Geometry Lesson 3 Constructing Parallel And Perpendicular .

Geometry 1Unit 3: Perpendicular andParallel Lines

Geometry 1 Unit 33.1 Lines and Angles

Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and donot intersect.

Some examples of parallel lines

Lines and Angles Skew Lines Lines that are not coplanar and do notintersect

Lines and Angles Parallel Planes Planes that do not intersectParallel capacitors

Lines and Angles Example 1ABDCName two parallellinesName two skewlinesAGEFName twoperpendicular lines

Lines and Angles Example 2 Think of each segment in the diagram as part of a line.Which of the lines appear fit the description?a.b.c.d.Parallel to TW and contains VPerpendicular to TW and contains VSkew to TW and contains VName the plane(s) that contain V and appear to beparallel to the plane TPQQRSPUTVW

Lines and Angles Parallel Postulate If there is a line and a point not on the line,then there is exactly one line through the pointparallel to the given line.PlThere is exactly one linethrough P parallel to l.

Lines and Angles Perpendicular Postulate If there is a line and a point not on the line,then there is exactly one line through the pointperpendicular to the given line.PThere is exactly one linethrough P perpendicular to l.l

Constructing Perpendicular LinesStep 1:Draw a line, and a point not on the line

Constructing Perpendicular LinesStep 2:

Constructing Perpendicular LinesStep 3:

Constructing Perpendicular LinesStep 4:

Constructing Perpendicular LinesStep 4 – completedthis is what your paper should look like

Constructing Perpendicular LinesDraw a line throughThe intersection andThe point not on theline

Constructing Perpendicular LinesYou now have two perpendicular lines – theyintersect at 90o – the line you constructedpasses through the point that you drew atthe beginningWay to Go!

Lines and Angles Transversal A line that intersects two or more coplanarlines at different points

Lines and Angles Corresponding Angles Two angles that occupy correspondingpositions357681 24

Lines and Angles Alternate Exterior Angles Angles that lie outside two lines on theopposite sides of the transversal357681 24

Lines and Angles Alternate Interior Angles Angles that lie between the two lines onopposite sides of the transversal357681 24

Lines and Angles Consecutive Interior Angles Also called same side interiorAngles that lie between two lines on the sameside of a transversal357681 24

Lines and Angles Transversal A line that intersects two or more coplanarlines at different points

Lines and Angles Corresponding Angles Two angles that are formed by two lines and atransversal and occupy correspondingpositions.LEAMTHVO

Lines and Angles Alternate Exterior Angles Two angles that are formed by two lines and atransversal and that lie between the two lineson opposite sides of the transversalLEAMTHVO

Lines and Angles Alternate Interior Angles Two angles that1 are formed by two lines and atransversal and that lie outside the two lineson opposite sides of the transversalLEAMTHVO

Lines and Angles Consecutive Interior Angles Two angles that are formed by two lines and atransversal and that lie between the two lineson the same side of the transversalalso called “same side interior angles”LEAMTHVO

Lines and Angles Example 3 Label your diagram.List all pairs of anglesthat fit thedescription.a. Transversalb. Correspondingc. Alternate exterioranglesd. Alternate interioranglesd. Consecutive interiorangles2 31 46 75 8

Geometry 1 Unit 33.2 Proof and PerpendicularLines

Proof and Perpendicular LinesReview of 2.5A two-column proof has numbered on one side, andthat show the logical order of an argument on theother. In the two-column proof, the reasons must use one of the following:;a ;a ;a ;or a

Proof and Perpendicular Lines 3 types of ProofsTwoColumnProofParagraphProofFlow ProofThe most formal type of proof. It lists numbered statements inthe left-hand column and a reason for each in the right handcolumn

Proof and Perpendicular Lines 3 types of ProofsTwo-ColumnProofParagraphProofFlow ProofThe most formal type of proof. It lists numbered statements inthe left-hand column and a reason for each in the right handcolumnDescribes the logical argument with sentences. It is moreconversational than a two-column proof.

