Vertical angles and Linear Pairs

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SWBAT recognize why vertical angles are always congruent and reinforce other angle relationships.

Big Idea

When two lines intersect, pairs of the four angles formed are either congruent vertical angles.or supplementary linear pairs .


10 minutes

As I begin, I project Launch Vertical Angles Image on the SmartBoard and tell the class to find similar intersections on the map of part of Manhattan, NYC. I call on volunteers to go up and circle an intersection on the map that is similar to the intersections in Inset_Images 1 through 7. I ask them to write its corresponding number in the circle. My students have seen all of these in previous lessons. Three of these images involve angle pairs studied in the last two lessons.

My students like this short introduction because 8th graders like to go up to the board. An alternate route is printing and handing them the image. After examples of all seven similar intersections are circled and numbered, I ask the class if anyone has any objections. If there are none, I ask the class to indicate which intersections involve angles formed by two lines and a transversal, and, which involve only two lines.

If necessary, I hint to students that they may want to indicate which special angle pairs are displayed by images 2, 5, and 6. I tell the class that we will be looking at angles formed by two intersecting lines in this lesson, learning their names and properties. 

New Info

10 minutes

For the New Info section I handout the Vertical angles and Linear pairs sheet. I tell the students that they are to work with an elbow partner and read the information before answering the questions. The non examples of vertical angles and linear pairs are those I've found students usually make errors with, so I included them here. I allow 10 to 15 minutes to silently read, complete, and discuss their answers with their partners before going on to the next section, where they will actually work with the angle measures. Then, I call on students to share some of their answers to the questions. I always like to project these with the document camera in case a student wants to go up to the board to use the image as an aid in their explanation. 



25 minutes

This activity allows students to recognize the angle measure relationships of Linear Pairs and Vertical Angles. I find that it is not enough for students to simply see these on paper.  The activity also asks students to measure the constructed angles with protractors. In addition, for this lesson, the use of Geometer's Sketchpad (or any other Dynamic Geometry System software) is well liked by my students.

Each pair should be given the Angle Activity Sheet. The video below explains the next steps.

Source URL (Accessed May 14, 2014)

I ask that each pair of students work together using a computer and open a sketchpad document. For the first task students construct a diagram just like Figure 1, on the Vertical and Linear Pairs handout. Students will  measure angles 1 and 2 and find their sum and then are directed to drag point M, moving ray PM, in a clockwise and counterclockwise fashion and observe the angle measures.  

When going over the sheet, I ask the class why shouldn't this be surprising?  I go around to check that students are on the right track and that they are able to answer question 4 correctly. Knowing that if one of the linear pairs measures x degrees, the other measures 180 – x, is essential for being able to figure out Task 2.

For the second task, students should determine that if two angles are the supplement of the same angle, then they must be equal.  As I walk around I ask students questions to make sure they actually used deduction to conclude that Vertical angles 1 and 2 are equal, and not just guessed or measured them with the software. This activity section is a good time to tell students about the phrase "is the supplement of",  if they haven't heard of it already. I ask students to state "angle 1 is the supplement of angle 5", angle 2 is also the supplement of angle 5". The use of the language can also help in the deduction that angles 1 and 2 are equal because they are the supplement of the same angle. 

For Questions 8 and 9, I ask the student volunteer to provide examples to prove their answers. I expect students to go to the board and draw angles justifying their responses.  



10 minutes

The closure of this lesson is basically to have student volunteers share the answers to the Angle Activity sheet with the whole class. Students love to show their sketchpad constructions on the whiteboard for all to see. This gives the whole class opportunities to make suggestions or signal corrections if pertinent. With questions 6 through 8, I have students go up to the board and draw the examples they used to justify their answers. The closure discussion here is quite important because there are various ideas being set in place, so I would try and time myself well. If time runs out, the discussion should definitely be held at the start of the next class.