The measure of the ∠Q is 41°.
What is the Law of Cosine?
The law of cosine helps us to know the third side of a triangle when two sides of the triangle are already known the angle opposite to the third side is given. It is given by the formula,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}[/tex] c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}
where
c is the third side of the triangle
a and b are the other two sides of the triangle,
and θ is the angle opposite to the third side, therefore, opposite to side c.
Given to us
Law of Cosine
According to the law of cosine,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}[/tex]
Substituting the values,
[tex]PR =\sqrt{PQ^2 + RQ^2 -2(PQ)(RQ)\cdot Cos(\angle Q)}[/tex]
[tex]4 =\sqrt{5^2 + 6^2 -2\times 5 \times 6\cdot Cos(\angle Q)}\\\\4^2 =5^2 + 6^2 -2\times 5 \times 6\cdot Cos(\angle Q)\\\\16 = 25 + 36 - 60 Cos(\angle Q)\\\\16 -25 -36= -60 Cos(\angle Q)\\\\-45 = -60 Cos(\angle Q)\\\\\dfrac{45}{60} = Cos(\angle Q)\\\\(\angle Q) = Cos^{-1} (\dfrac{45}{60} )\\\\(\angle Q) = 41.4^o \approx 41^o[/tex]
Hence, the measure of the ∠Q is 41°.
Learn more about the Law of Cosine:
https://brainly.com/question/17289163
