1. **State the problem:** Understanding the properties of inscribed angles and intercepted arcs in a circle based on given theorems and angle measures. 2. **Theorem 1:** The measure of an inscribed angle equals half the measure of its intercepted arc. Given: $AT=120^\circ$, intercepted arc $LACT=60^\circ$. This matches the theorem since $\frac{1}{2}\times 120^\circ = 60^\circ$. 3. **Theorem 2:** Two inscribed angles intercepting congruent arcs are congruent. Given: $LI=95^\circ$, $LL=35^\circ$ and $LI\cong LL$. Since $95^\circ \neq 35^\circ$, angles $LI$ and $LL$ cannot be congruent unless their intercepted arcs are equal; this suggests an error or different arcs intercepted by $LI$ and $LL$, or angle measures differ from arcs. 4. **Theorem 3:** An inscribed angle intercepting a semicircle is a right angle ($90^\circ$). Given: $LNTIE=90^\circ$ confirms this. Also, $NE=45^\circ$ is provided but not relevant here. 5. **Theorem 4:** Opposite angles of a cyclic quadrilateral sum to $180^\circ$ (are supplementary). Given angles sums: $LRDA + LKEA = ?$ $LPRA + LDAE = ?$ By the theorem, $$ LRDA + LKEA = 180^\circ $$ $$ LPRA + LDAE = 180^\circ $$ **Summary:** - Use $m(\text{inscribed angle}) = \frac{1}{2} m(\text{intercepted arc})$ - Congruent arcs subtend congruent angles - Angles intercepting a semicircle are right angles - Opposite angles in cyclic quadrilaterals are supplementary Thus: - $60^\circ$ is half of $120^\circ$ intercepted arc - If two inscribed angles intercept congruent arcs, then they are congruent - $LNTIE=90^\circ$ corresponds to interception of a semicircle - $LRDA + LKEA = 180^\circ$, $LPRA + LDAE = 180^\circ$. Final answers are the theorem applications stated.