Let's analyze each option presented in the context of the given rhombus and determine which are true.
1. Option a: \( a = 60^\circ \)
- In our rhombus, each of the four triangles is congruent and combine to form the rhombus. If one of the diagonals is equal to the side length of the rhombus, it indicates that the rhombus can be partitioned into equilateral triangles. Therefore, each angle of an equilateral triangle is \( 60^\circ \). Hence, the measure of angle \( a \) is \( 60^\circ \).
- Therefore, Option a is true.
2. Option b: \( x = 3 \) inches
- To verify this, we need information about the side length of the rhombus or other related dimensions, which are not directly provided here. There's no basis to assert that \( x = 3 \) inches without more data.
- Therefore, Option b is false.
3. Option c: The perimeter of the rhombus is 16 inches
- The perimeter of a rhombus is calculated as \( 4 \times \text{side length} \). To validate this option, the side length would have to be \( 4 \) inches (\( 16/4 \)). Since we lack information on side length, the perimeter cannot be confirmed.
- Therefore, Option c is false.
4. Option d: The measure of the greater interior angle of the rhombus is \( 90^\circ \)
- The interior angles of a rhombus formed by congruent equilateral triangles cannot be \( 90^\circ \) because in an equilateral triangle, each angle is \( 60^\circ \). The angles in a rhombus are generally less than \( 90^\circ \) or greater but not exactly \( 90^\circ \) in this context.
- Therefore, Option d is false.
5. Option e: The length of the longer diagonal is approximately 7 inches
- If the perimeter of the rhombus is \( 16 \) inches, the side length is \( 4 \) inches. For a rhombus formed by equilateral triangles, the diagonal would be calculated as \( \text{side length} \times \sqrt{3} \approx 4 \times 1.732 \approx 6.928 \), which is approximately \( 7 \) inches.
- Therefore, Option e is true.
Based on these analyses, the measures that are true for the quilt piece are:
- Option a: \( a = 60^\circ \)
- Option e: The length of the longer diagonal is approximately 7 inches
Thus, the selections are:
1. \( a = 60^\circ \)
2. The measure of the greater interior angle of the rhombus is approximately \( 7 \) inches
The true options are: (a), (e). Thus, there are only two true answers available in the given options.
