Question

the measures of the angles of a triangle are shown in the figure below. solve for x. 81° 53° (6x + 20)°

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(81^\circ + 53^\circ + (6x + 20)^\circ = 180^\circ\).

Step2: Simplify the left side

First, add \(81\), \(53\), and \(20\): \(81 + 53 + 20 = 154\). So the equation becomes \(154 + 6x = 180\).

Step3: Solve for \(x\)

Subtract \(154\) from both sides: \(6x = 180 - 154 = 26\)? Wait, no, wait: \(180 - 154 = 26\)? Wait, no, \(81 + 53 = 134\), \(134 + 20 = 154\). Then \(180 - 154 = 26\)? Wait, no, \(6x = 180 - 154\)? Wait, no, let's recalculate: \(81 + 53 = 134\), \(134 + (6x + 20) = 180\). So \(6x + 154 = 180\). Subtract \(154\): \(6x = 180 - 154 = 26\)? Wait, that can't be. Wait, no, \(81 + 53 = 134\), \(134 + 20 = 154\). Then \(180 - 154 = 26\), so \(6x = 26\)? No, that would make \(x\) a fraction, but maybe I made a mistake. Wait, no, the third angle is \((6x + 20)\). So sum is \(81 + 53 + 6x + 20 = 180\). Combine like terms: \(81 + 53 + 20 = 154\), so \(6x + 154 = 180\). Subtract 154: \(6x = 26\)? Wait, no, \(180 - 154 = 26\)? Wait, 154 + 26 = 180. So \(6x = 26\)? Then \(x = 26/6 = 13/3 \approx 4.33\)? That seems odd. Wait, maybe I misread the angles. Let me check again. The angles are 81°, 53°, and (6x + 20)°. So 81 + 53 = 134, 134 + 6x + 20 = 180. So 6x + 154 = 180. 180 - 154 = 26. So 6x = 26. x = 26/6 = 13/3 ≈ 4.33. Wait, but maybe the third angle is (6x + 2) instead of 20? Maybe a typo? But according to the given problem, it's (6x + 20). So perhaps that's correct. Wait, no, let's do it again:

Sum of angles in triangle: \( \angle A + \angle B + \angle C = 180^\circ \)

So \( 81 + 53 + (6x + 20) = 180 \)

Calculate \( 81 + 53 = 134 \)

Then \( 134 + 6x + 20 = 180 \)

\( 6x + 154 = 180 \)

Subtract 154 from both sides: \( 6x = 180 - 154 = 26 \)

Then \( x = \frac{26}{6} = \frac{13}{3} \approx 4.33 \). Wait, but that seems unusual. Maybe the problem was (6x + 2) instead of 20? If it were (6x + 2), then 81 + 53 + 6x + 2 = 146 + 6x = 180, so 6x = 34, x = 34/6 = 17/3 ≈ 5.66. Still odd. Alternatively, maybe the angles are 81°, 55°, and (6x + 20)°? Let's check: 81 + 55 = 136, 136 + 20 = 156, 180 - 156 = 24, 6x = 24, x = 4. Ah, maybe a typo in the problem, 53° should be 55°? But according to the given problem, it's 53°. So perhaps the original problem has a typo, but following the given numbers, x = 13/3 ≈ 4.33. But maybe I made a mistake. Wait, let's check again:

81 + 53 = 134

134 + 20 = 154

180 - 154 = 26

So 6x = 26

x = 26/6 = 13/3 ≈ 4.333...

So that's the solution.

Answer:

\( \frac{13}{3} \) (or approximately \( 4.33 \))