Question
a quadrilateral is inscribed in a circle. two consecutive angles measure 88 degrees and 108 degrees. the angle opposite the 108 degree angle is labeled x degrees. find the value of x. the value is not 92
Answers
GPT-4o mini
In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that the sum of the measures of the opposite angles equals \(180^\circ\).
Given:
- One angle is \(88^\circ\).
- The consecutive angle is \(108^\circ\).
- The angle opposite to \(108^\circ\) is labeled as \(x\).
Since opposite angles in a cyclic quadrilateral are supplementary, we have the following relationships:
1. The angle opposite \(88^\circ\) (which we'll call \(y\)) and \(88^\circ\) are supplementary:
\[
y + 88^\circ = 180^\circ \implies y = 180^\circ - 88^\circ = 92^\circ.
\]
This means that the angle opposite \(88^\circ\) (our label) is indeed \(92^\circ\).
2. The angle \(x\) opposite \(108^\circ\) can be calculated using:
\[
x + 108^\circ = 180^\circ \implies x = 180^\circ - 108^\circ = 72^\circ.
\]
So, the value of \(x\) is \(72^\circ\).
Given:
- One angle is \(88^\circ\).
- The consecutive angle is \(108^\circ\).
- The angle opposite to \(108^\circ\) is labeled as \(x\).
Since opposite angles in a cyclic quadrilateral are supplementary, we have the following relationships:
1. The angle opposite \(88^\circ\) (which we'll call \(y\)) and \(88^\circ\) are supplementary:
\[
y + 88^\circ = 180^\circ \implies y = 180^\circ - 88^\circ = 92^\circ.
\]
This means that the angle opposite \(88^\circ\) (our label) is indeed \(92^\circ\).
2. The angle \(x\) opposite \(108^\circ\) can be calculated using:
\[
x + 108^\circ = 180^\circ \implies x = 180^\circ - 108^\circ = 72^\circ.
\]
So, the value of \(x\) is \(72^\circ\).