Thales' Theorem

By Thales’s Theorem, when we have a circle with diameter AB and any point C on the semicircle, the angle $\angle ACB$ will always be a right angle, regardless of the location of point C on the semicircle.

By Thales’s Theorem, when we have a circle with a diameter AB and any point C on the semicircle, the angle $\angle ACB$ will always be a right angle, regardless of the location of point C on the semicircle.

Since AB is the diameter of the semicircle, Thales’s Theorem tells us that $\angle ACB=90^{\circ }$. We are given that $\angle CAB=65^{\circ }$. In any triangle, the sum of the angles is $180^{\circ }$. Therefore, to find $\angle ABC$, we use $\angle ABC=180^{\circ }-90^{\circ }-65^{\circ }=25^{\circ }$.

Thales’s Theorem tells us that $\angle ACB=90^{\circ }$. We are given that $\angle ABC=35^{\circ }$, and we know that the sum of the angles in any triangle is $180^{\circ }$. To find $\angle CAB$, we calculate $\angle CAB=180^{\circ }-90^{\circ }-35^{\circ }=55^{\circ }$.

Thales’s Theorem states that when AB is the diameter of a semicircle, $\angle ACB$ will always be $90^{\circ }$, no matter the values of the other angles.

By Thales’s Theorem, since AB is the diameter of the semicircle, $\angle ACB=90^{\circ }$. Then, in right-angled triangle ABC, we can calculate $\angle A=180^{\circ }-90^{\circ }-50^{\circ }=40^{\circ }$. Finally, in right-angled triangle ACD, we find $\angle ACD=180^{\circ }-90^{\circ }-40^{\circ }=50^{\circ }$.