To find the measure of angle 2, we need to start by considering the expressions given for the angles:
- The measure of angle 1 is [tex]\( (3x - 4)^\circ \)[/tex]
- The measure of angle 2 is [tex]\( (4x + 10)^\circ \)[/tex]
We are given the following options for the possible measure of angle 2: [tex]\(32^\circ\)[/tex], [tex]\(38^\circ\)[/tex], [tex]\(58^\circ\)[/tex], and [tex]\(66^\circ\)[/tex].
Step-by-Step Solution:
1. Test the options for angle 2:
- Try [tex]\(32^\circ\)[/tex]:
Solve [tex]\( 4x + 10 = 32 \)[/tex]
[tex]\[ 4x + 10 = 32 \][/tex]
[tex]\[ 4x = 22 \][/tex]
[tex]\[ x = \frac{22}{4} = 5.5 \][/tex]
Check the corresponding measure for angle 1 when [tex]\( x = 5.5 \)[/tex]:
[tex]\[ 3x - 4 = 3(5.5) - 4 = 16.5 - 4 = 12.5 \][/tex]
[tex]\( 12.5^\circ \)[/tex] could be the measure of angle 1. This does not obviously relate strongly to the measure of an angle.
- Try [tex]\(38^\circ\)[/tex]:
Solve [tex]\( 4x + 10 = 38 \)[/tex]
[tex]\[ 4x + 10 = 38 \][/tex]
[tex]\[ 4x = 28 \][/tex]
[tex]\[ x = 7 \][/tex]
Check the corresponding measure for angle 1 when [tex]\( x = 7 \)[/tex]:
[tex]\[ 3x - 4 = 3(7) - 4 = 21 - 4 = 17 \][/tex]
[tex]\( 17^\circ \)[/tex] could be the measure of angle 1. This again does not make any special relational sense.
- Try [tex]\(58^\circ\)[/tex]:
Solve [tex]\( 4x + 10 = 58 \)[/tex]
[tex]\[ 4x + 10 = 58 \][/tex]
[tex]\[ 4x = 48 \][/tex]
[tex]\[ x = 12 \][/tex]
Check the corresponding measure for angle 1 when [tex]\( x = 12 \)[/tex]:
[tex]\[ 3x - 4 = 3(12) - 4 = 36 - 4 = 32 \][/tex]
Here, [tex]\(32^\circ\)[/tex] corresponds well to the practical possibility.
- Try [tex]\(66^\circ\)[/tex]:
Solve [tex]\( 4x +10 = 66 \)[/tex]
[tex]\[ 4x + 10 = 66 \][/tex]
[tex]\[ 4x = 56 \][/tex]
[tex]\[ x = 14 \][/tex]
Check the corresponding measure for angle 1 when [tex]\( x = 14 \)[/tex]:
[tex]\[ 3x - 4 = 3(14) - 4 = 42 - 4 = 38 \][/tex]
Here, [tex]\(38^\circ\)[/tex] corresponds but does not seem special.
After evaluating these options, the one that best satisfies the given mathematical framework and options is:
The measure of angle 2 is [tex]\(58^\circ\)[/tex].
