Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Based on the diagram above, the following relations hold true.
This theorem will be proven in two parts.
Consider a circle and an inscribed quadrilateral
Since is not cyclic, the circle that passes through and does not pass through Let be the point of intersection of and the circle. Consider the quadrilateral
Because is inscribed in a circle, it can be concluded that the opposite angles and are supplementary. It was assumed that has supplementary opposite angles. Therefore, and are supplementary angles. By the Transitive Property of Equality, the above equations imply that and have equal measures. However, this is not possible. The reason is that the measure of the exterior angle of can not be the same as the measure of the interior angle
This contradiction proves that the initial assumption was false, and is a cyclic quadrilateral. Note that a similar argument can be used if lies inside the circle. The proof of Part is now complete.
Tiffaniqua is given a quadrilateral She wants to draw a circle that passes through all the vertices, but she does not know if it is possible. For that reason, she decided to measure the angles of
Compare the sums of the opposite angles' measures.
The Inscribed Quadrilateral Theorem can be used to determine whether is cyclic. According this theorem, the opposite angles of the quadrilateral need to be supplementary. Calculate the sum of opposite angles' measures and see if it is true.
The sum of the angle measures in each pair is not equal to Therefore, neither and nor and are supplementary. This finding implies that is not a cyclic quadrilateral.
Find the measure of Write your answer without the degree symbol.
In this case, is said to be the opposite interior angle. The relationship between these angles is described by the Cyclic Quadrilateral Exterior Angle Theorem.
Based on the diagram above, the following relation holds true.
Consider an inscribed quadrilateral with one side extended to point
From the diagram, it can be observed that and form a linear pair. Therefore, these angles are supplementary, which means that the sum of their measures is Also, by the Inscribed Quadrilateral Theorem, the opposite angles and are also supplementary.
Therefore, the following relation is true. After analyzing the equations, and show to be supplementary to the same angle Therefore, by the Congruent Supplements Theorem, they are congruent.
This relation is illustrated on the diagram below.
By the same logic, this theorem can be proven for any other extended side of The proof is now complete.
Davontay wants to go to a concert, but his parents say that he has to finish his homework first. In the last math exercise, he is asked to find the values of all variables.
Help Davontay solve the last exercise so that he can go to the concert.
It is worth mentioning that not only quadrilaterals can be inscribed in a circle. There can also be inscribed polygons with a different number of sides. Stonehenge is a real-world example of an inscribed polygon. Unfortunately, only some parts of it remain to this day.
However, when Stonehenge was built by ancient peoples about years ago, it had a cyclic polygon structure, as illustrated on the diagram below.