Angles formed by 2 Lines being cut by a Transversal
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1 Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa t If I asked you to ook at the figure above and find two anges that are on the sae side of the transversa, one an interior ange (between the ines), the other an exterior ange that were not adjacent, coud you do it? 2 and 4 are on the sae side of the transversa, one interior, the other is exterior whoops, they are adjacent. How about 2 and 6? Those two anges fit those conditions. We ca those anges corresponding anges. Can you nae any other pairs of corresponding anges? If you said 4 and 8, or 1 and 5, or 3 and 7, you d be right. Aternate Interior anges are on opposite sides of the transversa, both interior and not adjacent. 4 and 5 are a pair of aternate interior anges. Nae another pair. Based on the definition of aternate interior anges, how ight you define aternate exterior anges? How about sae side interior anges? Can you describe the? One pair of sae side interior anges is 3 and 5, can you nae another pair?
2 Sae Side Interior anges are two anges that are on the sae side of the transversa, nonadjacent, both interior anges. 3 and 5 are one pair and 4 and 6 is another pair of sae side interior anges. Parae ines are ines that are in the sae pane and have no points in coon. Skew ines are ines that are not parae and do not intersect, they do not ie in the sae pane. Anges Pairs fored by Parae ines Soething interesting occurs if the two ines being cut by the transversa happen to be parae. It turns out that every tie I easure the corresponding anges, they turn out to be equa. You ight use a protractor to easure the corresponding anges beow. Since that sees to be true a the tie and we can t prove it, we write it as an axio a stateent we beieve without proof Postuate If two parae ines are cut by a transversa, the corresponding anges are congruent. t Now et s take this inforation and put it together and see what we can coe up with. Knowing the corresponding anges are congruent, we can use our knowedge of vertica anges being congruent and our knowedge of anges whose exteriors sides ie in a ine to find the other anges.
3 If and we are given the 80 ange in the eft diagra, then we can fi in a the other 80 anges in the right diagra using corresponding anges are congruent and vertica anges are congruent. And two anges whose exteriors sides ie in a ine are suppeentary. Exape: Given the ines are parae and the ange easures 50, find the easure of a the other anges. 50 Begin by fiing in the corresponding anges, foowed by the vertica anges and finay the anges fored by their exterior sides being in a ine they are suppeentary NOTICE The aternate interior anges and the aternate exterior anges have the sae easure. Aso notice the sae side interior anges are suppeentary in these exapes.
4 In the ast coupe of exapes we saw the reationships between different anges fored by parae ines being cut by a transversa. Let s foraize that inforation. Proofs: Aternate Interior Anges Let s see, we ve aready earned vertica anges are congruent and corresponding anges are congruent, if they are fored by parae ines. Using this inforation we can go on to prove aternate interior anges are aso congruent, if they are fored by parae ines. What we need to reeber is drawing the picture wi be extreey hepfu to us in the body of the proof. Let s start. Theore If 2 parae ines are cut by a transversa, the aternate interior anges are congruent. By drawing the picture of parae ines being cut by a transversa, we abe the aternate interior anges. t 2 1 The question is, how do we go about proving 1 2? Now this is iportant. We need to ist on the picture things we know about parae ines. We, we just earned that corresponding anges are congruent when they are fored by parae ines. Let s use that inforation and abe an ange in our picture so we have a pair of corresponding anges. 2 t 3 1
5 Since the ines are parae, 1 and 3 are congruent. Oh wow, 2 and 3 are vertica anges! We have studied a theore that states a vertica anges are congruent. That eans 1 3 because they are corresponding anges and 2 3 are congruent because they are vertica anges, that eans 1 ust be congruent to That woud suggest that 1 2 by substitution. Now we have to write that in two couns, the stateents on the eft side, the reasons to back up those stateents on the right side. Let s use the picture and what we abeed in the picture and start with what has been given to us, ine is parae to. Stateents 1. Given 1 and 2 are at int s Reasons 2. 1 and 3 Def of corr. s are corr. s Two ines, cut by t, corr. s Vert s Transitive Prop Is there a trick to this? Not at a. Draw your picture, abe what s given to you, then fi in ore inforation based on your knowedge. Start your proof with what is given, the ast step wi aways be your concusion. Now, we have proved vertica anges are congruent, we accepted corresponding anges fored by parae ines are congruent, and we just proved aternate interior anges are congruent. Coud you prove aternate exterior anges are congruent? Try it. Write the theore, draw the picture, abe the aternate exterior anges, add ore inforation to your picture based on the geoetry you know, identify what has been given to you and what you have to prove.
