Let's solve the problems shown in the uploaded image one by one.

Problem 4: Collinear Points B, D, and E

Given:

• $BD = 30$
• $BE = 15$
• $E$ is between $B$ and $D$.

To find: Measure of $ED$.

Solution: Using the segment addition postulate, we know: $BD = BE + ED$ Substituting the given values: $30 = 15 + ED$ Solve for $ED$: $ED = 30 - 15 = 15$

Answer: $ED = 15$

Problem 5: Line contains $H, J,$ and $K$

Given:

• $HJ = 6x + 4$
• $JK = 7x + 9$
• $HK = 37$

To find: Solve for $x$ using the segment addition postulate.

Solution: Using the segment addition postulate: $HK = HJ + JK$ Substitute the given values: $37 = (6x + 4) + (7x + 9)$ Simplify: $37 = 6x + 4 + 7x + 9$ $37 = 13x + 13$ Solve for $x$: $13x = 37 - 13$ $13x = 24$ $x = \frac{24}{13} \approx 1.8$

Answer: $x = 1.8$ (rounded to the nearest tenth)

Problem 6: Line segment $MY$

Given:

• $MY = 30$
• $EY = 8$
• $OY = 23$
• $ON = NK = KE$

To find: Length of $NY$.

Solution:

• Since $ON = NK = KE$, let each segment be $x$.
• Then: $ON + NK + KE = OY$ $x + x + x = 23$ $3x = 23$ $x = \frac{23}{3} \approx 7.67$

Now, to find $NY$, we add $NK + KE + EY$: $NY = x + x + 8$ $NY = 7.67 + 7.67 + 8 \approx 23.34$

Answer: $NY \approx 23.34$

Do you want further clarification or details on any of these? Let me know!

Here are 5 related questions for further exploration:

1. How do you use the segment addition postulate in geometry?
2. What are the conditions for points to be collinear?
3. How can you set up equations involving line segments for more complex problems?
4. Why is rounding to the nearest tenth important in real-world applications?
5. What are some practical applications of segment addition postulates?

Tip: Always double-check your setup of equations when solving geometry problems to ensure all given values are correctly used.