Question

∠dmf is bisected by (overrightarrow{mg}) where (mangle dmg=(4x)^{circ}) and (mangle dmf=(2x + 24)^{circ}). 7. sketch a diagram that could be used to represent this situation. 8. find the value of (x). 4x = 2x+24 9. find (mangle dmg). 10. find (mangle dmf).

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{MG}$ bisects $\angle DMF$, we know that $m\angle DMG=\frac{1}{2}m\angle DMF$. So, $4x = \frac{1}{2}(2x + 24)$. Multiply both sides by 2 to get $8x=2x + 24$.

Step2: Solve for $x$

Subtract $2x$ from both sides of the equation $8x=2x + 24$. We have $8x-2x=2x + 24-2x$, which simplifies to $6x=24$. Then divide both sides by 6: $x = \frac{24}{6}=4$.

Step3: Find $m\angle DMG$

Substitute $x = 4$ into the expression for $m\angle DMG$. Since $m\angle DMG=(4x)^{\circ}$, then $m\angle DMG=4\times4^{\circ}=16^{\circ}$.

Step4: Find $m\angle DMF$

Substitute $x = 4$ into the expression for $m\angle DMF$. Since $m\angle DMF=(2x + 24)^{\circ}$, then $m\angle DMF=(2\times4 + 24)^{\circ}=(8 + 24)^{\circ}=32^{\circ}$.

Answer:

1. $x = 4$
2. $m\angle DMG=16^{\circ}$
3. $m\angle DMF=32^{\circ}$