Angle+Pairs

=LESSON #4 Angle Pairs and the Coordinate Grid = Last revision date - July 28 New revision date - Fall 2017

** Purpose: ** This lesson introduces a classification system for pairs of angles. You will also learn about the coordinate plane and how to position the turtle in the plane. ** Essential Questions: ** How can you use Logo to position the turtle in the coordinate plane? How are angles related to each other? A Section of the Lesson Follows

** The Coordinate Grid: ** The connection between geometry and algebra is the coordinate grid. A grid can be shown on a plane. An ordered pair will represent points in the plane. A plane has two dimensions, which is why points are represented by two values. The first coordinate is called the **abscissa** and moves left or right. The second coordinate is called the **ordinate** and moves up or down. It is important to use the correct order (abscissa, ordinate). The ordered pair (0,0) is called the **origin**. In Logo, the turtle’s home location is the origin. ** New Commands: ** The grid in the graphics window can be turned on by using the command **gridon** and can be turned off by typing **gridoff** in the listener window. The turtle command **setxy [x y]** moves the turtle to the (x, y) position in the coordinate plane. Note that you do not need a comma in the setxy Logo command. Figure 4.1 shows a grid in the graphics window. Segments were drawn from the origin to points in each corner of the grid. Recall that the ordered pair moves horizontally the amount designated by the first coordinate (x), and moves vertically the amount designated by the second coordinate (y). There are four quadrants in the coordinate plane. ** LESSON 4 - TASK 1: ** Predict the quadrant for the given coordinates in the chart. Check your work using Logo. setxy [-100 -100] setxy [92 82] setxy [300 -50] setxy [-50 -50]

** Adjacent Angles ** are positioned next to each other. They share a common ray between them and the respective angles’ interiors do **not** overlap. In figure 4.2 ∠ POQ is adjacent to ∠ QOR. Note that adjacent angles MUST share a common vertex. An alternate definition is: Angles that share a common vertex and edge but do not share any interior points are called **adjacent angles**. A few definitions use the term **non-adjacent angles**. Figure 4.3 shows an example of two angles that are not adjacent to each other because they do not share a common ray (segment). ** Vertical Angles: ** Two angles opposite each other, formed by two intersecting straight lines that form an X-like shape, are called ** vertical **** angles ** or **opposite** ** angles ** or **vertically opposite angles**. These angles are equal in measure. See figure 4.1 for an example of two pairs of vertical angles. An alternate definition is: When two lines intersect, the non-adjacent angles are called **vertical angles**. These angles share a vertex and are equal in measure. ** Common Mistake ** Many of my students have created graphics similar to what is shown in figure 4.4. These examples are not vertical angles! What part of the definition did they not pay attention to?