3.1 Identify Pairs Of Lines And Angles

3.1 Identify Pairs of Lines and AnglesObj.: Identify angle pairs formed by three intersecting lines.Key Vocabulary Parallel lines - Two lines are parallel lines if they do not intersect and arecoplanar. Skew lines - Two lines are skew lines if they do not intersect and are not coplanar. Parallel planes - Two planes that do not intersect are parallel planes. Transversal - A transversal is a line that intersects two or more coplanar lines atdifferent points. Corresponding angles Alternate interior anglesTwo angles are correspondingangles if they have correspondingpositions. For example, 2 and 6 are above the lines and to theright of the transversal t. Alternate exterior angles-Two angles are alternate interiorangles if they lie between the twolines and on opposite sides of thetransversal.Two angles are alternate exteriorangles if they lie outside the twolines and on opposite sides of thetransversal.Two angles are consecutiveinterior angles if they lie betweenthe two lines and on the sameside of the transversal. Consecutive interior angles -Lines m and n are parallel lines (m // n).Lines m and k are skew lines.Planes T and U are parallel planes (T // U).Lines k and n are intersecting lines, andthere is a plane (not shown) containing them.Parallel PostulateIf there is a line and a point not on the line, then there isexactly one line through the point parallel to the given line.There is exactly one line through P parallel to l.Perpendicular PostulateIf there is a line and a point not on the line, then there isexactly one line through the point perpendicular to the given line.There is exactly one line through P perpendicular to l.

EXAMPLE 1 Identify relationships in spaceThink of each segment in the figure as part of a line. Whichline(s) or plane(s) in the figure appear to fit the description?a. Line(s) parallel to AF and containing point Eb. Line(s) skew to AF and containing point Ec. Line(s) perpendicular to AF and containing point Ed. Plane(s) parallel to plane FGH and containing point ESolutionEXAMPLE 2 Identify parallel and perpendicular linesUse the diagram at the right to answer each question.a. Name a pair of parallel lines.b. Name a pair of perpendicular lines.c. Is AB BC ? Explain.SolutionEXAMPLE 3 Identify angle relationshipsIdentify all pairs of angles of the given type. a. Corresponding,b. Alternate interior, c. Alternate exterior, d. Consecutive interior,Solution(3.1 cont.)

3.2 Use Parallel Lines and TransversalsObj.: Use angles formed by parallel lines and transversals.Corresponding Angles PostulateIf two parallel lines are cut by a transversal, then thepairs of corresponding angles are congruent. 2 6Alternate Interior Angles TheoremIf two parallel lines are cut by a transversal, thenthe pairs of alternate interior angles are congruent. 4 5Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, thenthe pairs of alternate exterior angles are congruent. 1 8Consecutive Interior Angles TheoremIf two parallel lines are cut by a transversal,then the pairs of consecutive interior angles aresupplementary. 3 & 5 supplementaryEXAMPLE 1 Identify congruent anglesThe measure of three of the numbered angles is 125⁰. Identifythe angles. Explain your reasoning.Solution

EXAMPLE 2 Use properties of parallel linesALGEBRA Find the value of x.Solution33EXAMPLE 3 Solve a real-world problemRunways A taxiway is being constructed that intersects twoparallel runways at an airport. You know that m 2 98⁰ .What is the m 1? How do you know?Solution

3.3 Prove Lines are ParallelObj.: Use angle relationships to prove that lines are parallel.Key Vocabulary Paragraph proof - A proof can also be written in paragraph form, called aparagraph proof. Converse - To write the converse of a conditional statement, exchange thehypothesis and conclusion. Two-column proof - A two-column proof has numbered statements andcorresponding reasons that show an argument in a logical order.Corresponding Angles ConverseIf two lines are cut by a transversal so the correspondingangles are congruent, then the lines are parallel.j // kAlternate Interior Angles Converse

If two lines are cut by a transversal so the alternateinterior angles are congruent, then the lines are parallel.j // kAlternate Exterior Angles ConverseIf two lines are cut by a transversal so the alternateexterior angles are congruent, then the lines are parallel.j // kConsecutive Interior Angles ConverseIf two lines are cut by a transversal so the consecutiveinterior angles are supplementary, then the lines are parallelIf 3 and 5 aresupplementary, then j // k.Transitive Property of Parallel LinesIf two lines are parallel to the same line,then they are parallel to each other.If p // q and q // r, then p // r.EXAMPLE 1 Apply the Corresponding Angles Converse7ALGEBRA Find the value of x that makes m // n.SolutionEXAMPLE 2 Solve a real-world problemFlags How can you tell whether the sides of the flag of Nepal are parallel?SolutionEXAMPLE 3 Prove the Alternate Interior Angles Converse(3.3 cont.)

In the figure, a // b and 1 is congruent to 3. Prove x // y.SolutionEXAMPLE 4 Use the Transitive Property of Parallel LinesUtility poles Each utility pole shown is parallelto the pole immediately to its right. Explain whythe leftmost pole is parallel to the rightmost pole.Solution7

angles if they have corresponding angles if they lie between the two positions. For example, 2 and lines and on opposite sides of the 6 are above the lines and to the transversal. right of the transversal t. Alternate exterior angles- Consecutive interior angles - Two angles are alternate exterior Two