Answer :
Sure, let's solve this step by step.
We are given two angles with their measures expressed in terms of [tex]\( x \)[/tex]:
- Measure of angle 1: [tex]\( (3x - 4)^\circ \)[/tex]
- Measure of angle 2: [tex]\( (4x + 10)^\circ \)[/tex]
We need to find the value of [tex]\( x \)[/tex] that satisfies one of the given angle measurements and then determine the measure of angle 2.
### Check Each Given Option
#### Check [tex]\( 32^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 32 \][/tex]
[tex]\[ 3x = 32 + 4 \][/tex]
[tex]\[ 3x = 36 \][/tex]
[tex]\[ x = 12 \][/tex]
2. Verify [tex]\( x = 12 \)[/tex] in angle 2:
[tex]\[ 4x + 10 = 4(12) + 10 = 48 + 10 = 58 \][/tex]
This does not match [tex]\( 32^\circ \)[/tex], so [tex]\( 32^\circ \)[/tex] is not the answer.
#### Check [tex]\( 38^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 38 \][/tex]
[tex]\[ 3x = 38 + 4 \][/tex]
[tex]\[ 3x = 42 \][/tex]
[tex]\[ x = 14 \][/tex]
2. Verify [tex]\( x = 14 \)[/tex] in angle 2:
[tex]\[ 4x + 10 = 4(14) + 10 = 56 + 10 = 66 \][/tex]
This does not match [tex]\( 38^\circ \)[/tex], so [tex]\( 38^\circ \)[/tex] is not the answer.
#### Check [tex]\( 58^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 58 \][/tex]
[tex]\[ 3x = 58 + 4 \][/tex]
[tex]\[ 3x = 62 \][/tex]
[tex]\[ x = 20.\bar{6} \][/tex]
This value of [tex]\( x \)[/tex] is not an integer, so it is not likely necessary to verify it further.
#### Check [tex]\( 66^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 66 \][/tex]
[tex]\[ 3x = 66 + 4 \][/tex]
[tex]\[ 3x = 70 \][/tex]
[tex]\[ x = 23.\bar{3} \][/tex]
This value of [tex]\( x \)[/tex] is not an integer, so it is also not necessary to verify it further.
### Verifying Using Angle 2
We can also cross-check with angle 2 equation:
#### Check [tex]\( 32^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 32 \][/tex]
[tex]\[ 4x = 32 - 10 \][/tex]
[tex]\[ 4x = 22 \][/tex]
[tex]\[ x = 5.5 \][/tex]
2. Verify [tex]\( x = 5.5 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(5.5) - 4 = 16.5 - 4 = 12.5 \][/tex]
This does not match [tex]\( 32^\circ \)[/tex], so [tex]\( 32^\circ \)[/tex] is not the answer.
#### Check [tex]\( 38^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 38 \][/tex]
[tex]\[ 4x = 38 - 10 \][/tex]
[tex]\[ 4x = 28 \][/tex]
[tex]\[ x = 7 \][/tex]
2. Verify [tex]\( x = 7 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(7) - 4 = 21 - 4 = 17 \][/tex]
This does not match [tex]\( 38^\circ \)[/tex], so [tex]\( 38^\circ \)[/tex]is not the answer.
#### Check [tex]\( 58^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 58 \][/tex]
[tex]\[ 4x = 58 - 10 \][/tex]
[tex]\[ 4x = 48 \][/tex]
[tex]\[ x = 12 \][/tex]
2. Verify [tex]\( x = 12 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(12) - 4 = 36 - 4 = 32 \][/tex]
This matches [tex]\( 58^\circ \)[/tex]. So the measure of angle [tex]\( 2 \)[/tex] is:
[tex]\[ \boxed{58} \][/tex]
We are given two angles with their measures expressed in terms of [tex]\( x \)[/tex]:
- Measure of angle 1: [tex]\( (3x - 4)^\circ \)[/tex]
- Measure of angle 2: [tex]\( (4x + 10)^\circ \)[/tex]
We need to find the value of [tex]\( x \)[/tex] that satisfies one of the given angle measurements and then determine the measure of angle 2.
### Check Each Given Option
#### Check [tex]\( 32^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 32 \][/tex]
[tex]\[ 3x = 32 + 4 \][/tex]
[tex]\[ 3x = 36 \][/tex]
[tex]\[ x = 12 \][/tex]
2. Verify [tex]\( x = 12 \)[/tex] in angle 2:
[tex]\[ 4x + 10 = 4(12) + 10 = 48 + 10 = 58 \][/tex]
This does not match [tex]\( 32^\circ \)[/tex], so [tex]\( 32^\circ \)[/tex] is not the answer.
#### Check [tex]\( 38^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 38 \][/tex]
[tex]\[ 3x = 38 + 4 \][/tex]
[tex]\[ 3x = 42 \][/tex]
[tex]\[ x = 14 \][/tex]
2. Verify [tex]\( x = 14 \)[/tex] in angle 2:
[tex]\[ 4x + 10 = 4(14) + 10 = 56 + 10 = 66 \][/tex]
This does not match [tex]\( 38^\circ \)[/tex], so [tex]\( 38^\circ \)[/tex] is not the answer.
#### Check [tex]\( 58^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 58 \][/tex]
[tex]\[ 3x = 58 + 4 \][/tex]
[tex]\[ 3x = 62 \][/tex]
[tex]\[ x = 20.\bar{6} \][/tex]
This value of [tex]\( x \)[/tex] is not an integer, so it is not likely necessary to verify it further.
#### Check [tex]\( 66^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 1:
[tex]\[ 3x - 4 = 66 \][/tex]
[tex]\[ 3x = 66 + 4 \][/tex]
[tex]\[ 3x = 70 \][/tex]
[tex]\[ x = 23.\bar{3} \][/tex]
This value of [tex]\( x \)[/tex] is not an integer, so it is also not necessary to verify it further.
### Verifying Using Angle 2
We can also cross-check with angle 2 equation:
#### Check [tex]\( 32^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 32 \][/tex]
[tex]\[ 4x = 32 - 10 \][/tex]
[tex]\[ 4x = 22 \][/tex]
[tex]\[ x = 5.5 \][/tex]
2. Verify [tex]\( x = 5.5 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(5.5) - 4 = 16.5 - 4 = 12.5 \][/tex]
This does not match [tex]\( 32^\circ \)[/tex], so [tex]\( 32^\circ \)[/tex] is not the answer.
#### Check [tex]\( 38^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 38 \][/tex]
[tex]\[ 4x = 38 - 10 \][/tex]
[tex]\[ 4x = 28 \][/tex]
[tex]\[ x = 7 \][/tex]
2. Verify [tex]\( x = 7 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(7) - 4 = 21 - 4 = 17 \][/tex]
This does not match [tex]\( 38^\circ \)[/tex], so [tex]\( 38^\circ \)[/tex]is not the answer.
#### Check [tex]\( 58^\circ \)[/tex]
1. Solve for [tex]\( x \)[/tex] using angle 2:
[tex]\[ 4x + 10 = 58 \][/tex]
[tex]\[ 4x = 58 - 10 \][/tex]
[tex]\[ 4x = 48 \][/tex]
[tex]\[ x = 12 \][/tex]
2. Verify [tex]\( x = 12 \)[/tex] in angle 1:
[tex]\[ 3x - 4 = 3(12) - 4 = 36 - 4 = 32 \][/tex]
This matches [tex]\( 58^\circ \)[/tex]. So the measure of angle [tex]\( 2 \)[/tex] is:
[tex]\[ \boxed{58} \][/tex]
