Question

m∠gxf = 40° and m∠fxe = 30°. find each measure. m∠axf = m∠cxb = m∠axc = m∠bxf = m∠axd = m∠bxd =

Explanation:

Step1: Note vertical - angle property

Vertical angles are equal. $\angle GXF$ and $\angle DXE$ are vertical angles, so $m\angle DXE = 40^{\circ}$, and $\angle FXE$ and $\angle BXC$ are vertical angles, so $m\angle BXC=30^{\circ}$. Also, $\angle AXG$ and $\angle BXC$ are vertical angles, so $m\angle AXG = 30^{\circ}$.

Step2: Calculate $m\angle AXF$

Since $\angle AXG$ and $\angle GXF$ make up $\angle AXF$, then $m\angle AXF=m\angle AXG + m\angle GXF$. Substituting the values, $m\angle AXG = 30^{\circ}$ and $m\angle GXF = 40^{\circ}$, we get $m\angle AXF=30^{\circ}+40^{\circ}=70^{\circ}$.

Step3: Calculate $m\angle BXF$

$\angle BXF=\angle BXC+\angle CXF$. Since $\angle CXF = \angle DXE = 40^{\circ}$ and $\angle BXC = 30^{\circ}$, then $m\angle BXF=30^{\circ}+40^{\circ}=70^{\circ}$.

Step4: Calculate $m\angle AXC$

$\angle AXC=\angle AXB+\angle BXC$. Since $\angle AXB = 90^{\circ}$ and $\angle BXC = 30^{\circ}$, then $m\angle AXC=90^{\circ}+30^{\circ}=120^{\circ}$.

Step5: Calculate $m\angle AXD$

$\angle AXD=\angle AXB+\angle BXC+\angle CXD$. $\angle AXB = 90^{\circ}$, $\angle BXC = 30^{\circ}$, and $\angle CXD = 90^{\circ}$, so $m\angle AXD=90^{\circ}+30^{\circ}+90^{\circ}=150^{\circ}$.

Step6: Calculate $m\angle BXD$

$\angle BXD=\angle BXC+\angle CXD$. $\angle BXC = 30^{\circ}$ and $\angle CXD = 90^{\circ}$, so $m\angle BXD=30^{\circ}+90^{\circ}=120^{\circ}$.

Answer:

$m\angle AXF = 70^{\circ}$, $m\angle BXF = 70^{\circ}$, $m\angle CXB = 30^{\circ}$, $m\angle AXC = 120^{\circ}$, $m\angle AXD = 150^{\circ}$, $m\angle BXD = 120^{\circ}$