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). 9. find (mangle dmg). 10. find (mangle dmf).

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{MG}$ bisects $\angle DMF$, then $m\angle DMG=\frac{1}{2}m\angle DMF$. Given $m\angle DMG = 4x$ and $m\angle DMF=2x + 24$, we have the equation $4x=\frac{1}{2}(2x + 24)$. But the hand - written work has $4x=2x + 24$ which is incorrect. The correct equation based on the angle - bisector property is $4x=\frac{1}{2}(2x + 24)$. 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 get $8x-2x=2x+24 - 2x$, which simplifies to $6x=24$. Then divide both sides by 6: $x = 4$.

Step3: Find $m\angle DMG$

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

Step4: Find $m\angle DMF$

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

Answer:

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