Question

Given: KLIJ is inscribed in circle k(O)
m∠K = (9x+1)°,
m (arc) LI = (10x−1)°
m (arc) IJ = 59°,
m (arc) KJ =97°
Find: All angles of KLIJ

Answer 1

Check the picture below.

let's notice that the angle at K is an inscribed angle with an intercepted arc

$\bf \stackrel{\textit{using the inscribed angle theorem}}{K=\cfrac{\widehat{LI}+\widehat{IJ}}{2}}\implies 9x+1=\cfrac{(10x-1)+59}{2} \\\\\\ 9x+1=\cfrac{10x+58}{2}\implies 18x+2=10x+58\implies 8x+2=58 \\\\\\ 8x=56\implies x=\cfrac{56}{8}\implies x=7 \\\\[-0.35em] ~\dotfill\\\\ K=9x+1\implies K=9(7)+1\implies \boxed{K=64}$

now, let's notice something again, the angle at L is also an inscribed angle, intercepting and arc of 97 + 59 = 156, so then, by the inscribed angle theorem,

∡L is half that, or 78°.

now, let's take a look at the picture down below, to the inscribed quadrilateral conjecture, since ∡J and ∡I are both supplementary angles, then

∡I = 180 - 64 = 116°.

∡J = 180 - 78 = 102°.

Answer 2

The measure of all the angle of KLIJ which is inscribed in circle k(O) are, 64, 78, 116, 102 degrees.

What is inscribed angle theorem?

Inscribed angle theorem is the theorem, which state that the angle inscribed in a circle will be half of the angle which delimits the same arc on the circle.

The quadrilateral KLIJ is inscribed in circle k(O). In this the measure of the angle are given as,

$m\angle K = (9x+1)^o$

m (arc) LI = (10x−1)°

m (arc) IJ = 59°,

m (arc) KJ =97°

All angles of the quadrilateral KLIJ has to be found out. By the inscribed angle theorem,

$K=\dfrac{LI+IJ}{2}\\9x+1=\dfrac {10x-1+59}{2}\\18x+2=10x+58\\8x=56\\x=7$

Therefore, the value of the angle k is,

$m\angle K = (9(7)+1)^o\\m\angle K = 64^o$

Similarly, the measure of the angle L is,

$m\angle L=\dfrac{KJ+IJ}{2}\\m\angle L=\dfrac {97+59}{2}\\m\angle L=\dfrac{156}{2}\\m\angle L=78^o$

Now the angles I and J are the supplementary angles of the angle K and angle L respectively. Therefore,

$m\angle I=180-m\angle K=180-64=116^o\\ m\angle J=180-m\angle L=180-78=102^o$

Hence, the measure of all the angle of KLIJ which is inscribed in circle k(O) are, 64, 78, 116, 102 degrees.

Learn more about the inscribed angle theorem here;

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