Intent

Students explore via measurement some important relationships among angles formed by intersecting lines.

Mathematics

In the discussion of Very Special Triangles, students were introduced to complementary angles. Now they will explore the relationships among supplementary angles, straight angles, and vertical angles. Through their measurements, students might conjecture that vertical angles are equal in measure. The discussion of the activity will confirm that because vertical angles are supplements of the same angle, they must be equal.

Progression

In their groups, students measure angles in the given diagram and make conjectures about which angles are equal. They also investigate angle sum relationships and then generalize their results. A class discussion then introduces the terms supplementary angles, straight angles, and vertical angles.

Approximate Time

30 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Protractors

Rulers

Doing the Activity

The discussion that follows assumes that students are doing the activity by hand. If they have access to dynamic geometry software, however, they may want to work on this activity using the software rather than drawing diagrams on paper and measuring angles with a protractor.

You may want to spend a few minutes discussing angle notation. Point out that the standard practice is to identify an angle by naming three points in sequence, with the vertex as the middle point and one point from each ray forming the angle as the other two points.

In some cases, angles can be described unambiguously by a single letter. For example, in the diagram from the activity, reproduced below, writing A is clear because only one angle has A as its vertex. On the other hand, writing C is ambiguous, because it could refer to any of several angles. (We will use notation such as C, rather than m(C), the “measure” of angle C, to represent the size of an angle.) In this case, the angles must be named using three letters, such as ACD or DCG.

Figure 1

Warn students to draw the line segments long enough to allow them to measure the angles accurately.

Discussing and Debriefing the Activity

Ask several students to share their observations and explanations. As examples arise, introduce the appropriate terminology. For example, someone is likely to point out that ACD = BCG. If not, ask how each of these angles is related to BCA. How are angles ACD and BCG related to angle BCA? If needed, prompt students by then asking, If you knew that angle BCA was 110°, what other angles could you find? Students should be able to state these relationships:

ACD + BCA = 180° BCG + BCA = 180°

Explain that angles ACD and BCA are called supplementary angles and that each angle is the supplement of the other.

Also introduce the term straight angle for an angle of 180°—that is, an angle whose two sides go in opposite directions. Point out that a pair of angles that fit together to form a straight angle are supplementary.

Introduce the term verticle angles for a pair of “opposite” angles formed by the intersection of two lines. Ask if anyone can state a general principle about such angles. Students should be able to see that this statement holds true:

Vertical angles are equal.

Taken together, the two equations above prove that angles ACD and BCG are equal. You may want to introduce the following observation, which is essentially the proof of the statement above.

Angles that are supplements of the same angle are equal.

Key Questions

How are angles ACD and BCG related to angle BCA ?

If you knew that angle BCA was 110°, what other angles could you find?