Question

Julius needs to determine how much information needs to be given about an isosceles triangle to be able to draw an isosceles triangle. He is looking at six different cases and is trying to determine if that amount of detail will allow him to construct just one isosceles triangle or if it allows for more than one isosceles triangle.

Case 1. The length of the two congruent sides and the measure of the included angle.

Case 2. The lengths of two non-congruent sides and the measure of the included angle.

Case 3. The lengths of two non-congruent sides.

Case 4. The measure of the two congruent angles and the included side.

Case 5. The measure of two non-congruent angles.

Case 6. The measure of two non-congruent angles and the included side.

Forms only one isosceles triangle: Forms more than one isosceles triangle:

Answer 1

To draw an isosceles triangle, the amount of information needed depends on the given details. Cases involving congruent sides or angles, along with side lengths and angle measures, determine a unique triangle. Other cases may result in multiple possible triangles.

Step-by-step explanation:

An isosceles triangle is a triangle with at least two congruent sides. To be able to draw an isosceles triangle, we need to know certain information about the triangle. Let's analyze each case:

1. Case 1: The length of the two congruent sides and the measure of the included angle. This case will form only one isosceles triangle because specifying the congruent sides and the included angle uniquely determine the triangle.
2. Case 2: The lengths of two non-congruent sides and the measure of the included angle. This case will also form only one isosceles triangle because specifying the non-congruent sides and the included angle uniquely determine the triangle.
3. Case 3: The lengths of two non-congruent sides. This case will form more than one isosceles triangle because the lengths of the non-congruent sides alone do not uniquely determine the triangle.
4. Case 4: The measure of the two congruent angles and the included side. This case will form more than one isosceles triangle because the measure of the angles alone does not uniquely determine the triangle.
5. Case 5: The measure of two non-congruent angles. This case will form more than one isosceles triangle because the measure of the angles alone does not uniquely determine the triangle.
6. Case 6: The measure of two non-congruent angles and the included side. This case will form only one isosceles triangle because specifying the angles and the included side uniquely determine the triangle.

Therefore, Case 1 and Case 2 form only one isosceles triangle, while Case 3, Case 4, Case 5, and Case 6 form more than one isosceles triangle.