Proof and Perpendicular Lines 3 types of ProofsTwo-ColumnProofThe most formal type of proof. It lists numbered statements inthe left-hand column and a reason for each in the right handcolumnDescribes the logical argument with sentences. It is moreParagraphProofFlowProofconversational than a two-column proof.Uses the same statements as a two column proof, but the logicalflow connecting the statements are connected by arrows

Proof and Perpendicular LinesTheoremCongruentLinear PairTheoremAdjacentComplementaryAngle ationIf two lines intersect to form a linearpair of congruent angles, then thelines are perpendicularSketch

Proof and Perpendicular LinesTheoremExplanationCongruentLinear PairTheoremIf two lines intersect to form a linearpair of congruent angles, then thelines are perpendicularAdjacentComplementaryAngle TheoremIf two sides of two adjacent acuteangles are perpendicular then theangles are Sketch

Proof and Perpendicular LinesTheoremExplanationCongruentLinear PairTheoremIf two lines intersect to form a linearpair of congruent angles, then thelines are perpendicularAdjacentComplementaryAngle TheoremIf two sides of two adjacent acuteangles are perpendicular then theangles are If two angles are perpendicular thenthey intersect to form four rightanglesSketch

Proof and Perpendicular Lines Example 1-Method 1 Given: AB CDProve: AC BDAStatementsBCReasons1.1.2.2.3.3.4.4.D

Proof and Perpendicular Lines Example 1- Method 2 Given: AB CDProve: AC BDABCD

Proof and Perpendicular Lines Example 2- Method 1 Given: BA perpendicular to BCProve: 1 and 2 are 5.6.6.

Proof and Perpendicular Lines Example 2- Method 3 Given: BA perpendicular to BCProve: 1 and 2 are complementary21

Proof and Perpendicular Lines Example 3- Method 1 Two Column Proof Given: 5 and 6 are a linear pair 6 and 7 are a linear pairProve: 5 z 7StatementsReasons

Proof and Perpendicular Lines Example 3- Method 2 Paragraph Proof

Proof and Perpendicular Lines Example 3- Method 3 Flow Chart Proof

Geometry 1 Unit 33.3 Parallel Lines andTransversals

Parallel Lines and Transversals Activity:Measuring angles of parallel lines and their transversals Objective: Discover the relationships between the angles of parallel linesand their transversals Question: What is the relationship between the angles and the lines? Step 1: Construct a segment Step 2: Construct 2 parallel lines crossing that segment Step 3: Number the angles 1 – 8 Step 4: Measure each angle with a protractor, write that measure onthe figure Step 5: Write, in paragraph form, the relationships you see

Parallel Lines and Transversals Step 1: Construct a segment

Parallel Lines and Transversals Construct 2 parallel lines crossing thatSegment

Parallel Lines and Transversals Step 3: Number the angles 1 – 813245768

Parallel Lines and Transversals Step 4: Measure each angle with aprotractor, write that measure on thefigureo13oo o24o o5768oo

Parallel Lines and Transversals Step 5: Write, in paragraph form, therelationships you see

Parallel Lines and Transversals Corresponding Angles Postulate If two parallel lines are cut by a transversal,then the pairs of corresponding angles arecongruent.12 1 z 2

Parallel Lines and Transversals Alternate Interior Angles Theorem If two parallel lines are cut by a transversal,then the pairs of alternate interior angles arecongruent.34 3 z 4

Parallel Lines and Transversals Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal,then the pairs of consecutive interior anglesare supplementary.56m 5 m 2 180

Parallel Lines and Transversals Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal,then the pairs of alternate exterior angles arecongruent.78 7 z 8

Parallel Lines and Transversals Perpendicular Transversal Theorem If a transversal is perpendicular to one of twoparallel lines, then it is perpendicular to theother.hkj is perpendicular to k

Parallel Lines and Transversals Example 1 Given: p qProve: m 1 m 2 180 StatementsReasons1.1.2.2.3.3.4.4.3 15 67 84 2pq

