Help with Proofs and Theorems

[NamiNuitsuki]

Hi, I'd like to get some help with a question I have been stuck on for a few days. Here it is:

It is question number 36. Theorem 12-14 states the following:
"The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs".

Any sort of help would be greatly appreciated. I do not know how to do proofs so an explanation would be welcome, as well. Or even the completed answer. I am desperate for anything at this point. Please help. Thank you with all my heart.

 

[NamiNuitsuki]

Urgent! Help me, please!?

I'd please like the answer to question #36 of the image below. It is asking me to prove the "other two cases" of theorem 12-14 which states:

"The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs".

I have included an image of the definition and of the question itself. It is #36. I have also included my proof of question #35 if you need it.

please tell me the proof, i am desperate. please just help me. this is urgent. much love and thanks.

 

[stapel]

As near as I can tell (the sideways images are obviously difficult to read), you have the following:

---

Theorem 12-14: The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the two intercepted arcs.

Exercise 35:
. . .Given: a circle with center O and two secants CA and CE, intersecting the circle also at B between C and A and at D between C and E.

. . .Prove: the measure of angle ACE is equal to half the difference of the measures of arc AE and arc BD.

Exercise 36: Prove the other two cases of Theorem 12-14.

---

And the work you've shown appears to be for Exercise 35, not Exercise 36. Is this correct?

 

[NamiNuitsuki]

As near as I can tell (the sideways images are obviously difficult to read), you have the following:

---

Theorem 12-14: The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the two intercepted arcs.

Exercise 35:
. . .Given: a circle with center O and two secants CA and CE, intersecting the circle also at B between C and A and at D between C and E.

. . .Prove: the measure of angle ACE is equal to half the difference of the measures of arc AE and arc BD.

Exercise 36: Prove the other two cases of Theorem 12-14.

---

And the work you've shown appears to be for Exercise 35, not Exercise 36. Is this correct?

yes that is correct. Thanks for replying.

 

[NamiNuitsuki]

I just wanted to ask

I am sorry. If you'd like for me to re-post the images and try to make them right-sided, please tell me and i'll do so ASAP. I really appreciate your reply and I'd like to make this as comfortable as possible for you to help with. Thank you so much.

 

[stapel]

Theorem 12-14: The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the two intercepted arcs.

Exercise 35:
. . .Given: a circle with center O and two secants CA and CE, intersecting the circle also at B between C and A and at D between C and E.

. . .Prove: the measure of angle ACE is equal to half the difference of the measures of arc AE and arc BD.

Exercise 36: Prove the other two cases of Theorem 12-14.

yes that is correct.

Okay; good. Now: What have you done, in your attempts to do the other two cases (namely, with a secant and a tangent, and with two tangents)? How far have you gotten? Where are you stuck? From what other sources (your book, another book, a website, etc) have you gleaned possible steps? And so forth.

Please be complete. Thank you! :wink:

 

[NamiNuitsuki]

Not Far

Okay; good. Now: What have you done, in your attempts to do the other two cases (namely, with a secant and a tangent, and with two tangents)? How far have you gotten? Where are you stuck? From what other sources (your book, another book, a website, etc) have you gleaned possible steps? And so forth.

Please be complete. Thank you! :wink:

I have tried using the book, but it doesn't provide much other than a website that promises to help, but has since been shut down. When I have visited other math websites, I have never seen a question similar to the one I am stuck on. They are all usually proofs for theorem 12-15 which doesn't help in my case. I am stuck on virtually everything. I do not know where to begin, the steps after that and where to end. I have asked my teachers, but even they can't do anything to help because their answer key systems are down. I'd like to know what exactly I am trying to prove and how to prove it. I have gotten 0 things done that have to do with the question. My sister helped my prove number 35, but she doesn't know number 36. I feel like everywhere I look, no one knows anything about it at all.

 

[stapel]

I have tried using the book, but it doesn't provide much other than a website that promises to help, but has since been shut down. When I have visited other math websites, I have never seen a question similar to the one I am stuck on.... everywhere I look, no one knows anything about it at all.

Really? You typed the text of the Theorem into Google, and nobody had anything even vaguely on-topic?

 

[NamiNuitsuki]

Just textbook pdfs

Really? You typed the text of the Theorem into Google, and nobody had anything even vaguely on-topic?

When i typed it in all i got were pdfs of other geometry textbooks with the same question. That's it and some Slader links, but on there the question hadn't been answered either. I even went on websites like Chegg.com and Brainly, but still nothing. Nothing was shown. The next things that would come up if i switched around the words a little bit would be quizlet flashcards that were just definitions that i already knew or more pdfs. Nothing more than that was there.

 

[NamiNuitsuki]

Any Ideas

When i typed it in all i got were pdfs of other geometry textbooks with the same question. That's it and some Slader links, but on there the question hadn't been answered either. I even went on websites like Chegg.com and Brainly, but still nothing. Nothing was shown. The next things that would come up if i switched around the words a little bit would be quizlet flashcards that were just definitions that i already knew or more pdfs. Nothing more than that was there.

