To solve this right triangle, we’ll follow these steps to determine the values of $XY$, $\angle X$, and $\angle Z$:

1. Identify the triangle sides and angles:

   • Given $Y$ is a right angle (90°).
   • $XY$ is the unknown leg (opposite of $\angle Z$).
   • $YZ = 14$ (opposite of $\angle X$).
   • $XZ = 19$ (hypotenuse).

2. Calculate $XY$ using the Pythagorean theorem: $XY = \sqrt{XZ^2 - YZ^2} = \sqrt{19^2 - 14^2} = \sqrt{361 - 196} = \sqrt{165} \approx 13$

3. Calculate $\angle X$ using trigonometry (sine): $\sin(\angle X) = \frac{YZ}{XZ} = \frac{14}{19} \approx 0.7368$ $\angle X \approx \arcsin(0.7368) \approx 48^\circ$

4. Calculate $\angle Z$ since it complements $\angle X$: $\angle Z = 90^\circ - \angle X \approx 90^\circ - 48^\circ = 42^\circ$

Final Answers:

• $XY \approx 13$
• $m\angle X \approx 48^\circ$
• $m\angle Z \approx 42^\circ$

The lock code (in order of answers) is: 13, 48, 42

Would you like further details or have questions about this solution?

1. How is the Pythagorean theorem used to solve for side lengths?
2. What are the main trigonometric ratios in a right triangle?
3. How can you use complementary angles in right triangles?
4. Why is rounding necessary in trigonometric calculations?
5. How would the process change if $\angle X$ were given instead?

Tip: Remember that in a right triangle, the two acute angles add up to 90°.