PQ and RS intersect at O if d:c=5:7 find the angle a,b,c,d

To solve this problem, we need to understand the configuration of the intersecting lines and the given ratio of \( d:c = 5:7 \). 

### Understanding the Problem:

- \(PQ\) and \(RS\) are two intersecting lines that meet at point \(O\).
- Angles around the intersection are usually named \(a\), \(b\), \(c\), and \(d\).
- The ratio \(d:c = 5:7\) suggests that \(d\) and \(c\) are two adjacent angles formed by the intersecting lines at point \(O\).
- Let's assume the angles around point \(O\) are labeled as \(a, b, c,\) and \(d\) in a consecutive manner around the intersection.

### Step-by-step Solution:

1. **Identify Relationships Between Angles:**
   - When two lines intersect, opposite angles (also called vertically opposite angles) are equal. Therefore:
   \[
   a = c \quad \text{and} \quad b = d.
   \]

2. **Set up the Ratio and Equation for Adjacent Angles:**
   - The given ratio \(d:c = 5:7\) can be written as:
   \[
   \frac{d}{c} = \frac{5}{7}.
   \]
   - From the ratio, express \(d\) in terms of \(c\):
   \[
   d = \frac{5}{7}c.
   \]

3. **Apply the Angle Sum Property Around Point \(O\):**
   - The sum of angles around a point is \(360^\circ\):
   \[
   a + b + c + d = 360^\circ.
   \]
   - Substitute \(a = c\) and \(b = d\):
   \[
   c + d + c + d = 360^\circ \implies 2c + 2d = 360^\circ.
   \]
   - Simplify to:
   \[
   c + d = 180^\circ.
   \]

4. **Substitute the Ratio Equation:**
   - Using \(d = \frac{5}{7}c\):
   \[
   c + \frac{5}{7}c = 180^\circ.
   \]
   - Combine terms:
   \[
   \left(1 + \frac{5}{7}\right)c = 180^\circ \implies \frac{12}{7}c = 180^\circ.
   \]
   - Solve for \(c\):
   \[
   c = 180^\circ \times \frac{7}{12} = 105^\circ.
   \]

5. **Find \(d\) Using the Ratio:**
   - Using \(d = \frac{5}{7}c\):
   \[
   d = \frac{5}{7} \times 105^\circ = 75^\circ.
   \]

6. **Determine the Remaining Angles \(a\) and \(b\):**
   - Since \(a = c\) and \(b = d\):
   \[
   a = 105^\circ, \quad b = 75^\circ.
   \]

### Summary of Angles:

- \(a = 105^\circ\)
- \(b = 75^\circ\)
- \(c = 105^\circ\)
- \(d = 75^\circ\)

### Conclusion:
The angles \(a\) and \(c\) are both \(105^\circ\), and the angles \(b\) and \(d\) are both \(75^\circ\).

Would you like further details or have any questions?

### Related Questions:
1. How do vertically opposite angles work when two lines intersect?
2. What is the sum of adjacent angles formed by two intersecting lines?
3. Can you solve for the angles if the ratio given was \(c:d = 3:4\)?
4. How do you convert an angle ratio to actual angle values?
5. What happens to the angle measurements if one line rotates while the other stays fixed?

**Tip:** Always remember that vertically opposite angles are equal when two lines intersect.

Explore how to find angles a, b, c, and d at the intersection of lines PQ and RS given the ratio d:c = 5:7. Learn the application of the angle sum property around a point and the theorem on vertical angles. Suitable for high school students.

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