Question

1. ( mangle tuw=(5x + 3)^{circ}=)____( mangle wuv=(10x - 5)^{circ}=)__( mangle tuv=(17x - 16)^{circ}=)____

Explanation:

Step1: Use angle addition postulate

Since \(m\angle TUV=m\angle TUW + m\angle WUV\), we substitute the given expressions: \((17x - 16)=(5x + 3)+(10x - 5)\).

Step2: Simplify the equation

Simplify the right - hand side: \((5x + 3)+(10x - 5)=5x+10x+3 - 5=15x - 2\). So the equation becomes \(17x-16 = 15x-2\).

Step3: Solve for \(x\)

Subtract \(15x\) from both sides: \(17x-15x-16=15x - 15x-2\), which gives \(2x-16=-2\). Then add \(16\) to both sides: \(2x-16 + 16=-2 + 16\), so \(2x=14\). Divide both sides by \(2\): \(x = 7\).

Step4: Find \(m\angle TUW\)

Substitute \(x = 7\) into \(m\angle TUW=(5x + 3)^{\circ}\). Then \(m\angle TUW=(5\times7 + 3)^{\circ}=(35 + 3)^{\circ}=38^{\circ}\).

Step5: Find \(m\angle WUV\)

Substitute \(x = 7\) into \(m\angle WUV=(10x - 5)^{\circ}\). Then \(m\angle WUV=(10\times7-5)^{\circ}=(70 - 5)^{\circ}=65^{\circ}\).

Step6: Find \(m\angle TUV\)

Substitute \(x = 7\) into \(m\angle TUV=(17x - 16)^{\circ}\). Then \(m\angle TUV=(17\times7-16)^{\circ}=(119 - 16)^{\circ}=103^{\circ}\).

Answer:

\(m\angle TUW = 38^{\circ}\), \(m\angle WUV = 65^{\circ}\), \(m\angle TUV = 103^{\circ}\)