6 Let s write those as theores. Theore Theore Theore If two parae ines are cut by a transversa, the aternate exterior anges are congruent. If two parae ines are cut by a transversa, the sae side interior anges are suppeentary. If a transversa is perpendicuar to one of two parae ines, it is perpendicuar to the other ine aso. Suarizing, we have: If two parae ines are cut by a transversa, ABBA then the corresponding s are congruent. then the at. int. s are congruent. then the at. ext. s are congruent. then the sae side int. s are suppeentary. Another way to reeber that postuate and those three theores is by reebering ABBA. A the anges arked A have the sae easure, a the anges arked B have the sae easure, and A+B = 180 if. A B B A A B B A Exape: Given that n and the inforation in the diagra, find the vaue of x. (3x+20) (5x+10) n
7 Since the two ines are parae, we know the corresponding anges are congruent, so we know they have the sae easure. 5x + 10 = 3x x = 10 x = 5 So, to answer the question, x = 5. If I wanted to know the easure of each ange, I woud substitute 5 into each expression. Each ange easures 35 Exape: Given j k and the inforation in the diagra, find the vaue of x. j k (2x+10) 60 In this case, we have two parae ines being cut by a transversa, we know the aternate interior anges are congruent. We can set those two anges equa and sove the resuting equation. 2x + 10 = 60 2x = 50 x = 25 Exape: Given and the inforation in the diagra, find the easure of CAB and DBA. B A C (3x+10) (2x+5) D Sae Side Interior anges are suppeentary
8 Since CAB and DBA are sae side interior anges, we know their su is 180 (3x + 10) + (2x + 5) = 180 5x + 15 = 180 5x = 165 x = 33 Reeber, the question was NOT to find the vaue of x, we were to find the easure of each ange, CAB and DBA. Substituting 33 into those expressions, we have CAB = 109 and DBA = 71 Proving that Lines are Parae Earier we stated the converse of a theore is not necessariy true, but coud be true. As it turns out, any of the theores and postuates deaing with parae ines, the converses are aso true. We wi start with a postuate. This is the converse of the postuate that read; if two parae ines are cut by a transversa, the corresponding anges are congruent. Now what I wi accept as true is if the corresponding anges are congruent, the ines ust be parae. Postuate If to ines are cut by a transversa so that the corresponding anges are congruent, the ines are parae. The converse of a conditiona is not aways true, so this deveopent is fortunate. As it turns out, the other three theores we just studied about having parae ines converses are aso true. That eads to the foowing three theores. Theore Theore Theore If two ines are cut by a transversa so that the aternate interior anges are congruent, then the ines are parae. If two ines are cut by a transversa so that the aternate exterior anges are congruent, then the ines are parae. If two ines are cut by a transversa so that the sae side interior anges are suppeentary, then the ines are parae. Now I have four ways to show ines are parae; corresponding s congruent, aternate interior s congruent, aternate exterior s congruent, sae side interior s suppeentary. You shoud know those theores because if we know ines are parae, that wi give us equations.
9 Let s ook at a proof of the theore concerning aternate interior anges. The other proofs wi be very siiar. Exape: Prove the Theore If two ines are cut by a transversa so that the aternate interior anges are congruent, then the ines are parae. 2 3 k Given: 1 2 k and j cut by t Prove: j k 1 j t STATEMENTS REASONS 1. k and j cut by t Given Vertica anges Transitive Property 4. j k 2 ines cut by t & corr. s are Reeber, your proof does not have to ook exacty ike ine. For instance, after step 1, I coud have put in another step 2 and 3 are vertica anges. The reason the definition of vertica anges. The reason for step 3 coud have been substitution. My point is a proof is just an arguent (deductive reasoning) in which the concusion foows fro the arguent. In a nutshe, we have earned the postuates and theores deaing with parae ines are bi-conditiona. That is, the converses are aso true if the stateents were true. Suarizing and writing those as bi-conditionas, we have: Two ines cut by a transversa are parae if and ony if the corresponding anges are congruent. Two ines cut by a transversa are parae if and ony if the aternate interior anges are congruent.
10 Two ines cut by a transversa are parae if and ony if the aternate exterior anges are congruent. Two ines cut by a transversa ate parae if and ony if the sae side interior anges are suppeentary.
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Nae: Period: 9/ 0/ LINES & TRANSVERSALS ) I can define, identify and iustrate te foowing ters Transversa Corresponding anges Aternate exterior anges. Aternate interior anges Sae side interior anges Dates,
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