To find the measure of angle ∠OPQ in the cyclic quadrilateral OPQR that is inscribed in circle N, we can use the property that opposite angles in a cyclic quadrilateral are supplementary.
Given that:
- m∠O = \(x + 16\)
- m∠Q = \(6x - 4\)
- m∠R = \(2x + 16\)
We are looking for a way to relate the measures of the various angles in the quadrilateral involving ∠OPQ.
The most appropriate relationships to use for a cyclic quadrilateral are:
- The sum of the opposite angles equals \(180^\circ\).
- To find the relationship with neighboring angles around the vertex O or P.
Considering the options given:
-
m∠OPQ + (2x + 16)° = 180°: This would imply that angle ∠OPQ and angle ∠R (which is \(2x + 16\)) are supplementary. This is valid since they are opposite angles.
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m∠OPQ = (6x − 4)° + (2x + 16)°: This expression incorrectly suggests a direct equivalence that doesn't relate to the properties of cyclic quadrilaterals.
-
m∠OPQ + (x + 16)° + (6x − 4)°= 360°: This suggests summing all angles equal to \(360°\). This does not correctly use the nature of opposite angles.
-
m∠OPQ = (x + 16)° + (6x − 4)°: This also incorrectly suggests a direct equivalence that does not come from the properties of a cyclic quadrilateral.
Thus, the best choice that utilizes the property of supplementary opposite angles correctly is:
m∠OPQ + (2x + 16)° = 180°.
3 answers