To construct angle XYZ with XY = 8.3 cm and YZ = 11.9 cm, follow these steps:
1. Draw a line segment XY of length 8.3 cm.
2. At point Y, draw a ray in any direction to form an angle with XY.
3. Using a compass, draw an arc with center at point Y and radius 11.9 cm. This arc should intersect the ray drawn in step 2 at point Z.
4. Draw the line segment YZ of length 11.9 cm.
5. Using a compass, draw an arc with center at point X and radius equal to the length of segment YZ. This arc should intersect segment XY at two points. Label the point of intersection closest to Y as M.
6. Draw a line segment XM and a line segment ZM.
7. Angle XYZ is the angle formed by segments XY and YZ.
To confirm that angle XYZ is 60 degrees, we need to show that XMZ is also a 60-degree angle. Since M is the midpoint of XZ, we have:
XM = MZ
Therefore, triangles XMY and ZMY are congruent by the Side-Side-Side (SSS) criterion. Thus, angles XMY and ZMY are congruent. Since they form a straight line, we know that:
angle XMY + angle ZMY = 180 degrees
Therefore, each of these angles measures:
angle XMY = angle ZMY = 180 degrees / 2 = 90 degrees
Since angle XYZ is the sum of angles XMY and ZMY, we have:
angle XYZ = 90 degrees + 90 degrees = 180 degrees
This means that angle XYZ is a straight angle, which measures 180 degrees. However, we know that XYZ is a 60-degree angle, so we must have made an error in the construction. Double-check the construction steps to make sure that each step was performed accurately.