Question

1. graph line ab and find the midpoint (using the midpoint formula) a(5,3) b(-2,-4) 6) solve for x find the m∠axb find the m∠dxc 7) line ab is || to line cd solve for x find the m∠bgh find the m∠ghd find the m∠dhe 8) solve for x find the m∠bgd if m∠dgf = 115 what is the m∠egf

Explanation:

5. Finding the mid - point of line AB

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here, $x_1 = 5,y_1 = 3,x_2=-2,y_2 = - 4$.
$M(\frac{5+( - 2)}{2},\frac{3+( - 4)}{2})$

Step2: Calculate the coordinates

$\frac{5+( - 2)}{2}=\frac{3}{2}=1.5$ and $\frac{3+( - 4)}{2}=\frac{-1}{2}=-0.5$. So the mid - point is $(1.5,-0.5)$.

6. Solving for $x$ and finding angles

Step1: Use vertical angles property

Vertical angles are equal. So, $2x-17=x + 8$.
$2x-x=8 + 17$
$x = 25$

Step2: Find $m\angle AXB$

Substitute $x = 25$ into $x + 8$. So $m\angle AXB=25 + 8=33^{\circ}$

Step3: Find $m\angle DXC$

Since $\angle DXC$ and $\angle AXB$ are vertical angles, $m\angle DXC = 33^{\circ}$

7. Solving for $x$ and finding angles with parallel lines

Step1: Use corresponding angles property

Since line $AB\parallel CD$, corresponding angles are equal. So, $133+x=135$.
$x=135 - 133=2$

Step2: Find $m\angle BGH$

$\angle BGH$ and the $135^{\circ}$ angle are corresponding angles, so $m\angle BGH = 135^{\circ}$

Step3: Find $m\angle GHD$

$\angle BGH$ and $\angle GHD$ are same - side interior angles. Since $AB\parallel CD$, $m\angle BGH+m\angle GHD = 180^{\circ}$. So $m\angle GHD=180 - 135 = 45^{\circ}$

Step4: Find $m\angle DHE$

$\angle DHE$ and $\angle BGH$ are alternate exterior angles. So $m\angle DHE = 135^{\circ}$

8. Solving for $x$ and finding angles

Step1: Use the fact that angles around a point sum to $360^{\circ}$

We know that the sum of angles around point $G$ is $360^{\circ}$. If one angle is $59^{\circ}$ and another is $(5x + 4)^{\circ}$, and we assume the angles are part of a full - circle sum. But if we consider linear pairs or other angle relationships. Let's assume we use the fact that if we consider a linear pair or angle - addition postulate. If we assume that the angle $59^{\circ}$ and $(5x + 4)^{\circ}$ are related in a linear - pair (sum to $180^{\circ}$)
$5x+4+59 = 180$
$5x=180-(4 + 59)$
$5x=117$
$x=\frac{117}{5}=23.4$

Step2: Find $m\angle BGD$

We need more information about the position of $\angle BGD$ relative to the given angles to find its measure precisely. But if we assume some standard angle - relationships. Let's assume $\angle BGD$ is related to the $59^{\circ}$ angle. If they are vertical angles (not clear from the problem setup), $m\angle BGD = 59^{\circ}$

Step3: If $m\angle DGF=115^{\circ}$, find $m\angle EGF$

If $\angle DGF$ and $\angle EGF$ form a linear pair, then $m\angle EGF=180 - 115=65^{\circ}$

Answer:

• Mid - point of $AB$: $(1.5,-0.5)$
• 6. $x = 25,m\angle AXB = 33^{\circ},m\angle DXC = 33^{\circ}$
• 7. $x = 2,m\angle BGH = 135^{\circ},m\angle GHD = 45^{\circ},m\angle DHE = 135^{\circ}$
• 8. $x = 23.4,m\angle BGD$ (needs more context), $m\angle EGF=65^{\circ}$