Question

if ( tu = tw = 11 ), ( mangle tvu = 4x ), and ( mangle tvw = x + 45^{circ} ), what is ( mangle tvw )?
( mangle tvw = square^{circ} )

Explanation:

Step1: Use the Angle Bisector Theorem

Since \(TU = TW = 11\) and \(TV\) is the common side, by the Hypotenuse - Leg (HL) congruence theorem for right - triangles (\(\triangle TVU\) and \(\triangle TVW\) are right - triangles with right angles at \(U\) and \(W\) respectively), \(\triangle TVU\cong\triangle TVW\). Then \(m\angle TVU=m\angle TVW\).

Step2: Solve the equation for \(x\)

Set \(4x=x + 45\). Subtract \(x\) from both sides: \(4x-x=x + 45-x\), which gives \(3x=45\). Divide both sides by 3: \(x=\frac{45}{3}=15\).

Step3: Find \(m\angle TVW\)

Substitute \(x = 15\) into the expression for \(m\angle TVW\). \(m\angle TVW=x + 45\), so \(m\angle TVW=15+45\).

Answer:

\(60\)