Acute △ABC with angles α, β, and γ is inscribed in a circle. Tangents to the circle at points A, B, and C intersect in points M, N, and P. Find measures of angles of the △MNP.

Answer:The measures of angles of triangle MNP areStep-by-step explanation:The picture of the question in the attached figurestep 1Find the measure of arcs AB, BC and ACwe know thatThe inscribed angle is half that of the arc it comprises.so[tex]\gamma=\frac{1}{2}[arc\ AB] ----> arc\ AB=2\gamma\\\alpha=\frac{1}{2}[arc\ BC] ----> arc\ BC=2\alpha\\\beta=\frac{1}{2}[arc\ AC] ----> arc\ AC=2\beta[/tex] step 2Find the measure of angle Mwe know thatThe measurement of the outer angle is the semi-difference of the arcs it encompasses.[tex]M=\frac{1}{2}[arc\ AB+arc\ BC-arc\ AC][/tex]substitute[tex]M=\frac{1}{2}[2\gamma+2\alpha-2\beta]\\M=[\gamma+\alpha-\beta][/tex]step 3   Find the measure of angle Nwe know thatThe measurement of the outer angle is the semi-difference of the arcs it encompasses.[tex]N=\frac{1}{2}[arc\ AC+arc\ BC-arc\ AB][/tex]substitute[tex]N=\frac{1}{2}[2\beta+2\alpha-2\gamma]\\N=[\beta+\alpha-\gamma][/tex]step 4    Find the measure of angle Pwe know thatThe measurement of the outer angle is the semi-difference of the arcs it encompasses.[tex]P=\frac{1}{2}[arc\ AC+arc\ AB-arc\ BC][/tex]substitute[tex]P=\frac{1}{2}[2\beta+2\gamma-2\alpha]\\P=[\beta+\gamma-\alpha][/tex]