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Let's break down and solve the problems one by one:

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### **22. Given BD = 3, DA = 4. Find \( h \), \( a \), and \( b \).**
*Hint: You will be using proportions and the Pythagorean Theorem.*

This appears to be a right triangle setup where \( BD = 3 \), \( DA = 4 \), and \( D \) is the point on base \( AB \) of triangle \( ABC \). Since \( D \) divides \( AB \) into two segments, the goal is to find \( h \) (the height from \( C \) to \( AB \)), and the lengths of \( a = AD \) and \( b = BD \).

To solve:
1. The full length of \( AB = BD + DA = 3 + 4 = 7 \).
2. Use the Pythagorean Theorem:
   \[
   h^2 + b^2 = c^2
   \]
   Since the right triangle relationship applies, we use proportions to split it into two sub-triangles and then solve for \( h \), \( a \), and \( b \).

---

### **23. Given \( YZ = 5\sqrt{2} \, \text{cm} \) and \( XZ = 9 \, \text{cm} \), find the length of \( YZ \).**

From the problem description, it seems you're given a right triangle with known lengths for \( YZ \) and \( XZ \). To find the missing side, apply the Pythagorean Theorem. 

For a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):
\[
a^2 + b^2 = c^2
\]
Substitute the known values and solve for the unknown side.

---

### **24. Determine the measure of the angles \( x \), \( y \), and \( z \).**

Here, we see a set of angles with given values: one of the angles is \( 134^\circ \) and another is \( 112^\circ \). These angles form a straight line, so the sum of angles around a point is \( 180^\circ \).

- For angle \( x \), subtract from \( 180^\circ \) to find the complement of \( 112^\circ \).
- Angle \( z \) can be calculated similarly by subtracting from \( 180^\circ \) at that point.

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### **25. Determine if triangle \( ABC \) is similar to triangle \( ADE \). Justify completely, including calculations.**

To determine similarity, compare the ratios of corresponding sides or check if the triangles have the same angles.

Using the given side lengths of \( AB \), \( AC \), and the corresponding sides of triangle \( ADE \):
- Apply the properties of similar triangles: corresponding sides should be proportional.
- You can also verify angle relationships to confirm similarity.

### Next Steps:
Would you like more detailed steps for any of these solutions, or a breakdown of the calculations? Here are 5 additional related questions to consider:

1. How do we calculate the height \( h \) in a triangle using proportions?
2. What other geometric properties could help in determining triangle similarity?
3. How can angle bisectors be used to split a triangle and apply the Pythagorean theorem?
4. What are other methods to solve for missing triangle sides besides the Pythagorean Theorem?
5. How does the concept of supplementary angles help in angle calculation problems?

**Tip:** Always double-check calculations by verifying if the sum of angles or side lengths match known geometric properties (like \( 180^\circ \) in triangles).

Problems: Given BD = 3 and DA = 4. Find h, a, and b. Hint: You will be using proportions and Pythagorean Theorem!; Given YZ = 5√2 cm and XZ = 9 cm, find the length of YZ.; Determine the measure of angles x, y, and z.; Determine if triangle ABC is similar to ADE.

Learn how to find the height of a triangle using the Pythagorean theorem, solve for missing side lengths, determine unknown angles in geometric figures, and check for triangle similarity. This lesson covers key concepts like proportions, angle sums, and triangle similarity, suitable for students in Grades 8-10.

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