Answer:
m∠CAB = 66°
m∠CAD = 24°
Step-by-step explanation:
m∠CAB
The given parameters are;
The measure of arc m[tex]\widehat{ADC}[/tex] = 228°
The diameter of the given circle = [tex]\overline{AD}[/tex]
The tangent to the circle = [tex]\underset{AB}{\leftrightarrow}[/tex]
The measure of m∠CAB and m∠CAD = Required
By the tangent and chord circle theorem, we have;
m∠CAB = (1/2) × m[tex]\widehat{AC}[/tex]
However, we have;
m[tex]\widehat{AC}[/tex] + m[tex]\widehat{ADC}[/tex] = 360° the sum of angles at the center of a circle is 360°
∴ m[tex]\widehat{AC}[/tex] = 360° - m[tex]\widehat{ADC}[/tex]
Which gives;
m[tex]\widehat{AC}[/tex] = 360° - 228° = 132°
m[tex]\widehat{AC}[/tex] = 132°
Therefore;
m∠CAB = (1/2) × 132° = 66°
m∠CAB = 66°
m∠CAD
Given that [tex]\overline{AD}[/tex] is the diameter of the given circle, we have
The tangent, [tex]\underset{AB}{\leftrightarrow}[/tex], is perpendicular to the radius of the circle, and therefore [tex]\underset{AB}{\leftrightarrow}[/tex] is also perpendicular to the diameter of the circle
∴ m∠DAB = 90° which is the measure of the angle formed by two perpendicular lines
By angle addition property, we have;
m∠DAB = m∠CAB + m∠CAD
∴ m∠CAD = m∠DAB - m∠CAB
By substitution, we have;
m∠CAD = 90° - 66° = 24°
m∠CAD = 24°
