Question

circle f is congruent to circle j, and ∠efd ≅ ∠gjh. m∠dfe = 80°. what is the measure of arc (widehat{gh})?

Explanation:

Step1: Use the property of congruent circles

Since circle \(F\) is congruent to circle \(J\), their radii are equal. Also, given \(\angle EFD\cong\angle GJH\).
The sum of angles in a circle - centered triangle: In \(\triangle DFE\), we know that the measure of an arc is related to the central - angle that subtends it. The sum of angles around a point is \(360^{\circ}\). In circle \(F\), \(\angle DFE + \angle EFD=360^{\circ}- \text{(sum of other non - relevant angles in the circle - centered triangle setup)}\). But more simply, for the arc - central angle relationship: The measure of an arc \(\overset{\frown}{DE}\) is equal to the measure of its central angle \(\angle DFE\). Similarly, for circle \(J\), the measure of arc \(\overset{\frown}{GH}\) is equal to the measure of its central angle \(\angle GJH\).
We know that the sum of angles in a triangle formed by two radii and a chord (in a circle) and the fact that \(\angle EFD\cong\angle GJH\). Also, using the angle - sum property in the circle - centered triangles (where the sum of angles in a triangle is \(180^{\circ}\), but for central angles and arcs: The measure of an arc is equal to the measure of its central angle.
We know that \(\angle DFE = 80^{\circ}\). The sum of angles in the circle - centered triangle \(\triangle DFE\) (where \(FD = FE\) as radii) and \(\triangle GJH\) (where \(JG=JH\) as radii).
Since \(\angle EFD\cong\angle GJH\) and using the formula for the measure of an arc \(s = r\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians, but when comparing congruent circles (\(r_F=r_J\)) and using degree - measure for arcs (where the degree - measure of an arc is equal to the degree - measure of its central angle).
The measure of an arc \(\overset{\frown}{GH}\) is found by first finding the central angle \(\angle GJH\).
We know that in circle \(F\), if we consider the full - circle angle relationships. The measure of an arc is equal to the measure of its central angle.
Since \(\angle EFD\cong\angle GJH\) and in circle \(F\), if we assume the triangle \(DFE\) (with \(FD = FE\)) and circle \(J\) with \(JG = JH\).
The measure of arc \(\overset{\frown}{GH}\) is calculated as follows:
The sum of angles in a circle - centered triangle (where two sides are radii). Let's use the fact that the measure of an arc is equal to the measure of its central angle.
We know that \(m\angle DFE=80^{\circ}\). The sum of angles in \(\triangle DFE\) (\(FD = FE\)): \(\angle DFE + 2\angle FDE=180^{\circ}\) (isosceles triangle property). But for the arc - central angle relationship:
The measure of arc \(\overset{\frown}{DE}\) is \(m\angle DFE\) (central - angle theorem: the measure of an arc is equal to the measure of its central angle).
Since circle \(F\cong\) circle \(J\) and \(\angle EFD\cong\angle GJH\).
We know that \(m\angle DFE + m\angle EFD=180^{\circ}\) (linear - pair - like relationship in the circle - centered triangle setup, but more accurately, using the property that the measure of an arc is equal to the measure of its central angle.
The measure of arc \(\overset{\frown}{GH}\) is \(100^{\circ}\) because \(m\angle DFE = 80^{\circ}\) and in the circle - centered triangle (sum of angles in a triangle \(=180^{\circ}\), and the central angle for arc \(\overset{\frown}{GH}\) is \(180 - 80=100^{\circ}\))

Answer:

\(100\)