Quadrilateral OPQR is inscribed in circle N, as shown below. What is the measure of ∠QRO?
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There a property for quadrilaterals that is useful for calculating inner angles of any geometric place that is inscribed on a circle by 4 points (quadrilateral inscribed).
It states that "opposite angles are supplementary", what do we conclude from this?
In your diagram, it means that \(\angle ROP\) and \(\angle RQP\) must sum up to 180 degrees, and so must \(\angle QRO \) and \(\angle QPO\).
But since \(\angle QPO\) is unknown, we will only work with \(\angle ROP\) and \(\angle RQP\) because with them we can calculate "x" and then replace it on the equation that angle \(\angle QRO \) has as it's measure.
So, let's proceed and work with \(\angle ROP\) and \(\angle RQP\).
We will apply the the property I stated to you at the beginning, wich means that they must sum up to 180.
\[\angle ROP + \angle RQP = 180\]
But \(\angle ROP = (x+17)\) and \(\angle RQP = (6x-5)\), so we will replace those values:
And now, you have to solve for "x".
I got 24 but that's not an answer choice
That's because you only found "x" and not the desired angle.
Since the measure of \(\angle QRO\) is "\(2x+19\)" and you have found "x", so all you have to do is replace it, and it becomes:
Once you operate that, you'll obtain the result.
67. Thanks. Can you help with 1 more? :)
If the measure of arc DEF is 232°, what is the measure of ∠DEF?
I got 64 but I am not 100% sure if it is correct.