reasoning Conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given info) Conclusion MUST be true if the hypothesis is true Inductive Reasoning Conclusion based on several past observations Conclusion is PROBABLY true, but necessarily true So far we have only used deductive reasoning

The End