In the diagram below, start overline, A, D, end overline, is perpendicular to, start overline, D, C, end overline, comma

Question

In the diagram below,   start overline, A, D, end overline, is perpendicular to, start overline, D, C, end overline, comma 
AD
 ⊥ 
DC
 ,   m, angle, C, equals, 55, degreesm∠C=55 
∘
  and m, angle, C, A, B, equals, 78, degrees, .m∠CAB=78 
∘
 . Find m, angle, D, A, B, .m∠DAB.

GPT-4o mini · Answer

To find the measure of angle $ DAB $, we start by using the information given in the problem. We know the following: - $ AD \perp DC $ implies that angle $ ADC = 90^\circ $. - $ m\angle CAB = 78^\circ $. - $ m\angle C = 55^\circ $. We will use the facts about the angles around point $ A $. In triangle $ ACD $: 1. The sum of the angles in triangle $ ACD $ is $ 180^\circ $. 2. We have $ m\angle ACD = 55^\circ $ and $ m\angle ADC = 90^\circ $. Let $ m\angle DAB $ be represented as $ x $. From the angle sum property, we can write: $$ m\angle DAC + m\angle ACD + m\angle ADC = 180^\circ $$ Substituting the known values: $$ x + 55^\circ + 90^\circ = 180^\circ $$ Combining the angles on the left side: $$ x + 145^\circ = 180^\circ $$ Now, we can solve for $ x $: $$ x = 180^\circ - 145^\circ $$ $$ x = 35^\circ $$ Therefore, $ m\angle DAB = 35^\circ $. So, the measure of angle $ DAB $ is $ \boxed{35^\circ} $.