Question

what is the value of u? u = \boxed{^\circ}

Explanation:

Step1: Recall exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(u=(u - 48^{\circ})+(u - 49^{\circ})\)

Step2: Simplify the equation

Expand the right - hand side: \(u=u - 48^{\circ}+u - 49^{\circ}\)
Combine like terms: \(u = 2u-(48^{\circ}+49^{\circ})\)
\(u=2u - 97^{\circ}\)
Subtract \(u\) from both sides: \(0=u - 97^{\circ}\)

Step3: Solve for \(u\)

Add \(97^{\circ}\) to both sides: \(u = 97^{\circ}+u - u\)? Wait, no. Let's re - do the equation solving.
Starting from \(u=2u - 97^{\circ}\)
Subtract \(2u\) from both sides: \(u-2u=2u - 97^{\circ}-2u\)
\(-u=- 97^{\circ}\)
Multiply both sides by \(- 1\): \(u = 97^{\circ}\)? Wait, that can't be right. Wait, no, the exterior angle theorem: the exterior angle \(u\) is equal to the sum of the two remote interior angles. The two remote interior angles are \((u - 48^{\circ})\) and \((u - 49^{\circ})\), and also, we know that a straight angle is \(180^{\circ}\), but maybe I made a mistake. Wait, actually, the sum of angles in a triangle is \(180^{\circ}\), and the adjacent angle to \(u\) is \(180 - u\). So, \((u - 48)+(u - 49)+(180 - u)=180\)
Simplify: \(u-48+u - 49+180 - u=180\)
\((u + u - u)+(-48-49 + 180)=180\)
\(u+(83)=180\)
\(u=180 - 83=97\)? Wait, no, let's check again.
Wait, the sum of interior angles of a triangle is \(180^{\circ}\). Let the three angles be \(A=(u - 48)^{\circ}\), \(B=(u - 49)^{\circ}\), and \(C = 180 - u\) (since \(u\) and \(C\) are supplementary). Then \(A + B+C=180\)
So, \((u - 48)+(u - 49)+(180 - u)=180\)
\(u-48+u - 49+180 - u=180\)
\(u+( - 48-49 + 180)=180\)
\(u + 83=180\)
\(u=180 - 83 = 97\)? Wait, but if \(u = 97\), then the two interior angles are \(97 - 48 = 49^{\circ}\) and \(97 - 49 = 48^{\circ}\), and the third angle is \(180 - 97=83^{\circ}\), and \(49 + 48+83=180\), which works. Wait, but also, the exterior angle \(u\) should be equal to the sum of the two non - adjacent interior angles. The two non - adjacent interior angles are \(49^{\circ}\) and \(48^{\circ}\), and \(49 + 48 = 97^{\circ}\), which is equal to \(u\). So that's correct.

Wait, maybe my first approach was wrong. Let's use the exterior angle theorem correctly. The exterior angle \(u\) is equal to the sum of the two remote interior angles. The two remote interior angles are \((u - 48)^{\circ}\) and \((u - 49)^{\circ}\). So:

\(u=(u - 48)+(u - 49)\)

\(u=u - 48+u - 49\)

\(u=2u-97\)

Subtract \(u\) from both sides:

\(0 = u - 97\)

Add 97 to both sides:

\(u = 97\)

Yes, that's correct.

Answer:

\(97\)