Question

1. in the following diagram, it is known that ∠ehg is a right - angle, m∠ehf=x + 1 and m∠ghf=5x + 5. find the value of x, the numerical values of m∠ehf and m∠ghf, and mark them on the diagram.

Explanation:

Step1: Set up equation

Since $\angle EHG$ is a right - angle and $\angle EHG=\angle EHF+\angle GHF$, we have $(x + 1)+(5x + 5)=90$.

Step2: Simplify the left - hand side

Combining like terms, we get $x+1 + 5x+5=6x + 6$. So the equation becomes $6x+6 = 90$.

Step3: Solve for $x$

Subtract 6 from both sides: $6x=90 - 6=84$. Then divide both sides by 6: $x=\frac{84}{6}=14$.

Step4: Find $m\angle EHF$

Substitute $x = 14$ into the expression for $m\angle EHF$. So $m\angle EHF=x + 1=14 + 1=15$.

Step5: Find $m\angle GHF$

Substitute $x = 14$ into the expression for $m\angle GHF$. So $m\angle GHF=5x + 5=5\times14+5=70 + 5=75$.

Answer:

$x = 14$, $m\angle EHF=15$, $m\angle GHF=75$