Question

1. how many unique triangles are there with angle measures 45°, 45°, and 90°? explain.

Explanation:

Step1: Recall triangle similarity

Triangles with the same angle measures are similar by the AA (Angle - Angle) similarity criterion. For triangles with angles \(45^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\), we can have different side lengths.

Step2: Analyze uniqueness based on angles

When we only fix the angle measures (and not the side lengths), we can create an infinite number of triangles. This is because we can scale the triangle (multiply all side lengths by a positive real number \(k>0\)) and the angles will remain the same. For example, if we have a right - isosceles triangle with legs of length \(1\) and hypotenuse \(\sqrt{2}\), we can also have a triangle with legs of length \(2\) and hypotenuse \(2\sqrt{2}\), and both will have angle measures \(45^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\).

Answer:

There are infinitely many unique triangles with angle measures \(45^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\) because triangles with these angle measures are similar, and we can scale the side lengths (multiply all side lengths by a positive real number) to get different (non - congruent) triangles with the same angle measures.