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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:
-
bisects . -
. - If
, then .
Step-by-Step Proof:
-
Understanding the Angle Bisector:
Sincebisects , we can denote: Therefore, we have: -
Using the Given Angles:
From the condition, we can denote: Thus, we have: -
Finding the Sum of Opposite Angles:
Now, we need to find the sum of the opposite anglesand : -
Using the Right Angle Condition:
From the condition, we can express: -
Substituting Back:
From, we can express in terms of : -
Substituting
into the Sum of Angles:
Now substitutingback into the sum of opposite angles: -
Finding the Other Opposite Angle:
We also need to consider the angle. Since (as they are supplementary), we can express: -
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.
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Mind Expander
To show that quadrilateral
is cyclic, we need to establish that opposite angles sum to
.
-
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. -
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 . -
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
.