Question

find the measure of ∠joi. after you enter your answer press go. m∠joi =

Explanation:

Step1: Note vertical - angle property

Vertical angles are equal. $\angle GOF$ and $\angle HOJ$ are vertical angles. Also, $\angle FOI = 90^{\circ}$.

Step2: Set up an equation for vertical angles

Since vertical angles are equal, $2x - 3=3x - 2$.
Solve for $x$:

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Step3: Find $\angle HOJ$

Substitute $x = - 1$ into the expression for $\angle HOJ$ which is $3x-2$.
$\angle HOJ=3(-1)-2=-3 - 2=-5^{\circ}$ (This is incorrect. Let's use the fact that $\angle GOF+\angle FOI+\angle IOJ = 180^{\circ}$)
Since $\angle GOF$ and $\angle HOJ$ are vertical - angles, and $\angle FOI = 90^{\circ}$, and $\angle GOF+(2x - 3)$ and $\angle HOJ=(3x - 2)$ and $\angle GOF+\angle FOI+\angle IOJ = 180^{\circ}$.
We know that $\angle GOF$ and $\angle HOJ$ are vertical angles, so $2x-3 + 90+\angle IOJ=180$. Also, since vertical angles are equal, we can use the fact that the sum of angles around a point is $360^{\circ}$. But a simpler way is to note that $\angle GOF$ and $\angle HOJ$ are vertical angles and $\angle FOI = 90^{\circ}$.
We know that $\angle GOF+\angle HOJ = 90^{\circ}$ (because of the right - angle formed by the perpendicular lines). So $(2x - 3)+(3x - 2)=90$.

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Step4: Find $\angle HOJ$

Substitute $x = 19$ into the expression for $\angle HOJ$: $\angle HOJ=3x-2=3\times19-2=57 - 2 = 55^{\circ}$

Step5: Find $\angle JOI$

Since $\angle FOI = 90^{\circ}$ and $\angle GOF$ and $\angle HOJ$ are vertical angles, and $\angle JOI$ and $\angle HOJ$ are complementary (because of the perpendicular lines).
$\angle JOI = 90^{\circ}-\angle HOJ$.
Substitute $\angle HOJ = 55^{\circ}$, we get $\angle JOI=35^{\circ}$

Answer:

$35^{\circ}$