Depending on a given set of conditions and the properties of triangles, any of these four outcomes is possible when constructing triangles: No triangles fit the condition.
One unique triangle fits the condition.
Two triangles fit the condition.
Infinitely many triangles fit the condition.
Complete the steps below to find the number of triangles that can be constructed based on the following conditions: one side measures 7 centimeters, another side measures 9 centimeters, and the angle between them measures 74°.

Part A
What does the dotted line in the diagram represent?

Part B
Now think about changing the triangle. What happens to angle Z if the unknown side length increases while keeping the known side lengths (XZ and ZY) the same? Will the triangle still fit the given conditions?

Part C
What happens to angle Z if the unknown side length decreases while keeping the known side lengths (XZ and YZ) the same? Will the triangle still fit the given conditions?

Part D
Based on your conclusions in parts B and C, can the length of the unknown side be changed in any way without changing the given conditions for the triangle?

Part E
You know the given conditions for the triangle are fixed. You also know the unknown side length is fixed. What does this tell you about the two angles adjacent to the unknown side?

Part F
From your conclusions in part E, how many triangles can be constructed based on the given conditions?

Part G
Let c be the unknown side of the triangle. Use this triangle calculator to solve for c. Under Sides, enter 7 for side a and 9 for side b. Under Angles, enter 74 for angle C. Click Calculate once you have entered the information. What is the length of side c?

Part H
Now try to construct a triangle using a different set of measurements. This time, you’ll enter three angle measurements. Return to the Calculator tab, and click the Clear button to begin a new calculation.

Under Angles, enter 45 for A, 40 for B, and 95 for C. Then click Calculate. What happened? What message did the tool deliver? Explain the message in terms of the properties of a triangle and the given angles.

Part I
Click Clear in the tool to begin a new calculation. This time, you’ll check for valid triangles given two sides and the angle opposite one of the sides.

Under Sides, enter 9 for a and 6 for b. Under Angles, enter 30 for B. Then click Calculate. What happened? What message did the tool deliver? Click “Show other solution” in the tool and explain the message in terms of the angle measurements and the given information.

Part J
Return to the Calculator tab, and click the Clear button to begin a new calculation. This time, you’ll check for valid triangles given two angles and the side between them.

Under Sides, enter 5 for a. Under Angles, enter 30 for B and 50 for C. Then click Calculate. How many triangles can be created from the given conditions?

Part K
Return to the Calculator tab, and click the Clear button to begin a new calculation. This time, you’ll check for valid triangles given three sides of specified length.

Under Sides, enter 6 for a, 7 for b, and 13 for c. Then click Calculate. What happened? What message did the tool deliver? Explain the message in terms of the properties of a triangle and the given side lengths.