Need help on two practice problems

JSZ2005

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Jun 11, 2019
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Hi everyone I am studying for a Geometry test out and i was doing the problems on the online practice quizzes and i had 2 that i got stuck on
Can someone explain them that would be super helpful
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Dr.Peterson

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#34 is a horrible question. I don't like the word "must"; there are many ways to prove an isosceles trapezoid, and none of the methods listed is directly related to the definition, or is sufficient as a proof.

Take it just as asking, "Which of the following are true?" Sketch the diagram and determine the various lengths and slopes, then compare with the statements.

#40 is a fairly direct application of some circle theorems. Focus on x first. Label the diagram with the given arc measures, ignoring the tangent at C. What theorem applies?

Be sure to read and follow our submission guidelines: Tell us what you have learned and what ideas you have, at least, so we can see where you really need help.
 

pka

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Hi everyone I am studying for a Geometry test out and i was doing the problems on the online practice quizzes and i had 2 that i got stuck on
Can someone explain them that would be super helpful
View attachment 12532
View attachment 12533
In addition to Prof Peterson's comments, see here for properties of the isosceles trapezoid. It seems the question is going for the diagonals have equal lengths. However, depending upon the sizes of \(\displaystyle a,~\&~b\) the formulae are ambiguous.

For the other, let \(\displaystyle D\) be the point of intersection of \(\displaystyle \overrightarrow {AC} \) with the circle.
Then \(\displaystyle y\) is one-half the measure of \(\displaystyle arc{CD}\).
And \(\displaystyle m(\angle A)=\frac{m(arc{BC})-x}{2}\)
 

JSZ2005

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Jun 11, 2019
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thanks for the explanations they were very helpful
i have a different problem which i think is right but it says my answer is wrong
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i'm pretty sure there is a theorem that says that two inscribed angles that intercept the same arc are congruent
here angle BAC and BOC are the inscribed angles and they both share arc BC but why are they not congruent angles?
maybe i'm being stupid
 
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Dr.Peterson

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BOC is not inscribed; O is not on the circle, but at its center.

Do these angles LOOK congruent to you?

A theorem says that an inscribed angle (like BAC) is half of the central angle (BOC). That's where the 30 comes from.

[There is another theorem that an external angle of a triangle (like BOC) is the sum of the other two internal angles (BAO and ABO). You can use that here instead (or as part of a proof of the first theorem).]
 
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