The image shows a problem where we need to solve for the values of $x$ and $z$ given two pairs of vertical angles.

The equations are based on the properties of vertical angles:

1. Vertical angles are equal, meaning that: $(6x + 7)^\circ = (5x - 14)^\circ$

2. Also, the angle labeled $z^\circ$ is vertically opposite the angle labeled $(6x + 7)^\circ$, so: $z^\circ = (6x + 7)^\circ$

Step 1: Solve for $x$

We start with the equation: $6x + 7 = 5x - 14$

Subtract $5x$ from both sides: $x + 7 = -14$

Now subtract 7 from both sides: $x = -21$

Step 2: Solve for $z$

Since $z = 6x + 7$, substitute $x = -21$ into the equation for $z$: $z = 6(-21) + 7 = -126 + 7 = -119$

Thus, the values are: $x = -21, \quad z = -119$

Would you like more details or have any questions?

5 Related Questions:

1. How do we solve equations involving supplementary angles?
2. What is the general property of vertical angles?
3. Can vertical angles ever be unequal?
4. How do we identify corresponding angles in parallel line problems?
5. What if the angles were complementary instead of vertical?

Tip:

When solving equations involving angles, always check if the angles are vertical, supplementary, or complementary, as this helps in setting up the correct relationships.