Properties of circles

Question

I'm told to find angle CAD and I got stuck...

Given that CD = BC

My workings ..

CAT = ACT = $180-40/2 = 70$

CAP = $180-70-60=50$

Answer 1

I am not sure if you are familiare with using arcs or not, but this is a way that you might follow.

Indeed, to find the angle CAD enough to find the value of the arc CD, then $$ \hat{CAD}= \frac{1}{2} \overset{\frown}{CD} $$

Note that, $\overset{\frown}{CD} = 90$, since $BC=CD$. Thus $\hat{CAD}= \frac{90}{2}=45 $

Answer 2

The angle formed by the ray $Cy$ ($Cy$ and $CT$ rays are opposite) and $\hat{yCA}$ is $110^\circ$, which implies $\hat{ABC} =110^\circ$. The previous result follows from the fact:

the angle which is between a tangent and a chord is equal to the inscribed angle which corresponds to the chord.

Due to $\hat{ABC} = 110^\circ \implies \hat{CBP} = 70^\circ \implies \hat{PCB} = 50^\circ$.

Using the fact, we have that $\hat {CDB} = 50^\circ = \hat{DBC}, $ since $\triangle BCD$ is isosceles. Thus, $\hat{DCB} = 80^\circ$.

But $ABCD$ is a cyclic (or inscribed) quadrilateral, which means that the opposite angles are supplementary angles, thus $\hat{DAB} = 100^\circ$.

But $\hat{DAC} = \hat{DAB} - \hat{CAP} = 100^\circ - 50^\circ = 50^\circ$.