In triangle ABC the perpendicular side opf line AB intersects the extenstion of line AC at poinmt D. Find the measure of anglke ABC if CBD is 16 degrees and ACB is 118 degrees

Answer 1

To find the measure of angle \( ABC \) in triangle \( ABC \), we can use the information given about angles \( CBD \) and \( ACB \).

1. In triangle \( ABC \), we know:
   • \( \angle ACB = 118^\circ \)
   • \( \angle CBD = 16^\circ \)

Since point \( D \) is on the extension of line \( AC \), angle \( ACD \) does not affect the measurements of angles in triangle \( ABC \) but is important for \( \angle CBD \).

2. Recognizing that angles \( ACB \) and \( CBD \) form a linear pair (they lie on a straight line at point \( C \)), we can find angle \( ACD \): \[ \angle ACD = 180^\circ - \angle ACB = 180^\circ - 118^\circ = 62^\circ \]

3. Next, we can find angle \( ABC \) by using the fact that the sum of angles in triangle \( ABC \) is \( 180^\circ \): \[ \angle ABC + \angle ACB + \angle BAC = 180^\circ \]

4. To determine angle \( BAC \), we utilize angle \( CBD \): Since \( \angle BAC + \angle CBD = \angle ACD \), we can express it as: \[ \angle BAC + 16^\circ = 62^\circ \] Therefore, \[ \angle BAC = 62^\circ - 16^\circ = 46^\circ \]

5. Now we substitute the known angles back into the triangle angle sum equation: \[ \angle ABC + 118^\circ + 46^\circ = 180^\circ \] Simplifying this gives: \[ \angle ABC + 164^\circ = 180^\circ \]

6. Solving for \( \angle ABC \): \[ \angle ABC = 180^\circ - 164^\circ = 16^\circ \]

Thus, the measure of angle \( ABC \) is \( \boxed{16^\circ} \).