Do you have any idea of what the answer may be. I've spent all this time just searching and searching haha. Nothing's come of it

 

[NamiNuitsuki]

Have you found anything?

When i typed it in all i got were pdfs of other geometry textbooks with the same question. That's it and some Slader links, but on there the question hadn't been answered either. I even went on websites like Chegg.com and Brainly, but still nothing. Nothing was shown. The next things that would come up if i switched around the words a little bit would be quizlet flashcards that were just definitions that i already knew or more pdfs. Nothing more than that was there.

I have exhausted all search possibilities and I don't know what to put for an answer. Do you have any ideas of what the answer may be? I'll accept anything at this point. This is so urgent, I am desperate. Please help me....

 

[hndalama]

Do you have any idea of what the answer may be. I've spent all this time just searching and searching haha. Nothing's come of it

The answer is very similar to the proof your sister determined. I'm surprised she said she didn't know. The slight difference is how you draw the triangle. In your proof for Q 35, you derived a triangle by drawing line BE. Notice that line BE forms a triangle by connecting the second point of intersection with the circle for one secant to the first point of intersection for the other secant. In the case where the outside angle is formed by a secant and a tangent line, then line "BE" would connect the secants second point of intersection and the point of tangency. From here the proof is the same as what your sister showed you. Can you see how?

In the case where the outside angle is formed by two tangents, the triangle is formed by connecting the points of tangency. you also need to know the theorem that says the measure of an angle formed by a chord and a tangent line that intersect at the point of tangency is *HALF the measure of the intercepted arc.* Do you understand that? Can you see why from here the proof is the same as what your sister showed you?
Draw a diagram with labels to help you understand the proofs (like Q 35). See if you can solve it now. If you're still stuck,Perhaps show your sister my explanation and see if she understands. If you still need help, post a picture of your attempt to draw the diagrams and say what you don't understand.

hope this helps

 

[stapel]

The answer is very similar to the proof your sister determined. I'm surprised...

It's also very simple to do a Google search (here), as was earlier strongly suggested, and quickly find complete worked solutions.

Some people just don't seem to want to put in any effort of their own, relying on others (sisters, online strangers, etc) to do the write-ups for them. :shock:

 

[NamiNuitsuki]

Here is what I got

The answer is very similar to the proof your sister determined. I'm surprised she said she didn't know. The slight difference is how you draw the triangle. In your proof for Q 35, you derived a triangle by drawing line BE. Notice that line BE forms a triangle by connecting the second point of intersection with the circle for one secant to the first point of intersection for the other secant. In the case where the outside angle is formed by a secant and a tangent line, then line "BE" would connect the secants second point of intersection and the point of tangency. From here the proof is the same as what your sister showed you. Can you see how?

In the case where the outside angle is formed by two tangents, the triangle is formed by connecting the points of tangency. you also need to know the theorem that says the measure of an angle formed by a chord and a tangent line that intersect at the point of tangency is *HALF the measure of the intercepted arc.* Do you understand that? Can you see why from here the proof is the same as what your sister showed you?
Draw a diagram with labels to help you understand the proofs (like Q 35). See if you can solve it now. If you're still stuck,Perhaps show your sister my explanation and see if she understands. If you still need help, post a picture of your attempt to draw the diagrams and say what you don't understand.

hope this helps

First of all, Thank you so much for the tremendous help! I truly truly appreciate it. Here is where I am. I did the first proof with a tangent and secant, but I am stuck on the one with two tangents.

I'm sorry they are sideways, I'm still trying to figure out how to solve that. The one I'm stuck on is the second image. I don't know what to put in for what angles ACE and AEC equal to. Maybe you can help with that? Thank you

 

[NamiNuitsuki]

I got it

The answer is very similar to the proof your sister determined. I'm surprised she said she didn't know. The slight difference is how you draw the triangle. In your proof for Q 35, you derived a triangle by drawing line BE. Notice that line BE forms a triangle by connecting the second point of intersection with the circle for one secant to the first point of intersection for the other secant. In the case where the outside angle is formed by a secant and a tangent line, then line "BE" would connect the secants second point of intersection and the point of tangency. From here the proof is the same as what your sister showed you. Can you see how?

In the case where the outside angle is formed by two tangents, the triangle is formed by connecting the points of tangency. you also need to know the theorem that says the measure of an angle formed by a chord and a tangent line that intersect at the point of tangency is *HALF the measure of the intercepted arc.* Do you understand that? Can you see why from here the proof is the same as what your sister showed you?
Draw a diagram with labels to help you understand the proofs (like Q 35). See if you can solve it now. If you're still stuck,Perhaps show your sister my explanation and see if she understands. If you still need help, post a picture of your attempt to draw the diagrams and say what you don't understand.

hope this helps

Nevermind my last post, I finally understood this. I fixed my work and finally finished! Thank you for all of your and stapel's help. I appreciate it so much! <3
Have a wonderful day, guys!