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Why is it that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary?
Because of one of the Circle Theorems that states that the angle subtended by any arc at the centre of the circle is half that at the circumference.
A rough version of the proof follows:
Suppose the quadrilateral ABCD is inscribed in a circle with centre O.
Join AO and CO. This partitions the circumference into two arcs - both AC but going around different sides of the centre.
One of the arcs AC, subtends angle B at the circumference and suppose the angle subtended by the same arc at O is X. Then 2B = X
The other arc AC subtends angle D at the circumference and suppose the angle subtended by the same arc at O is Y. Then 2D = Y
So 2B + 2D = X + Y
But X + Y = 360 degrees.
So B + D = 180 that is, B and D are supplementary.
And then, since A+B+C+D = 360, A + C = 180.
The converse can be proved similarly.
What else can I help you with?
Opposite angles of an inscribed quadrilateral are what?
Supplementary (they add to 180 degrees).
True or false If opposite angles of a quadrilateral are supplementary then the quadrilateral is a parallelogram?
True
If a quadrilateral is a parallelogram then its opposite angles are always supplementary?
False because it will have 2 equal opposite obtuse angles and 2 equal opposite acute angles with the 4 angles adding up to 360 degrees.
What are supplementary angles that are NOT a linear pair?
Any two angles whose measures add up to 180 degrees. For example, opposite angles of a cyclic quadrilateral (quadrilateral whose vertices are on a circle).
If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is not a parallelogram?
False. If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a parallelogram.
What the opposite angles of an inscribed quadrilateral?
Supplementary
Opposite angles of an inscribed quadrilateral are what?
Supplementary (they add to 180 degrees).
Which property is always true for a quadrilateral inscribed in a circle?
opposite angles are supplementary
The opposite angles of a quadrilateral inscribed in a circle are?
The opposite angles of a quadrilateral inscribed in a circle are supplementary, meaning they add up to 180 degrees. This is due to the property that the sum of the opposite angles of any quadrilateral inscribed in a circle is always 180 degrees. This property can be proven using properties of angles subtended by the same arc in a circle.
What are the opposite angles of a quadrilateral inscribed in a circle?
A quadrilateral is inscribed in a circle it means all the vertices of quadrilateral are touching the circle. therefore it is a cyclic quadrilateral and sum of the opposite angles in cyclic quadrilateral is supplementary. suppose if one angle is A then another will be 180 degree - angle A.
Are The Opposite Angles Of A Quadrilateral In A Circumscribed Circle Are Always Supplementary.?
Yes, the opposite angles of a quadrilateral inscribed in a circumscribed circle (cyclic quadrilateral) are always supplementary. This means that the sum of each pair of opposite angles equals 180 degrees. This property arises from the fact that the inscribed angles subtend the same arc, leading to their supplementary relationship. Thus, if one angle measures (x), the opposite angle will measure (180 - x).
Do all supplementary angles from a linear pair Are all linear pair supplementary?
All supplementary angles do not form a linear pair. The opposite angles of any quadrilateral inscribed in a circle (a cyclic quadrilateral) are supplementary but they are not a linear pair. However, all linear pair are supplementary.
If a parallelogram is inscribed in a circle it must be a rectangle?
Yes, a parallelogram inscribed in a circle must be a rectangle. This is because a circle's inscribed angle theorem states that the opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) must be supplementary. In a parallelogram, opposite angles are equal, which can only hold true if all angles are right angles, thus making the parallelogram a rectangle.
If opposite angles of a quadrilateral are supplementary is the quadrilateral a parallelogram?
No. A quadrilateral is a parallelogram when consecutiveangles are supplementary.
True or false If opposite angles of a quadrilateral are supplementary then the quadrilateral is a parallelogram?
True
The opposite angles of a quadrilateral in a circumscribed circle must be?
supplementary
True or False- If opposite angles of a quadrilateral are supplementary then the quadrilateral is a parallelogram?
false
