When more than one transversal is involved, an angle can be corresponding to more than one other angle. The same is true for the other types of angles. In the diagram below, ∠2 and ∠3 are same-side interior on transversal L, and ∠2 and ∠5 are also same-side interior on transversal n.
Complete the worksheet below, then check your answers.
Parallel Lines Cut by a Transversal
Recall that parallel lines have the same slope (rate of change) and will never intersect each other. So when a transversal cuts through any set of parallel lines it forms sets of congruent angles and supplementary angles. If you think of it as one parallel line is the exact copy of the other, it make since that the angles formed from the cut of the transversal will be an exact copy as well.
Before we get started with the angle relationships, lets review how to identify parallel lines. You CANNOT assume lines are parallel, even if they look as if they are. It has to be given or proven. Below are two ways parallel lines will be notated. The symbol "||" means parallel, or they can be notated with arrows.
Before we get started with the angle relationships, lets review how to identify parallel lines. You CANNOT assume lines are parallel, even if they look as if they are. It has to be given or proven. Below are two ways parallel lines will be notated. The symbol "||" means parallel, or they can be notated with arrows.
We learned the types of angles formed from transversals in the section above. The diagram below states which angles will be congruent and which angles will be supplementary when the lines being cut are parallel.
Copy the diagram into your notes.
Copy the diagram into your notes.
Now lets use this information to find missing angles. In the Khan academy skill linked below you will find basic angle measures. Make sure to use the hints and videos if you need more reminders/instruction.
Remember when there are multiple transversals, identify the transversal that is forming the angle that you are trying to find. Try the example below then check your answers.
Now we will use the relationships in other ways. In the two examples below angle are assigned an algebraic value. We can use the theorems and postulates of parallel lines cut by a transversal to set up equations to find the missing angles. Copy the examples into your notes.
Proving Lines Parallel
We can easily prove that lines are parallel by using the special angles from above. Read through the theorems to the right.
Notice that if you know, or can prove that any of the special angle relationships are true, then it is true that the lines are parallel.
You will be proving these with two column proofs. (See Example)
In a two column proof, a statement is made that work toward the final conclusion. For each statement you must give a reason (or proof) that the given statement is true.
Watch the 2 videos below as an introduction to 2 column proofs.
In order to give the reasons for the statements you will need to review the postulates and theorems of algebra and geometry. Some you have not seen yet, some you will not use in this course, but having the list will be extremely helpful. Make sure you download the list and save it. Also make sure to look study the Algebraic Properties of Congruence. These will be used often and it is important that you understand the concepts.