A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

Quadrilateral EFGH is inscribed inside a circle as shown below. Write a proof showing that angles H and F are supplementary.

  • This Question is Open
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    1 Attachment
  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @ganeshie8

  3. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do you need a two-column proof or just an outline of the proof?

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    just outline should be fine

  5. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok. Here's the idea for this proof. A full circle is an arc of measure 360 deg.

  6. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434114713677:dw|

  7. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Let's look at two arcs on the figure.

  8. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so would we have to figure they both have to equal 180 right?

  9. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434114801663:dw|

  10. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    No. We don't know how much either arc measures. We do know that the sum of their measures is 360. Also, if we make an arc 180, and the other one has to also be 180, then this proof would only work when the arcs are 180 deg. We want a proof that works for all inscribed quadrilaterals, no matter what any of their angle measures are.

  11. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Because of the above, we call the measure of one arc x. Since the sum of the measures of the arc is the entire circle, the arcs add up to 360 degrees.

  12. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so if one arc is x, the other arc must be 360 - x. Ok so far?

  13. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434114997252:dw|

  14. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so we would do 360 minus x for both angles?

  15. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    We are not dealing with angles yet. So far, we are only dealing with arc measures. The measure of arc EHG is x. The measure of arc EFG is 360 - x. This is all we have so far. Do you follow this so far?

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh yeah I get it now.

  17. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Great. Now we start with the angles. What do you know about an inscribed angle and its corresponding arc? Look at the figure below. What is the m<A? |dw:1434115295031:dw|

  18. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the arc is two times the measure of the angle correct?

  19. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Exactly. An inscribed angle is half the measure of its arc. (A central angle is the same measure as the arc, but we're not dealing with a central angle here.)

  20. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434115526244:dw|

  21. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok. Let's go back to our problem, and look at the inscribed angles of the two arcs we are dealing with.

  22. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I marked in the figure the two angles of the two arcs we are dealing with. |dw:1434115587886:dw|

  23. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Let's look at angle EFG first. Its arc is arc EHG. Arc EHG measures x. What is the measure of angle EFG?

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    half of x?

  25. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Exactly. It is half of the arc.

  26. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434115776669:dw|

  27. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Now let's look at the other arc and its corresponding inscribed angle. Arc EFG measures 360 - x. What is the measure of the inscribed angle EHG?

  28. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that would be half of 360- x so 1/2(360-x)

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    right?

  30. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Exactly. Let me write that down.

  31. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434116074608:dw|

  32. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    We are trying to prove that opposite angles are supplementary. Angles EFG and EHG are opposite angles, so let's add their measures.

  33. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1434116159157:dw| Ok?

  34. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  35. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Now let's simplify the right side.

  36. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    We can factor out 1/2 |dw:1434116259666:dw|

  37. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Now we combine the x's in the parentheses. x - x = 0, so we get: |dw:1434116311737:dw|

  38. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Finally we get: |dw:1434116353183:dw|

  39. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    The definition of supplementary angles is: Two angles are supplementary if their measures add up top 180 degrees. We have just shown that the two opposite angles have measures that add up to 180 degrees, so the opposite angles are supplementary.

  40. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    gtg, bye

  41. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you!

  42. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yw

  43. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.