Question

if m∠agf = 35° and gd bisects ∠egc, determine the measure of all the angles. m∠agb = ______ m∠bgc = ____ m∠cgd = ____ m∠dge = ____ m∠egf = ____ m∠fgd = ____ m∠bgd = ____ m∠bge = ____ how many angles share the same measure? ______

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. $\angle AGF$ and $\angle BGE$ are vertical angles, so $m\angle BGE = 35^{\circ}$. Also, $\angle AGB$ is a right - angle, so $m\angle AGB=90^{\circ}$.

Step2: Find $\angle BGC$

Since $\angle AGB = 90^{\circ}$ and $\angle AGF = 35^{\circ}$, and $\angle AGB+\angle BGC+\angle CGE+\angle EGF = 180^{\circ}$, and $\angle AGF=\angle BGE = 35^{\circ}$, then $m\angle BGC = 90^{\circ}-35^{\circ}=55^{\circ}$.

Step3: Use angle - bisector property

Since $GD$ bisects $\angle EGC$, and $\angle BGC = 55^{\circ}$, then $\angle CGE = 180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}$, so $m\angle CGD=m\angle DGE=\frac{55^{\circ}}{2}=27.5^{\circ}$.

Step4: Find other angles

$m\angle EGF = 35^{\circ}$ (vertical to $\angle AGB$). $m\angle FGD=m\angle FGE + m\angle EGD=35^{\circ}+27.5^{\circ}=62.5^{\circ}$. $m\angle BGD=m\angle BGC + m\angle CGD=55^{\circ}+27.5^{\circ}=82.5^{\circ}$.

Answer:

$m\angle AGB = 90^{\circ}$
$m\angle BGC = 55^{\circ}$
$m\angle CGD = 27.5^{\circ}$
$m\angle DGE = 27.5^{\circ}$
$m\angle EGF = 35^{\circ}$
$m\angle FGD = 62.5^{\circ}$
$m\angle BGD = 82.5^{\circ}$
$m\angle BGE = 35^{\circ}$
The angles with the same measure: $\angle AGF$ and $\angle BGE$ and $\angle EGF$ (3 angles)