To solve for the missing side lengths in this triangle, we need to break down the information:

Given:

• $xy = 10$
• $\angle y = 60^\circ$
• $\angle z = 45^\circ$

It appears the triangle is divided into two smaller right triangles at point $w$.

Let's solve for the side lengths step-by-step using trigonometric relationships.

Step 1: Label the known angles and distances:

1. $\triangle xyw$ is a right triangle with:

   • $\angle ywx = 90^\circ$
   • $\angle xyw = 60^\circ$
   • $xy = 10$

2. $\triangle yzw$ is another right triangle with:

   • $\angle zwy = 45^\circ$
   • $\angle wyz = 90^\circ$

We will now use trigonometric relationships (sine, cosine, tangent) to find the other side lengths.

Step 2: Use trigonometry in $\triangle xyw$

For triangle $xyw$:

• $xy = 10$
• The angle $\angle xyw = 60^\circ$

We can use the sine and cosine functions to find the lengths of sides $yw$ and $xw$.

• $\sin(60^\circ) = \frac{yw}{xy}$ $yw = xy \cdot \sin(60^\circ) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66$

• $\cos(60^\circ) = \frac{xw}{xy}$ $xw = xy \cdot \cos(60^\circ) = 10 \cdot \frac{1}{2} = 5$

Step 3: Use trigonometry in $\triangle yzw$

For triangle $yzw$:

• The angle $\angle wyz = 45^\circ$

We know that in a 45-45-90 right triangle, the legs are equal. Therefore:

• $yw = wz$

Thus, $wz = yw = 8.66$.

Using the Pythagorean theorem to find $yz$: $yz = \sqrt{yw^2 + wz^2} = \sqrt{(8.66)^2 + (8.66)^2} \approx \sqrt{149.86} \approx 12.25$

Final Side Lengths:

1. $xy = 10$ (given)
2. $yw = 8.66$
3. $xw = 5$
4. $wz = 8.66$
5. $yz = 12.25$

Would you like any further clarifications on these steps?

Here are five questions to further explore:

1. How can we calculate the area of the full triangle using the side lengths?
2. How does the use of the sine rule differ from the cosine rule in this context?
3. Can we verify the lengths using the Law of Cosines for non-right triangles?
4. How would the problem change if $\angle y$ were different?
5. What is the importance of knowing the Pythagorean theorem in right triangles?

Tip: Remember to always check the triangle type before applying trigonometric rules (right triangle, acute, obtuse, etc.).