To determine the measure of angle 1 given that angle 2 is [tex]\( (5x + 14)^\circ \)[/tex] and angle 3 is [tex]\( (7x - 14)^\circ \)[/tex], we need to remember that the sum of the angles in a triangle is always [tex]\( 180^\circ \)[/tex]. Therefore, we have the equation:
[tex]\[ \text{angle1} + \text{angle2} + \text{angle3} = 180^\circ \][/tex]
We are given:
[tex]\[ \text{angle2} = (5x + 14)^\circ \][/tex]
[tex]\[ \text{angle3} = (7x - 14)^\circ \][/tex]
We also need to determine the measure of angle 1 from the given options [tex]\( 88^\circ, 89^\circ, 90^\circ, 91^\circ \)[/tex].
Let's express angle 1 as [tex]\( \text{angle1} = 90^\circ \)[/tex]:
Substituting the given expressions for angle 2 and angle 3, we get:
[tex]\[ 90^\circ + (5x + 14)^\circ + (7x - 14)^\circ = 180^\circ \][/tex]
Simplifying inside the equation:
[tex]\[ 90^\circ + 5x + 14^\circ + 7x - 14^\circ = 180^\circ \][/tex]
Combine like terms:
[tex]\[ 90^\circ + 12x = 180^\circ \][/tex]
Now, isolate [tex]\( x \)[/tex]:
[tex]\[ 12x = 180^\circ - 90^\circ \][/tex]
[tex]\[ 12x = 90^\circ \][/tex]
[tex]\[ x = \frac{90^\circ}{12} \][/tex]
[tex]\[ x = \frac{15}{2} = 7.5 \][/tex]
With [tex]\( x = 7.5 \)[/tex]:
Calculating the measures of angle 2 and angle 3:
[tex]\[ \text{angle2} = 5x + 14 = 5(7.5) + 14 = 37.5 + 14 = 51.5^\circ \][/tex]
[tex]\[ \text{angle3} = 7x - 14 = 7(7.5) - 14 = 52.5 - 14 = 38.5^\circ \][/tex]
It's useful to verify:
[tex]\[ \text{angle1} + \text{angle2} + \text{angle3} = 90^\circ + 51.5^\circ + 38.5^\circ \][/tex]
[tex]\[ 180^\circ = 180^\circ \][/tex]
So, the measure of angle 1 that fits the provided options is [tex]\( 90^\circ \)[/tex].
Therefore, the correct measure of angle 1 is:
[tex]\[ \boxed{90^\circ} \][/tex]
