Question

find ( mangle wvz ) if ( mangle wvz=(2x - 24)^{circ},mangle zvu=(x + 12)^{circ} ), and ( mangle wvu = 105^{circ} ).
find ( mangle efg ) if ( mangle efz=(3x + 29)^{circ},mangle zfg=(79 - x)^{circ} ), and ( mangle efg = 7x^{circ} ).

Explanation:

Step1: Use the angle - addition postulate

For the first problem, by the angle - addition postulate \(m\angle WVZ+m\angle ZVU = m\angle WVU\).
Substitute the given expressions: \((2x - 24)+(x + 12)=105\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(2x-24+x + 12=105\), which simplifies to \(3x-12 = 105\).

Step3: Solve for \(x\)

Add 12 to both sides: \(3x=105 + 12\), so \(3x=117\).
Divide both sides by 3: \(x=\frac{117}{3}=39\).

Step4: Find \(m\angle WVZ\)

Substitute \(x = 39\) into \(m\angle WVZ=(2x - 24)^{\circ}\).
\(m\angle WVZ=(2\times39-24)^{\circ}=(78 - 24)^{\circ}=54^{\circ}\).

For the second problem:

Step1: Use the angle - addition postulate

By the angle - addition postulate \(m\angle EFZ+m\angle ZFG=m\angle EFG\).
Substitute the given expressions: \((3x + 29)+(79 - x)=7x\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(3x+29 + 79-x=7x\), which simplifies to \(2x+108 = 7x\).

Step3: Solve for \(x\)

Subtract \(2x\) from both sides: \(108=7x-2x\), so \(5x = 108\).
Divide both sides by 5: \(x=\frac{108}{5}=21.6\).

Step4: Find \(m\angle EFG\)

Substitute \(x = 21.6\) into \(m\angle EFG = 7x^{\circ}\).
\(m\angle EFG=7\times21.6^{\circ}=151.2^{\circ}\).

Answer:

\(m\angle WVZ = 54^{\circ}\), \(m\angle EFG=151.2^{\circ}\)