To solve this problem, we need to understand that the two lines $WZ$ and $XY$ intersect at point $Q$. Based on the given information:

• $YQ = 63$
• $WQ = 70$
• $WX = 60$
• $XQ = 120$

The problem is asking for the length of $YZ$.

Since $WZ$ and $XY$ are intersecting lines, and we see angles labeled $a^\circ$ on opposite sides, we can assume these are vertical (opposite) angles, meaning they are equal by definition of intersecting lines.

We can apply the Angle Bisector Theorem, which states that if a point lies on the angle bisector of an angle, the segments created by the intersection of the two lines are proportional.

Solution:

Since the two lines intersect and we are given segments on each side of $Q$: $\frac{WQ}{QZ} = \frac{YQ}{QX}$

1. Substitute known values into the proportion: $\frac{70}{QZ} = \frac{63}{120}$

2. Cross-multiply to solve for $QZ$: $70 \times 120 = 63 \times QZ$ $8400 = 63 \times QZ$ $QZ = \frac{8400}{63} = 133.33$

3. Now, to find $YZ$: $YZ = YQ + QZ = 63 + 133.33 = 196.33$

Final Answer:

The length of $YZ$ is approximately $196.33$.

Would you like more details on any step, or have any questions?

1. How is the Angle Bisector Theorem applied in intersecting lines?
2. Why are opposite angles equal in intersecting lines?
3. How can cross-multiplication be used to solve proportions?
4. What other geometric properties can help solve intersecting line problems?
5. How does changing one segment affect the total line length in intersecting problems?

Tip: When dealing with intersecting lines, check for equal opposite angles and apply the Angle Bisector Theorem if given ratios.