Question

1. ∠dmf is bisected by (overline{mg}) where (mangle dmg=(4x)^{circ}) and (mangle dmf=(2x + 24)^{circ}). draw a sketch of the diagram and label with measurements.

Explanation:

Step1: Set up equation from angle - bisector property

$4x=2x + 24$

Step2: Solve for $x$

$2x=24$, $x = 12$

Step3: Find measure of $\angle DMG$

$m\angle DMG=48^{\circ}+48^{\circ}=96^{\circ}$

Step4: Sketch the angle

Draw $\angle DMG$ and $\overrightarrow{MF}$ with labels

Answer:

1. First, since $\overrightarrow{MF}$ bisects $\angle DMG$, we know that $m\angle DMF=m\angle FMG$. So, $4x = 2x + 24$.
2. Solve the equation $4x=2x + 24$ for $x$:

• Subtract $2x$ from both sides: $4x-2x=2x + 24-2x$, which gives $2x=24$.
• Divide both sides by 2: $x = 12$.

1. Find $m\angle DMG$:

• $m\angle DMG=m\angle DMF + m\angle FMG$.
• Substitute $x = 12$ into the expressions. $m\angle DMF=4x=4\times12 = 48^{\circ}$ and $m\angle FMG=2x + 24=2\times12+24=48^{\circ}$. So $m\angle DMG=48^{\circ}+48^{\circ}=96^{\circ}$.

1. Sketch: Draw an angle $\angle DMG$ with a ray $\overrightarrow{MF}$ inside it such that $\angle DMF$ is labeled $48^{\circ}$ and $\angle FMG$ is labeled $48^{\circ}$.