1. **Problem Statement:** We have a circle with center O, tangent FE at point E, and points D, E, F, G on or around the circle. Given that EG = GF and angle Ê₃ = x, we need to: 10.1 Name two other angles equal to x with reasons. 10.2 Prove DE = EF. 10.3 Express angle DÔE in terms of x. 2. **Key Properties and Formulas:** - Tangent to a circle is perpendicular to the radius at the point of contact: $$\angle OEF = 90^\circ$$. - Equal chords subtend equal angles at the center. - In an isosceles triangle, angles opposite equal sides are equal. - Angles in the same segment of a circle are equal. --- ### 10.1 Find two other angles equal to $x$: - Given $\angle Ê_3 = x$. - Since EG = GF, triangle EGF is isosceles with $\angle EGF = \angle EFG$. - By the tangent property, $\angle EFD = \angle Ê_3 = x$ (alternate segment theorem). - Also, $\angle DÊF = x$ because angles in the same segment are equal. **Answer:** The two other angles equal to $x$ are $\angle EFD$ and $\angle DÊF$. --- ### 10.2 Prove $DE = EF$: - Triangle DEF has $\angle DÊF = \angle EFD = x$ (from 10.1). - Two angles equal implies two sides opposite those angles are equal. - Therefore, $DE = EF$. Intermediate step showing cancellation: $$\cancel{\angle DÊF} = \cancel{\angle EFD} = x \implies DE = EF$$ --- ### 10.3 Express $\angle DÔE$ in terms of $x$: - Triangle ODE has $\angle OEF = 90^\circ$ (tangent-radius). - Since $DE = EF$, triangle DEF is isosceles with base $DF$. - Using the circle properties, $\angle DÔE$ is twice the angle at the circumference subtended by the same chord DE. - The angle at circumference is $x$, so: $$\angle DÔE = 2x$$ --- **Final answers:** 10.1 $\angle EFD = x$ and $\angle DÊF = x$. 10.2 $DE = EF$. 10.3 $\angle DÔE = 2x$.