_____(0-10 pts) Describe the perpendicular transversal theorem. Give at least 3 examples.

Perpendicular Transversal Theorem:

• In a plane if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

Examples:

1.

2. Given: <--> parallel <--><--> perpendicular<-->

               BC                 DE,  AB                          BC

Prove: <--> perpendicular <-->

            AB                          DE

Proof : It is given that <--> parallel <-->

                                   BC                 DE,  so <ABC congruent <BDE by corresponding angles postulate. It is also given the <--> perpendicular <-->

     AB                           BC, so m<ABC = 90°. By the defenition of congruent angles, m<ABC = m<BDE, so m<BDE=90° by the transitive property of equality. By definition or perpendicular lines <--> perpendicular <-->

                                                                                                               AB                          DE.  

3. Given: K || L, T perpendicular K

Prove: T perpendicular L

By: Maria Jose Diaz-Duran  

 _____(0-10 pts) Describe how the transitive property also applies to parallel and perpendicular lines. Include a discussion about theorems 3-4-3 and 3-4-1. Give at least 3 examples of each.

Theorm 3-4-1 states that "If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of congruent supplementary lines then the lines are perpendicular.)"

This means that if to lines intersect and they form two side to side congruent angles, then the lines are perpendicular to each other. This is because in order for both angles to add up to 180 degrees, and be congruent, they both must equal 90 degrees. Making them perpendicular.

For example: If line CE intersects line AB in such way that they form a linear pair in which <CDB and <ADC are congruent angles, then they have to be right angles. This is because this two angles form a linear pair. So if the two angles form 90 degrees each then the opposite angles are also 90 degrees each by the vertical angles theorem. So if all angles equal 90 degrees than the two lines (lines AB and CE) are perpendicular to each other.

Theorem 3-4-3 states that "If two coplaner lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines perpendicular to the same line means the two lines are parallel.)"

This means that if two lines on the same plane are perpendicular to the same transversal. Then the the two lines are parallel to each other.

For example: Given that line EF is perpendicular to line AB and to line CD. We can deduce that m<AGH is 90 degrees by the vertical angles theorem. And if <AGH and <GHD are congruent, then the two lines are parallel due to the converse of the alternate interior angles theorem.

An example using both theorems: Given m<EGA is 90, m<GHD is 90 degrees.

                                                  Prove: line AB is parallel to line CD.

Given that m<EGA is equal to 90 degrees, and that m<GHD is also 90 degrees we can deduce that angle m<EGB is 90 degrees too by the Linear pair postulate. By knowing they both are 90 degrees we can see that lines EF and AB are perpendicular because of the theorem 3-4-1. By knowing that m<EGB is 90 degrees we can deduce that m<AGH is also 90 degrees by the vertical angles theorem. So if m<AGH is 90 degrees and m<GHD is also 90 degrees, we can see by the converse of the alternate interior angles theorem that line AB is parallel to line CD.

By: Rodrigo Maselli