Name each shaded angle in three different ways. 6. X Y Z 7. A B C 8. K J 2
The Correct Answer and Explanation is:
An angle is formed by two rays sharing a common endpoint, known as the vertex. There are three standard methods for naming an angle:
- By Three Points: You can name an angle using a point on one ray, the vertex, and a point on the other ray. The vertex letter must always be the middle letter. For example, an angle with vertex P and points Q and R on its rays can be named ∠QPR or ∠RPQ.
- By the Vertex: If there is only one angle associated with a vertex, you can name the angle simply by the vertex letter, such as ∠P. This method should not be used when multiple angles share the same vertex, as it would be ambiguous.
- By a Number: An angle can be named using a number written inside it, such as ∠1. This is often used in complex diagrams to clearly identify specific angles.
Let’s apply these conventions to name each shaded angle.
6.
The shaded angle has its vertex at point Y. The two rays that form the angle pass through points X and Z.
- Using the Vertex: Since there is only one angle shown at vertex Y, we can name it using just the vertex letter: ∠Y.
- Using Three Points: We can name the angle using the three points X, Y, and Z, ensuring the vertex Y is in the middle. This gives us two names: ∠XYZ and ∠ZYX.
Thus, three different ways to name the shaded angle are ∠Y, ∠XYZ, and ∠ZYX.
7.
The shaded angle has its vertex at point B. The rays pass through points A and C. The angle is also labeled with the number 1.
- Using a Number: The number inside the angle provides a simple name: ∠1.
- Using the Vertex: As there is only one angle shown at vertex B, we can call it ∠B.
- Using Three Points: With B as the vertex, we can name the angle ∠ABC (or ∠CBA).
Therefore, three different ways to name the shaded angle are ∠1, ∠B, and ∠ABC.
8.
The vertex for the shaded angle is K. This vertex is shared by multiple angles (∠JKM, ∠MKL, and the larger ∠JKL). The shaded angle is the one between rays KJ and KM, and it is labeled with the number 2.
- Using a Number: The angle is clearly labeled with a 2, so we can call it ∠2.
- Using Three Points: The rays forming the shaded angle pass through points J and M, with K as the vertex. Placing the vertex in the middle, we can name it ∠JKM or ∠MKJ.
- Note: We cannot call this angle ∠K, because it would be ambiguous. It would not be clear if we were referring to ∠JKM, ∠MKL, or ∠JKL.
So, three different ways to name the shaded angle are ∠2, ∠JKM, and ∠MKJ.
