Question
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Prove that :
is a cyclic quadrilateral
10.2

Ask by Ortiz Alexander. in South Africa
Jan 04,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

Quadrilateral is cyclic because the sum of its opposite angles equals .

Solution

To prove that quadrilateral is cyclic, we need to show that the opposite angles of the quadrilateral sum up to .

Given Conditions:

  1. bisects .
  2. .
  3. If , then .

Step-by-Step Proof:

  1. Understanding the Angle Bisector:
    Since bisects , we can denote:
    Therefore, we have:
  2. Using the Given Angles:
    From the condition , we can denote:
    Thus, we have:
  3. Finding the Sum of Opposite Angles:
    Now, we need to find the sum of the opposite angles and :
  4. Using the Right Angle Condition:
    From the condition , we can express:
  5. Substituting Back:
    From , we can express in terms of :
  6. Substituting into the Sum of Angles:
    Now substituting back into the sum of opposite angles:
  7. Finding the Other Opposite Angle:
    We also need to consider the angle . Since (as they are supplementary), we can express:
  8. Final Check for Cyclic Condition:
    Now we check the sum of the opposite angles:
    Since , we conclude:

Conclusion:

Since the sum of the opposite angles , we have shown that quadrilateral is cyclic. Thus, the proof is complete.

Answered by UpStudy AI and reviewed by a Professional Tutor

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Mind Expander

To show that quadrilateral is cyclic, we need to establish that opposite angles sum to .
  1. From the statement , we know that angle and angle relate in such a way that they are equal, which is an important condition for cyclic quadrilaterals.
  2. Next, we refer to the fact that . This implies that angles and are supplementary due to corresponding properties of cyclic quadrilaterals where angles opposite each other must sum to .
  3. Finally, with bisecting , we can conclude that confirms the relationship between all angles in correspondence to a cyclic quadrilateral’s properties.
Thus, quadrilateral is cyclic if the above conditions are satisfied with the angles appropriately summing to .

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