Angles Relationship (Adjacent Angles - Vertical Angles - Angles At A Point)

Definition

Two angles is called adjacent angles if they share the same vertex and the same side/leg. Look at the following figure.

Note: The same vertex A and same side AD.

Vertical angles are angles which is constructed by two intersecting lines.

Note:

The intersecting lines AB and CD at point E.
The angle p is vertically opposite with angle r and angle q vertically opposite with angle s.

"Two opposite vertical angles are equal".

Proof (Based on the figure above)

Angle p and angle q are supplementary angles, then p + q = 180º and the magnitude of q = 180º - p .... (1)
Angle p and angle s are supplementary angles, then p + s = 180º and the magnitude of s = 180º - p .... (2)

From (1) and (2) we can conclude that q = s (q.e.d)

Angles at a point are angles which share the same vertex and add up to 360º.
Look at the following figure.

Note: the same vertex B and x + y + z = 360º.

Example.
Two adjacent angles ABC and CBD is add up to 138º as in the following figure.

Determine the value of x and magnitude of angle ABE.
Solution:
it is given:
ABC + CBD = 138º , ABC = 90º then
CBD = 138º - 90º = 48º.

CBD = 2x then
2x = 48º
x = 48º/2
x = 24º

DBE = 3x
DBE = 3 . 24º
DBE = 72º

ABE + DBE + CBD + ABC = 360º (angles at a point)
ABE + 72º + 48ºº + 90º = 360º
ABE = 360º - 72º - 48º - 90º
ABE = 150º

The value of x is 24º and the magnitude of angle ABE is 150º.º