Question

algebra find the value of each variable and the measure of each angle. 33. (y + x)° 2x° (y - x)° 34. (x + y + 5)° y° 2x° 35.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. In the first figure (33), we have the equation based on vertical - angle equality: $2x=y - x$. Also, since $(y + x)+(y - x)=180^{\circ}$ (linear - pair of angles), we first simplify the linear - pair equation.
$(y + x)+(y - x)=180^{\circ}$ simplifies to $2y = 180^{\circ}$, so $y = 90^{\circ}$.

Step2: Solve for $x$

Substitute $y = 90^{\circ}$ into $2x=y - x$. We get $2x=90 - x$. Add $x$ to both sides: $2x+x=90$, $3x = 90$, so $x = 30^{\circ}$.
The angles are: $2x=60^{\circ}$, $y - x=60^{\circ}$, $y + x=120^{\circ}$.

Step3: For the second figure (34)

Set up equations using vertical - angle and linear - pair properties. Vertical angles give $2x=y$. And from the linear - pair, $(x + y+5)+y=180$.
Substitute $y = 2x$ into $(x + y+5)+y=180$. We have $(x + 2x+5)+2x=180$.

Step4: Simplify the equation

Combine like terms: $x+2x + 5+2x=180$, $5x+5 = 180$. Subtract 5 from both sides: $5x=175$. Divide by 5: $x = 35^{\circ}$.

Step5: Solve for $y$

Since $y = 2x$, then $y = 70^{\circ}$.
The angles are: $y = 70^{\circ}$, $2x=70^{\circ}$, $x + y+5=110^{\circ}$.

Answer:

For problem 33: $x = 30^{\circ}$, $y = 90^{\circ}$, angles are $60^{\circ},60^{\circ},120^{\circ},120^{\circ}$.
For problem 34: $x = 35^{\circ}$, $y = 70^{\circ}$, angles are $70^{\circ},70^{\circ},110^{\circ},110^{\circ}$.