In the circle below, AD is a diameter and AB is tangent at A. suppose mADC=228. Find the measures of mCAB and mCAD. Type your numerical answers without units in each blank

m∠CAB = 66°

m∠CAD = 24°

Step-by-step explanation:

m∠CAB

The given parameters are;

The measure of arc m = 228°

The diameter of the given circle =

The tangent to the circle =

The measure of m∠CAB and m∠CAD = Required

By the tangent and chord circle theorem, we have;

m∠CAB = (1/2) × m

However, we have;

m + m = 360° the sum of angles at the center of a circle is 360°

∴ m = 360° - m

Which gives;

m = 360° - 228° = 132°

m = 132°

Therefore;

m∠CAB = (1/2) × 132° = 66°

m∠CAB = 66°

m∠CAD

Given that   is the diameter of the given circle, we have

The tangent, , is perpendicular to the radius of the circle, and therefore 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°

i'm not entirely sure

step-by-step explanation:

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