A circle is a curve sketched out by a point moving in a plane. The measurements of the angle ∠CQJ and ∠LIJ are 99° and 118° respectively.
What is a circle?
A circle is a curve sketched out by a point moving in a plane so that its distance from a given point is constant; alternatively, it is the shape formed by all points in a plane that are at a set distance from a given point, the centre.
Given that measure of KJ is 124°, therefore, the measure of the ∠KOJ is 124°. Since the angle formed at the centre and at any point on the circle form the same chord are in the ratio of 2:1. Thus,
[tex]2\angle KCJ = \angle KOJ = 124^o\\\\\angle KCJ = 62^o[/tex]
Similarly, the measure of IC is 38°, therefore, the measure of ∠IOC is 38°. Further, it can be written as,
[tex]\angle IOC = 2\angle IJC = 38 ^o\\\\\angle IJC = 19^o[/tex]
Now, In the triangle CQJ, the sum of all the angles can be written as,
[tex]\angle CQJ + \angle QJC + \angle QCJ = 180^o\angle CQJ = 99^o[/tex]
As the ∠IKC and ∠IJC are formed from the same points in the circle their measurement will be equal as well, therefore,
[tex]\angle IKC = \angle IJC = 19^o[/tex]
Since ∠KCL and ∠KCj are supplementary angles, their measure can be written as,
[tex]\angle KCL + \angle KCJ = 180^o\\\angle KCL = 118^o[/tex]
In the triangle LCK, the sum of the angles can be written as,
[tex]\angle KCL+ \angle CKL+ \angle CLK= 180^o\\\\118^o + 19^o + \angle CLK= 180^o\\\\\angle CLK= 43^o[/tex]
In the triangle LIJ, the sum of the angles can be written as,
[tex]\angle LIJ+ \angle IJL+ \angle JLI= 180^o\\\\ \angle LIJ + 19^o + 43^o= 180^o\\\\\angle CLK= 118^o[/tex]
Thus, the measurements of the angle ∠CQJ and ∠LIJ are 99° and 118° respectively.
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