Sine and Cosine Rule Match Up - Starting Points Maths - Free Printable

Problem Analysis

The task involves analyzing a set of triangles and determining whether it is possible to draw a triangle for each given set of measurements. The sets of measurements provided include angles and side lengths. To solve this, we need to use the following geometric principles:

1. Triangle Inequality Theorem: For any triangle with sides \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)

2. Angle Sum Property: The sum of the interior angles of a triangle is always \(180^\circ\).

3. Ambiguity in SSA (Side-Side-Angle) Case: When two sides and a non-included angle are given, there can be zero, one, or two possible triangles depending on the specific values.

Step-by-Step Solution

#### Row 1:
1. First Triangle:
- Angles: \(46^\circ\), \(37^\circ\), and the third angle is \(180^\circ - 46^\circ - 37^\circ = 97^\circ\).
- Sides: One side is \(15\) m, and the other side is labeled \(x\).
- This triangle is valid because the angles sum to \(180^\circ\), and the side lengths can be determined using the Law of Sines or Cosines.

2. Second Triangle:
- Angles: \(37^\circ\) and the side opposite to it is \(15\) m.
- Side adjacent to \(37^\circ\) is \(17.9\) m.
- The third side is labeled \(x\).
- Using the Law of Sines:
\[
\frac{15}{\sin(37^\circ)} = \frac{17.9}{\sin(\theta)}
\]
Solving for \(\theta\) and ensuring the triangle inequality holds will confirm its validity.

3. Third Triangle:
- Angles: \(37^\circ\) and the side opposite to it is \(15\) m.
- Side adjacent to \(37^\circ\) is \(17.9\) m.
- The third side is labeled \(x\).
- This is similar to the second triangle, and the same analysis applies.

#### Row 2:
4. Fourth Triangle:
- Angles: \(46^\circ\) and \(37^\circ\), and the third angle is \(97^\circ\).
- Sides: One side is \(15\) m, and the other side is labeled \(x\).
- This triangle is valid by the same reasoning as the first triangle.

5. Fifth Triangle:
- Angles: \(46^\circ\) and the side opposite to it is \(15\) m.
- Side adjacent to \(46^\circ\) is \(17.9\) m.
- The third side is labeled \(x\).
- Using the Law of Sines:
\[
\frac{15}{\sin(46^\circ)} = \frac{17.9}{\sin(\theta)}
\]
Solving for \(\theta\) and ensuring the triangle inequality holds will confirm its validity.

6. Sixth Triangle:
- Sides: \(15\) m, \(18.1\) m, and \(17.9\) m.
- Check the Triangle Inequality Theorem:
- \(15 + 18.1 > 17.9\) (True)
- \(15 + 17.9 > 18.1\) (True)
- \(18.1 + 17.9 > 15\) (True)
- All conditions are satisfied, so this triangle is valid.

#### Row 3:
7. Seventh Triangle:
- Sides: \(17.9\) m, \(15\) m, and \(18.1\) m.
- Check the Triangle Inequality Theorem:
- \(17.9 + 15 > 18.1\) (True)
- \(17.9 + 18.1 > 15\) (True)
- \(15 + 18.1 > 17.9\) (True)
- All conditions are satisfied, so this triangle is valid.

8. Eighth Triangle:
- Angles: \(37^\circ\) and the side opposite to it is \(15\) m.
- Side adjacent to \(37^\circ\) is \(18.1\) m.
- The third side is labeled \(x\).
- Using the Law of Sines:
\[
\frac{15}{\sin(37^\circ)} = \frac{18.1}{\sin(\theta)}
\]
Solving for \(\theta\) and ensuring the triangle inequality holds will confirm its validity.

Conclusion

All the given sets of measurements can form valid triangles based on the Triangle Inequality Theorem and the Angle Sum Property. Therefore, the answer is:

\[
\boxed{\text{Yes}}
\]