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.

The measures of angles of triangle MNP are

Step-by-step explanation:

step 1

Find the measure of arcs AB, BC and AC

we know that

The inscribed angle is half that of the arc it comprises.

so

---->

---->

---->

step 2

Find the measure of angle M

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

substitute

step 3    

Find the measure of angle N

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

substitute

step 4    

Find the measure of angle P

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

substitute

x = 17

step-by-step explanation:

in the given figure, two triangles are inscribed in a circle where ∠bae ≅ ∠dae which makes the side be and de equal so we can write it as:

3x - 24 = x + 10

solving this equation for x to get:

3x - x = 10 + 24

2x = 34

x = 34/2

x = 17

therefore, the value of x = 17.