Question

quadrilaterals wrft and ysmk are shown. wrft \sim mkys, m\angle wrf = (18x + 8.6)\degree, m\angle rft = (9x + y)\degree, m\angle twr = (11y + 19.7)\degree, m\angle smk = (16y - 5.3)\degree, and m\angle ykm = (19x + 2.6)\degree. what is the measure of \angle rft? if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Use the property of similar quadrilaterals (corresponding angles are equal)

Since \(WRFT\sim MKYS\), \(\angle WRF=\angle YKM\) and \(\angle TWR = \angle SMK\).
For \(\angle WRF=\angle YKM\), we have the equation \(18x + 8.6=19x+2.6\).
Subtract \(18x\) from both sides: \(18x + 8.6-18x=19x+2.6 - 18x\), which gives \(x = 6\).
For \(\angle TWR=\angle SMK\), substitute \(x = 6\) into the expressions.
\(m\angle TWR=(11y + 19.7)^{\circ}\) and \(m\angle SMK=(16y-5.3)^{\circ}\).
Set up the equation \(11y + 19.7=16y-5.3\).
Subtract \(11y\) from both sides: \(11y + 19.7-11y=16y-5.3-11y\), so \(19.7 = 5y-5.3\).
Add \(5.3\) to both sides: \(19.7+5.3=5y-5.3 + 5.3\), then \(25=5y\), and \(y = 5\).

Step2: Calculate \(m\angle RFT\)

Substitute \(x = 6\) and \(y = 5\) into \(m\angle RFT=(9x + y)^{\circ}\).
\(m\angle RFT=(9\times6 + 5)^{\circ}=(54 + 5)^{\circ}=59^{\circ}\)

Answer:

\(59^{\circ}\)