Question

1. if m∠dag=(5x - 4)°, m∠gaf=(7x - 9)°, m∠fah=(9y + 3)°, find the values of x and y.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle DAG$ and $\angle FAE$ are vertical angles, so $\angle DAG=\angle FAE$. Also, $\angle GAF$ and $\angle DAH$ are vertical angles, so $\angle GAF = \angle DAH$.
We have $\angle DAG=(5x - 4)^{\circ}$, $\angle GAF=(4x - 9)^{\circ}$ and $\angle DAH=(y + 3)^{\circ}$.
Since $\angle GAF=\angle DAH$, we get the equation $4x-9=y + 3$.
Since $\angle DAG+\angle GAF = 180^{\circ}$ (linear - pair of angles), we have the equation $(5x - 4)+(4x - 9)=180$.

Step2: Solve the equation for x

First, simplify the left - hand side of the equation $(5x - 4)+(4x - 9)=180$.
Combine like terms: $5x+4x-4 - 9=180$, which gives $9x-13 = 180$.
Add 13 to both sides: $9x=180 + 13=193$.
Divide both sides by 9: $x=\frac{193}{9}\approx21.44$.

Step3: Solve the equation for y

Substitute $x$ into the equation $4x-9=y + 3$.
$4\times\frac{193}{9}-9=y + 3$.
$\frac{772}{9}-9=y + 3$.
$\frac{772}{9}-\frac{81}{9}=y + 3$.
$\frac{772 - 81}{9}=y + 3$.
$\frac{691}{9}=y + 3$.
Subtract 3 from both sides: $y=\frac{691}{9}-3=\frac{691}{9}-\frac{27}{9}=\frac{691 - 27}{9}=\frac{664}{9}\approx73.78$.

Answer:

$x=\frac{193}{9},y=\frac{664}{9}$