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# Tangent question with inscribed angles. Please explain.

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Lines $XTQ$ and $XUR$ are tangent to a circle, as shown below.

If $$\angle ATQ = 41^\circ$$ and $$\angle AUR = 63^\circ,$$ then find $$\angle QXR,$$ in degrees.

Jun 12, 2020

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Lines $$XTQ$$ and $$XUR$$ are tangent to a circle, as shown below.

If $$\angle ATQ = 41^\circ$$ and $$\angle AUR = 63^\circ$$, then find $$\angle QXR$$, in degrees.

$$\text{Let \angle QXR = x  }$$

$$\angle TCX = 360^\circ -2*90^\circ = 180^\circ - x$$

$$\begin{array}{|rcll|} \hline \mathbf{360^\circ} &=& \mathbf{(180^\circ - x)+82^\circ+126^\circ} \\ 360^\circ &=& 388^\circ - x \\ x &=& 388^\circ - 360^\circ \\ \mathbf{x} &=& \mathbf{28^\circ} \\ \hline \end{array}$$

The $$\angle QXR$$ is $$\mathbf{28^\circ}$$

Jun 12, 2020
edited by heureka  Jun 12, 2020