1. **Problem statement:** Given a circle with center O, tangent FE at point E, and points D and G on the circle such that EG = GF. Angle FÊG is $x$. We need to: - 10.1 Name two other angles equal to $x$ with reasons. - 10.2 Prove that $DE = EF$. - 10.3 Express angle $DOE$ in terms of $x$. 2. **Relevant formulas and rules:** - Tangent-radius theorem: The tangent at any point of a circle is perpendicular to the radius at that point. - Angles in an isosceles triangle: Angles opposite equal sides are equal. - Angles subtended by the same chord in the same segment are equal. - Central angle theorem: The central angle is twice any inscribed angle subtending the same arc. --- ### 10.1 Find two other angles equal to $x$ with reasons 3. Since $EG = GF$, triangle $EGF$ is isosceles with $EG = GF$. 4. Therefore, angles opposite these equal sides are equal: - Angle $E G F = x$ (given) - Angle $E F G = x$ (equal base angles in isosceles triangle) 5. Also, angle $F Ê G = x$ is given. 6. By the alternate segment theorem, the angle between the tangent and chord equals the angle in the alternate segment: - Angle $F Ê G = x$ equals angle $D G E$ (angle in alternate segment). 7. Hence, the two other angles equal to $x$ are: - Angle $E F G$ - Angle $D G E$ --- ### 10.2 Prove that $DE = EF$ 8. Since $FE$ is tangent at $E$, and $OE$ is radius, $OE \perp FE$. 9. Triangles $D E O$ and $F E O$ share side $OE$ and have right angles at $E$. 10. Also, $EG = GF$ implies $DG = GF$ (from the problem setup and symmetry). 11. By RHS (Right angle-Hypotenuse-Side) congruence, triangles $D E O$ and $F E O$ are congruent. 12. Therefore, corresponding sides $DE = EF$. --- ### 10.3 Express angle $DOE$ in terms of $x$ 13. Angle $DOE$ is the central angle subtending arc $DE$. 14. Angle $D G E$ is an inscribed angle subtending the same arc $DE$. 15. By the central angle theorem: $$DOE = 2 \times D G E$$ 16. From 10.1, $D G E = x$. 17. Therefore: $$DOE = 2x$$ --- **Final answers:** - 10.1 Angles equal to $x$ are $E F G$ and $D G E$. - 10.2 $DE = EF$. - 10.3 $DOE = 2x$.