Question

which statement(s) is/are true. circle all that apply.

a) all 45-45-90 triangles are similar
b) a 45-45-90 triangle is an isosceles right triangle
c) a 45-45-90 triangle is half a square
d) if a right triangle has two sides congruent, then the acute angles = 45

Explanation:

Step 1: Analyze Statement (a)

All 45 - 45 - 90 triangles have angles \(45^{\circ},45^{\circ},90^{\circ}\). By the AA (Angle - Angle) similarity criterion, if two triangles have two pairs of congruent angles, they are similar. Since all 45 - 45 - 90 triangles have two angles of \(45^{\circ}\) and one angle of \(90^{\circ}\), they are similar. So statement (a) is true.

Step 2: Analyze Statement (b)

A 45 - 45 - 90 triangle has two angles of \(45^{\circ}\), so the sides opposite these angles are congruent (by the Isosceles Triangle Theorem: if two angles of a triangle are congruent, then the sides opposite them are congruent). And since one angle is \(90^{\circ}\), it is a right triangle. So a 45 - 45 - 90 triangle is an isosceles right triangle. Statement (b) is true.

Step 3: Analyze Statement (c)

If we draw a diagonal in a square, we divide the square into two congruent triangles. Each of these triangles has angles \(45^{\circ},45^{\circ},90^{\circ}\) (because the square has right angles and the diagonal bisects the right angles). So a 45 - 45 - 90 triangle is half of a square (when the square is divided by its diagonal). Statement (c) is true.

Step 4: Analyze Statement (d)

In a right triangle, if two sides are congruent, the two non - right angles (acute angles) must be equal. Let the measure of each acute angle be \(x\). We know that the sum of the angles in a triangle is \(180^{\circ}\), and one angle is \(90^{\circ}\). So \(x + x+90^{\circ}=180^{\circ}\), \(2x = 90^{\circ}\), \(x = 45^{\circ}\). So if a right triangle has two congruent sides, the acute angles are \(45^{\circ}\). Statement (d) is true.

Answer:

a. All 45 - 45 - 90 triangles are similar, b. A 45 - 45 - 90 Triangle is an isosceles right triangle, c. A 45 - 45 - 90 triangle is half a square, d. If a right triangle has two sides congruent, then the acute angles \( = 45\)