Question

1. if (mangle tuv=(17x - 16)^{circ}), (mangle wuv=(10x - 5)^{circ}), and (mangle tuv=(5x + 3)^{circ}), find each measure.

(x =)
(mangle tuv=)
(mangle wuv=)
(mangle tuv=)

5x + 3+10x - 5=17x - 16
5x+10x + 3- 5=17x - 16
15x - 2=17x - 16

Explanation:

Step1: Set up the equation

Since \(m\angle TUV=m\angle TUW + m\angle WUV\), we have the equation \((5x + 3)+(10x-5)=17x-16\).
Combining like - terms on the left - hand side gives \(5x+10x + 3-5=15x - 2\). So the equation becomes \(15x-2 = 17x-16\).

Step2: Solve for x

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

Step3: Find \(m\angle TUW\)

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

Step4: Find \(m\angle WUV\)

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

Step5: Find \(m\angle TUV\)

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

Answer:

\(x = 7\)
\(m\angle TUW=38^{\circ}\)
\(m\angle WUV=65^{\circ}\)
\(m\angle TUV=103^{\circ}\)