Question

fghij ~ tuvrs. what are ( mangle f ), ( mangle r ), and ( mangle u )?
( mangle f = square^circ )
( mangle r = square^circ )
( mangle u = square^circ )

Explanation:

Step1: Recall properties of similar polygons

Similar polygons have corresponding angles equal, and the sum of interior angles of a pentagon is \((5 - 2)\times180^\circ= 540^\circ\).

Step2: Find \(m\angle F\)

Since \(FGHIJ\sim TUVRS\), corresponding angles are equal. \(\angle F\) corresponds to \(\angle T\), so \(m\angle F = m\angle T=100^\circ\).

Step3: Find \(m\angle R\)

\(\angle R\) corresponds to \(\angle I\), so \(m\angle R = m\angle I = 90^\circ\).

Step4: Find \(m\angle U\)

First, find the sum of known angles in \(TUVRS\): \(m\angle T = 100^\circ\), \(m\angle S=101^\circ\), \(m\angle V = 132^\circ\), \(m\angle R = 90^\circ\). Let \(m\angle U=x\). Then \(100 + 101+132 + 90+x=540\). Calculate the sum of known angles: \(100 + 101=201\), \(201+132 = 333\), \(333 + 90=423\). So \(x=540 - 423=117^\circ\). Alternatively, since \(\angle U\) corresponds to \(\angle G\), \(m\angle U=m\angle G = 117^\circ\).

Answer:

\(m\angle F=\boxed{100}\)°
\(m\angle R=\boxed{90}\)°
\(m\angle U=\boxed{117}\)°