Certainly! Let's break down each part of the question:
### Part 1: Find all angles of the first triangle
We are given that two angles of a triangle are in the ratio [tex]\(5:4\)[/tex] and the third angle is [tex]\(90^\circ\)[/tex].
1. Recall that the sum of all angles in any triangle is [tex]\(180^\circ\)[/tex].
2. Let the two unknown angles be represented as [tex]\(5x\)[/tex] and [tex]\(4x\)[/tex]. The third angle is [tex]\(90^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[
5x + 4x + 90^\circ = 180^\circ
\][/tex]
4. Combine like terms:
[tex]\[
9x + 90^\circ = 180^\circ
\][/tex]
5. Subtract [tex]\(90^\circ\)[/tex] from both sides:
[tex]\[
9x = 90^\circ
\][/tex]
6. Divide by [tex]\(9\)[/tex]:
[tex]\[
x = 10^\circ
\][/tex]
7. Now, calculate the actual angles:
[tex]\[
5x = 5 \times 10^\circ = 50^\circ
\][/tex]
[tex]\[
4x = 4 \times 10^\circ = 40^\circ
\][/tex]
So, the angles of the first triangle are:
[tex]\[
50^\circ, 40^\circ, 90^\circ
\][/tex]
### Part 2: Find all angles of the second triangle
We are given that one angle of a triangle is [tex]\(50^\circ\)[/tex] and the ratio of the remaining two angles is [tex]\(7:8\)[/tex].
1. Recall again that the sum of all angles in a triangle is [tex]\(180^\circ\)[/tex].
2. Let the remaining two angles be represented as [tex]\(7y\)[/tex] and [tex]\(8y\)[/tex]. The given angle is [tex]\(50^\circ\)[/tex].
3. The sum of all angles in the triangle:
[tex]\[
50^\circ + 7y + 8y = 180^\circ
\][/tex]
4. Combine like terms:
[tex]\[
50^\circ + 15y = 180^\circ
\][/tex]
5. Subtract [tex]\(50^\circ\)[/tex] from both sides:
[tex]\[
15y = 130^\circ
\][/tex]
6. Divide by [tex]\(15\)[/tex]:
[tex]\[
y \approx 8.6667^\circ
\][/tex]
7. Now, calculate the actual angles:
[tex]\[
7y = 7 \times 8.6667^\circ \approx 60.6667^\circ
\][/tex]
[tex]\[
8y = 8 \times 8.6667^\circ \approx 69.3333^\circ
\][/tex]
So, the angles of the second triangle are:
[tex]\[
50^\circ, 60.6667^\circ, 69.3333^\circ
\][/tex]
In summary, the angles are:
1. For the first triangle: [tex]\(50^\circ, 40^\circ, 90^\circ\)[/tex]
2. For the second triangle: [tex]\(50^\circ, 60.6667^\circ, 69.3333^\circ\)[/tex]
