Question

the figure below shows a circle with center x, diameter \\( \overline{lg} \\), and secant \\( \overleftrightarrow{ls} \\). which of the angles must be right angles? select all that apply.

Explanation:

Step1: Recall the Thales' theorem

Thales' theorem states that an angle inscribed in a semicircle is a right - angle. In a circle with center \(X\) and diameter \(\overline{LG}\), the inscribed angle subtended by the diameter \(\overline{LG}\) will be a right - angle.

Step2: Identify the inscribed angles subtended by the diameter

For an inscribed angle \(\angle T\) and \(\angle E\) in the circle with diameter \(\overline{LG}\), since the measure of an inscribed angle \(\theta\) is given by \(\theta=\frac{1}{2}\text{ measure of the arc it subtends}\). The arc subtended by \(\angle T\) and \(\angle E\) is the semicircle (arc \(LG\)). The measure of a semicircle is \(180^{\circ}\). So, \(\angle T=\frac{1}{2}\times180^{\circ} = 90^{\circ}\) and \(\angle E=\frac{1}{2}\times180^{\circ}=90^{\circ}\)

Answer:

\(\angle T\) and \(\angle E\)