To classify the triangles formed by a diagonal of a square, let's follow these steps:

Step 1: Geometry of the Square

In a square:

• All four sides are equal in length.
• All interior angles are $90^\circ$ (right angles).
• The diagonal divides the square into two equal triangles.

Step 2: Triangles Formed by the Diagonal

When you draw a diagonal in a square, it divides the square into two right triangles. These two triangles have:

1. One right angle ($90^\circ$) because the interior angles of the square are $90^\circ$.
2. Two other angles that are both $45^\circ$ because the diagonal splits the angles of the square equally.
3. Two equal sides that are half the length of the square’s side. These are the legs of the right triangle.
4. The diagonal itself, which is the hypotenuse of the right triangle.

Step 3: Classifying the Triangles

1. By Angle Measures: Each triangle has:

   • One $90^\circ$ angle.
   • Two $45^\circ$ angles.

   This makes both triangles right-angled isosceles triangles, because the two non-right angles are equal, and the two legs are of the same length.

2. By Side Lengths:

   • The two legs of the triangle are equal in length.
   • The hypotenuse is the diagonal of the square. If the side of the square is $s$, then by the Pythagorean theorem: $\text{Diagonal} = \sqrt{s^2 + s^2} = \sqrt{2} s$ Therefore, the two legs are equal, and the hypotenuse is longer.

Conclusion:

The triangles are right-angled isosceles triangles because:

• They have a $90^\circ$ angle and two $45^\circ$ angles.
• The two legs are equal in length, and the hypotenuse (diagonal of the square) is longer than the legs.

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Would you like more details on any of the steps, or have further questions? Here are 5 related questions you might find interesting:

1. How does the Pythagorean theorem apply to right-angled triangles?
2. How do you calculate the area of a right-angled isosceles triangle?
3. What is the relationship between the side length of a square and its diagonal?
4. Can a square have any other type of triangle formed by a diagonal?
5. How can we use the properties of right-angled isosceles triangles in real-world applications?

Tip: When dealing with right-angled isosceles triangles, remember that the lengths of the legs are always equal, and the hypotenuse can be found using the formula $\text{Hypotenuse} = \sqrt{2} \times \text{Leg Length}$.