Select all that are true
With which information can you construct more than one triangle?
The measurements of two angles
The measurements of two angles and the length of the included side
The measurement of all angles
The lengths of two sides and the measurement of the included angle
The correct options are:
- The measurements of two angles and the length of the included side
- The lengths of two sides and the measurement of the included angle
To construct a triangle, you need to know certain information about its angles and sides. Let's analyze each option to determine which ones are true:
1. The measurements of two angles: This is not enough information to construct a triangle. When you know only the measurements of two angles, the third angle can have multiple values, which may make it impossible to form a triangle.
2. The measurements of two angles and the length of the included side: This is enough information to construct a triangle. When you have the measurements of two angles and the length of the side between them (included side), you can use the Angle-Side-Angle (ASA) postulate to construct a triangle.
3. The measurement of all angles: This is not enough information to construct a triangle. Knowing only the measurements of all three angles doesn't provide any information about the lengths of the sides.
4. The lengths of two sides and the measurement of the included angle: This is enough information to construct a triangle. When you know the lengths of two sides and the measurement of the included angle, you can use the Side-Angle-Side (SAS) postulate to construct a triangle.
Therefore, the true statement is: "The measurements of two angles and the length of the included side" and "The lengths of two sides and the measurement of the included angle" are both sufficient to construct more than one triangle.
The measurements of two angles - Many triangles
The measurements of two angles and the length of the included side - Only one triangle
The measurement of all angles - Many triangles
The lengths of two sides and the measurement of the included angle - Only one triangle