Question

1. 43° 24° 57°

Explanation:

Step1: Find \( z \)

Use the exterior - angle property of a triangle (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles).
For the smaller triangle with angles \( 24^{\circ}\) and \( 57^{\circ}\), \( z=24 + 57\)
\( z = 81\)

Step2: Find \( y \)

Since \( y+z = 180\) (linear pair of angles), then \( y=180 - z\)
Substitute \( z = 81\) into the equation: \( y=180 - 81\)
\( y = 99\)

Step3: Find \( x \)

Use the angle - sum property of a triangle (\(x + 43+(y - 24)=180\))
First, \(y - 24=99 - 24=75\)
Then \(x+43 + 75=180\)
\(x=180-(43 + 75)\)
\(x=180 - 118\)
\(x = 62\)

Answer:

\(x = 62\), \(y = 99\), \(z = 81\)