Concepts and Connections

Shape

More Triangle Recipes--Concepts & Connections

Directed activities Angle-Side-Angle and Angle-Angle-Side focus on two additional strategies for constructing and/or duplicating triangles (See Figure 1.20).

Figure 1.20: Angle-Side-Angle vs. Angle-Angle-Side

In the case of Angle-Side-Angle, given triangle with known side AB between known angles and (ASA), the task is to construct a triangle congruent to. As a context for this question, imagine that the purpose of the construction is to determine the location (C’) of a buried treasure based on an identical triangle at another location. The strategy is to construct identical to and identify the location of C’ in the gray region of Figure 1.21. The steps in the construction are presented in Table 1.6. Using The Geometers Sketchpad (See 1_ASA.gsp), students may verify this result directly by measuring the sides and angles and that the treasure is buried at point C’.

Step 1	A random point A’ is plotted on a working line.
Step 2	Using A’ as the center and segment AB in as the radius, a circle is drawn. An intersection of this circle and the working line is labeled point B’.
Step 3	Using the angle copying strategy demonstrated in Table 1.5, a ray is constructed with endpoint A’.
Step 4	Using the angle copying strategy demonstrated in Table 1.5, a ray is constructed with endpoint B’.
Step 5	The intersection of these two rays is labeled point C’,creating.

Table 1.6: Angle-Side-Angle Construction

Figure 1.21: Angle-Side-Angle

In the case of Angle-Angle-Side, given triangle with known angles and , and with known side BC (AAS), the task is to construct a triangle congruent to? As a context for this question, imagine that a surveyor forgets to record the location of point A on a map. He does recall that, seen from point A, there is an angle between telephone poles at locations B and C. He also recalls that, seen from point B, there is and angle between point A and C. Finally, he knows the distance from B to C. The strategy is to construct congruent to and to identify the location of A’ in Figure 1.22. The steps in the construction are presented in Table 1.7. Using The Geometers Sketchpad (See 1_AAS.gsp), students may verify this result directly by measuring the angles.

Step 1	A random point B’ is plotted on a working line.
Step 2	Using B’ as the center and segment BC in as the radius, a circle is drawn. An intersection of this circle and the working line is labeled point C’.
Step 3	Using the angle copying strategy demonstrated in Table 1.5, is copied at B’.
Step 4	Using the angle copying strategy demonstrated in Table 1.5, is copied at an arbitrary point P, forming ray PQ.
Step 5	Using the Sketchpad’s construction tools, a segment parallel to ray PQ containing point C’ is drawn, creating .

Table 1.7: Angle-Angle-Side

Figure 1.22: Angle-Angle-Side

A somewhat different situation is encountered in directed activities Angle-Angle-Angle and Side-Side-Angle. In both of these activities, a task is posed for which the given information is insufficient to specify a unique triangle. For instance, given three angles with a sum of 180°, infinitely many triangles are possible, two of which are seen in Figure 1.23. That being the case, there is no way to know which triangle is the intended result.

Figure 1.23: Angle-Angle-Angle

A different situation is encountered in directed activity Side-Side-Angle (See Figure 1.24). Given two sides and a non-included angle (e.g., CB, BA, and ÐCBA), two different triangles may be possible (e.g., !CBA and !DBA). These findings provide a basis for answering the fourth and final question asked in the introduction to Chapter One, “Is it ever possible to construct more than one triangle from a given set of segments and/or angles?” The answer is yes.

Figure 1.24: Side-Side-Angle