if the triangle has angles alpha=45, beta=45, gamma=90, a=4cm, b=4cm, c=root of (a*a+b*b), and we construct the axes of the angles of each angle, we get the triangle KLM , what are the sizes of the angles?

Given the problem, we have a right-angled isosceles triangle with angles \(\alpha = 45^\circ\), \(\beta = 45^\circ\), and \(\gamma = 90^\circ\). The sides opposite these angles are \(a = 4 \, \text{cm}\), \(b = 4 \, \text{cm}\), and the hypotenuse \(c = \sqrt{a^2 + b^2} = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2} \, \text{cm}\).

When the angle bisectors of a triangle are constructed, they meet at the incenter of the triangle. The angles of the new triangle formed by the intersection points of the angle bisectors, typically denoted as triangle \(KLM\), are half the sum of the angles of the original triangle.

### Calculating the angles of triangle \(KLM\):

1. **Angle at \(K\)** (opposite \(\alpha = 45^\circ\)):
   - The angle at \(K\) will be half the sum of the other two angles \(\beta + \gamma\):
   \[
   \angle K = \frac{1}{2} \times (45^\circ + 90^\circ) = \frac{135^\circ}{2} = 67.5^\circ
   \]

2. **Angle at \(L\)** (opposite \(\beta = 45^\circ\)):
   - The angle at \(L\) will be half the sum of the other two angles \(\alpha + \gamma\):
   \[
   \angle L = \frac{1}{2} \times (45^\circ + 90^\circ) = \frac{135^\circ}{2} = 67.5^\circ
   \]

3. **Angle at \(M\)** (opposite \(\gamma = 90^\circ\)):
   - The angle at \(M\) will be half the sum of the other two angles \(\alpha + \beta\):
   \[
   \angle M = \frac{1}{2} \times (45^\circ + 45^\circ) = \frac{90^\circ}{2} = 45^\circ
   \]

### Therefore, the angles of the triangle \(KLM\) are:
- \(\angle K = 67.5^\circ\)
- \(\angle L = 67.5^\circ\)
- \(\angle M = 45^\circ\)

These angles add up to \(180^\circ\), confirming the calculation.

Would you like more details on this topic, or do you have any other questions?

### Related Questions:
1. What is the significance of the incenter in a triangle?
2. How do you calculate the length of the incenter in an isosceles right triangle?
3. What is the relationship between the circumcenter and the incenter in a right triangle?
4. How do the angles of a triangle affect the properties of its angle bisectors?
5. How do you construct the triangle \(KLM\) geometrically?

### Tip:
In any triangle, the angle bisectors always intersect at the incenter, which is equidistant from all the sides of the triangle.

If the triangle has angles alpha=45, beta=45, gamma=90, a=4cm, b=4cm, c=root of (a*a+b*b), and we construct the axes of the angles of each angle, we get the triangle KLM. What are the sizes of the angles?

Discover how to find the angles of triangle KLM, formed by the angle bisectors of a right-angled isosceles triangle with given sides and angles. Learn key geometric concepts and formulas, including the Pythagorean Theorem and the Angle Bisector Theorem. Suitable for students in Grades 9-12.

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