Parallel Lines and Transversals Solve for x1. 75 x2. 2x 1503. 12x 544. 2x 1 151

Parallel Lines and Transversals5. (2x 1) 1516. (7x 15) 81

Parallel Lines and Transversals Answers1. X 154.X 752. X 755.X 753. X 4.56.X 8

Parallel Lines and Transversals a.Example 2Given that m 5 65 ,find each measure. Tellwhich postulate or theoremyou used to find each one.m 6b.m 7qp967c.m 8d.m 958

Parallel Lines and Transversals Example 3 How many other angles have a measure of100 ?AB CDAC BDBA100 DC

Parallel Lines and Transversals Example 4 Use properties of parallel lines to find the valueof x.(x – 8) 72

Parallel Lines and Transversals Example 5 Find the value of x.x 70 (x – 20)

Geometry 1 Unit 33.4 Proving Lines are Parallel

Proving Lines are Parallel Corresponding Angle Converse Postulate If two lines are cut by a transversal so thatcorresponding angles are congruent, then thelines are paralleljkj k

Proving Lines are Parallel Alternate Interior Angles Converse If two lines are cut by a transversal so thatalternate interior angles are congruent thenthe lines are parallel.j31If 1 z 3, then j kk

Proving Lines are Parallel Consecutive Interior Angles Converse If two lines are cut by a transversal so thatconsecutive interior angles are supplementary,then the two lines are parallelj21kIf m 1 m 2 180 , then j k.

Proving Lines are Parallel Alternate Exterior Angles Converse If two lines are cut by a transversal so thatalternate exterior angles are congruent, thenthe lines are parallel.4jk5If 1 z 3, then j k.

Proving Lines are Parallel mExample 1 Given: m p, mProve: p qp1q2StatementsReasons1.1.2.2.3.3.4.4.q

Proving Lines are Parallel Example 2 Given: 5 z 6, 6 z 4 Prove: AD BCA4B56DC

Proving Lines are Parallel Example 3 Find the value of x that makes m n.mn(2x 1) (3x – 5)

Proving Lines are Parallel Example 4 Is AB DC?Is BC AD?155 D65 40 65 AB115 C

Proving Lines are Parallel Example 5 When the lines r and s are cut by atransversal, 1 and 2 are same side interiorangles. If m 1 is three times m 2, can r beparallel to line s? Explain

Proving Lines are Parallel The sum of the interior degrees of a triangle is180 .The sum of the degrees of a pair ofcomplementary angles is 90 .The sum of the degrees of a pair ofsupplementary angles is 180 .The sum of the degrees of consecutive interiorangles if transversal crosses parallel lines is180 .Parallel lines have slopes that are congruent.

Geometry 1 Unit 33.5 Using Properties of ParallelLines

Using Properties of Parallel Lines Lines Parallel to a Third Line Theorem If two lines are parallel to the same line, thenthey are parallel to each other.pqrIf p q and q r, then p r

Using Properties of Parallel Lines Lines Perpendicular to a Third LineTheorem In a plane, if two lines are perpendicular to thesame line, then they are parallel to each other.mnIf mpp and np, then m n

Using Properties of Parallel Lines 1Example 1 2Given: r s and s tProve: r t43StatementsReasons1.2.1.2.3.4.5.3.4.5.6.6.rst

Using Properties of Parallel Lines Example 2 The flag of the United States has 13alternating red and white stripes. Each stripe isparallel to the stripe immediately below it.Explain why the top stripe is parallel to thebottom stripe.S1S2S3S4S5S6S7S8S9S10S11S12S13Describe your thinking as you prove thatS1 and s13 are parallel

Using Properties of Parallel LinesExample 3You are building a CD rack. You cut the sides, bottom, and top so that each corner iscomposed of two 45o angles. Prove that the top and bottom front edges of the CD rackare parallel.Given: m 1 450m 2 450m ABC m 1 m 2Prove:m 1 450m 2 450Angle Addition PostulateGivenSubstitution PropertyDefinition of a right angleDefinition of perpendicular linesAngle Addition PostulateGivenSubstitution PropertySubstitution PropertyDefinition of perpendicular linesIn a plane, 2 lines to the same line are