Triangle ABC has vertices B and C on a semicircle centered at O, as shown, with AB tangent to the semicircle at B and AC intersecting the semicircle at point D.

If angle BAC = 78 degrees (in red) and angle ODC = 74 degrees (in green), what is the measure of angle BCA (in blue) in degrees?

Guest May 15, 2020

#1**+1 **

**Triangle ABC has vertices B and C on a semicircle centered at O, as shown, with AB tangent to the semicircle at B and AC intersecting the semicircle at point D.**

**If angle BAC = 78 degrees (in red) and angle ODC = 74 degrees (in green), **

**what is the measure of angle BCA (in blue) in degrees?**

\(\begin{array}{|rcll|} \hline \mathbf{x+\varphi} &=& \mathbf{74^\circ} \quad | \quad \text{in triangle ODC} \\ \mathbf{\varphi} &=& \mathbf{74^\circ-x } \\\\ \mathbf{ 78^\circ +(90^\circ-\varphi) +x} &=& \mathbf{180^\circ} \quad | \quad \text{in triangle BAC} \\ 78^\circ +\Big(90^\circ-(74^\circ-x)\Big) +x &=& 180^\circ \\ 78^\circ +(90^\circ-74^\circ+x) +x &=& 180^\circ \\ 78^\circ + 90^\circ-74^\circ+x +x &=& 180^\circ \\ 94^\circ+2x &=& 180^\circ \quad | \quad :2 \\ 47^\circ+ x &=& 90^\circ \\ x &=& 90^\circ -47^\circ \\ \mathbf{x} &=& \mathbf{43^\circ} \\ \hline \end{array}\)

The measure of angle BCA is \(\mathbf{43^\circ}\)

heureka May 15, 2020