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The diagram you've shared shows a geometric figure involving angles and lines intersecting at points \( P \), \( Q \), \( S \), and \( R \). Key details include:

- There is an angle of \( 154^\circ \) at point \( Q \), formed between lines \( PQ \) and \( QR \).
- There is another marked angle of \( 48^\circ \) at point \( S \) between lines \( US \) and \( SR \).
- The diagram includes four unknown angles, labeled as \( a \), \( b \), \( c \), and \( d \).

To determine these unknown angles:

### 1. Angle \( c \):
The angle \( c \) is supplementary to the \( 154^\circ \) angle at point \( Q \), meaning:
\[
c = 180^\circ - 154^\circ = 26^\circ
\]

### 2. Angle \( a \):
Angle \( a \) is vertically opposite to the \( 48^\circ \) angle at point \( S \). Since vertically opposite angles are equal, we have:
\[
a = 48^\circ
\]

### 3. Angle \( b \):
Angle \( b \) and angle \( d \) are part of a straight line, meaning they are supplementary (their sum is \( 180^\circ \)). However, to determine angle \( b \), we first need to calculate angle \( d \).

### 4. Angle \( d \):
The angles around point \( S \) include \( a \) and \( d \), which form a straight line with the \( 48^\circ \) angle. Since a straight line equals \( 180^\circ \):
\[
d = 180^\circ - 48^\circ = 132^\circ
\]

Now, we can solve for angle \( b \), which is supplementary to \( d \):
\[
b = 180^\circ - 132^\circ = 48^\circ
\]

### Summary of angles:
- \( a = 48^\circ \)
- \( b = 48^\circ \)
- \( c = 26^\circ \)
- \( d = 132^\circ \)

Would you like further clarification or more details on any of these steps?

### Related Questions:
1. How can we determine angles in more complex geometric figures with multiple intersecting lines?
2. What is the significance of vertically opposite angles in geometry?
3. Can you explain why angles on a straight line always add up to \( 180^\circ \)?
4. How are supplementary and complementary angles different in geometry?
5. How would the problem change if the lines in the diagram were not straight or parallel?

### Tip:
Always look for supplementary and vertically opposite angles when working with intersecting lines—they often provide the easiest way to solve for unknown angles.

How to solve for angles a, b, c, and d in the geometric diagram, where given angles include 154° and 48°?

Learn how to solve for the unknown angles in a geometric diagram involving a 154° and 48° angle. This problem includes key geometric concepts such as supplementary and vertically opposite angles, suitable for students in Grades 7-9.

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