Quadrilateral OPQR is inscribed in circle N, as shown below. What is the measure of ∠QRO?

Question

unofficialllyy · Answer

attachment

unofficialllyy · Answer

I'm not sure how to do this at all :(

unofficialllyy · Answer

@Owlcoffee could you help?

owlcoffee · Answer

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:
$$(x+17)+(6x-5)=180$$
And now, you have to solve for "x".

unofficialllyy · Answer

I got 24 but that's not an answer choice

unofficialllyy · Answer

@Owlcoffee

owlcoffee · Answer

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:
$$2(24)+19$$
Once you operate that, you'll obtain the result.

unofficialllyy · Answer

67. Thanks. Can you help with 1 more? :)

unofficialllyy · Answer

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.

unofficialllyy · Answer

@Owlcoffee

owlcoffee · Answer

What kind of angle do you think $\angle DEF $ is?