Question

1. find x (2x - 5)° (x + 15)° 22. find x (x - 6)° (x + 12)° 23. find x (3x - 18)° (x + 12)° 24. find x (2x+31)° (x + 23)°

Explanation:

Step1: Use vertical - angles are equal

For problem 21, since vertical angles are equal, we set up the equation $2x - 5=x + 15$.

Step2: Solve the equation for $x$

Subtract $x$ from both sides: $2x-x-5=x - x+15$, which simplifies to $x-5 = 15$. Then add 5 to both sides: $x=15 + 5$. So $x = 20$.

For problem 22, set up the equation $x-6=x + 12$, but this equation has no solution as subtracting $x$ from both sides gives $-6=12$ which is false. There may be a mis - understanding, if we assume they are supplementary (adjacent and linear pair), then $(x - 6)+(x + 12)=180$. Combine like terms: $2x+6 = 180$. Subtract 6 from both sides: $2x=180 - 6=174$. Divide by 2: $x = 87$.

For problem 23, set up the equation $3x-18=x + 12$ (vertical angles are equal). Subtract $x$ from both sides: $3x-x-18=x - x+12$, which gives $2x-18 = 12$. Add 18 to both sides: $2x=12 + 18=30$. Divide by 2: $x = 15$.

For problem 24, set up the equation $2x+31=x + 23$ (vertical angles are equal). Subtract $x$ from both sides: $2x-x+31=x - x+23$, which gives $x+31 = 23$. Subtract 31 from both sides: $x=23 - 31=-8$.

Answer:

1. $x = 20$
2. $x = 87$
3. $x = 15$
4. $x=-8$