Alright, let's solve the problem step-by-step!
1. Identify the known angle:
We know that one angle in the triangle is two-thirds of a right angle. A right angle is [tex]\( 90^\circ \)[/tex].
So, the known angle is:
[tex]\[
\text{Known angle} = \frac{2}{3} \times 90^\circ = 60^\circ
\][/tex]
2. Understand the angle sum property of a triangle:
In any triangle, the sum of all three internal angles is always [tex]\( 180^\circ \)[/tex].
3. Express the sum of the remaining two angles:
Since we know that the total sum of the angles must be [tex]\( 180^\circ \)[/tex], we can calculate the sum of the other two angles:
[tex]\[
\text{Sum of remaining angles} = 180^\circ - 60^\circ = 120^\circ
\][/tex]
4. Identify the greater angle:
We are given that the unknown third angle is the greatest of the three angles. Hence, one of the remaining two unknown angles must be smaller than the other.
5. Equations for the remaining angles:
Let's denote the second angle as [tex]\( x \)[/tex] and the third angle as [tex]\( y \)[/tex], where [tex]\( y \)[/tex] is the larger angle.
We know:
[tex]\[
x + y = 120^\circ
\][/tex]
6. Recognize the condition for the largest angle:
[tex]\( y \)[/tex] is greater than [tex]\( 60^\circ \)[/tex] and greater than [tex]\( x \)[/tex].
7. Solving for the angles:
Since [tex]\( x + y = 120^\circ \)[/tex] and [tex]\( y \)[/tex] is the largest angle, [tex]\( y > 60^\circ \)[/tex].
To maintain the condition that [tex]\( y \)[/tex] is the largest:
[tex]\[
y > 60^\circ \quad \text{and} \quad 60^\circ > x
\][/tex]
8. Identify a suitable pair for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given their sum is [tex]\( 120^\circ \)[/tex]:
To have [tex]\( y \)[/tex] as the largest, we can set [tex]\( y > 60^\circ \)[/tex], so [tex]\( x < 60^\circ \)[/tex].
If [tex]\( y \)[/tex] is very close to [tex]\( 120^\circ / 2 \)[/tex], say [tex]\( 70^\circ\)[/tex], then [tex]\( x \)[/tex] becomes:
[tex]\[
x = 120^\circ - 70^\circ = 50^\circ
\][/tex]
Checking:
[tex]\[
50^\circ + 70^\circ = 120^\circ \quad (\text{True})
\][/tex]
Therefore, the three angles in the triangle are:
[tex]\[
60^\circ, \, 50^\circ, \, \text{and} \, 70^\circ
\][/tex]
This satisfies all the conditions given: [tex]\( 60^\circ \)[/tex] is two-thirds of a right angle, the sum of angles is [tex]\( 180^\circ \)[/tex], and [tex]\( 70^\circ \)[/tex] is the largest angle.
