# SSA with Numbers (part 1)

- Author:
- Roger Gemberling

- Topic:
- Congruence, Numbers

This activity is part 1 of a two-part investigation on Side-Side-Angle (SSA).
For the first GeoGebra sketch, answer the 5 questions about the given right triangles.

## Hypotenuse-Leg

## Question 1

How would you determine the length of x for either triangle?

## Question 2

When solving for x for the triangle on the left, how many answers do you expect? Explain your answer.

## Question 3

How would you determine the measure of angle E?

## Question 4

How would you determine the measure of angle J?

## Question 5

When solving for the measure of angle E (or J), how many answers do you expect? Explain your answer.

Hopefully you indicated one answer is possible when answering questions 2 and 5. We have right triangle trigonometry rules and the Pythagorean Theorem to help us find the missing side or angle measure. Since only one answer is possible, this SSA pattern

**guarantees**one triangle. This triangle pattern is called**Hypotenuse-Leg**since you are given the hypotenuse length and a leg length. In the next investigation, you will look at three additional cases of SSA. The GeoGebra sketch illustrates the three cases you will investigate.In each case, you will use the sliders to create the SSA situation given in the spreadsheet. Then you will move the

**Green point**to determine the number of triangles that are possible with the given information. (Hint: When the Green point intersects the side of the angle, you will have a triangle region.) If one triangle is possible, continue to move the**Green point**to check for a second triangle. In the # column, click on the spreadsheet cell and place the number of possible triangles with the given information. The first problem in Case 1 has been completed. Move the Green point to see why the answer is 0 triangles. Now complete the tables for all 3 cases.## Question 6

Which case did you get only 1 triangle for each of the 5 examples?

In this investigation, you will try additional examples of Case 3. In Case 3, the length of side

**a is greater than**side**c**. So create ten different examples including different angle settings. Enter the information for each example into the spreadsheet. In the**# column**, indicate the number of possible triangles for each of your examples.## Question 7

How many triangles are possible for each example in Case 3?

Hopefully, you found only

**ONE**triangle is possible for each example. If you didn't get only ONE triangle for each of your case 3 examples, talk to your teacher or a classmate before continuing. We will refer to case 3 as**SsA**. It is read**Long Side-short side-Angle**. (The capital S represents the long side and the lower case s will represent the short side.) And**SsA***will only create ONE possible triangle*.## Question 8

We began with a special SSA case involving a right triangle. These triangles were called **Hypotenuse-Leg** triangles.
Should** Hypotenuse-Leg** triangles be considered a special **SsA** situation?

In the next activity, you will continue this investigation by exploring Cases 1 and 2.