Question

Triangle ABC is shown with its inscribed circle drawn. The measure of angle ECF is 72 degrees. What is the measure of angle EGF? Explain or show your reasoning.

Answer 1

108°

Explanation:

CB and CA are tangents to the circle at points E and F respectively. GE and GF are radii of the circle with center G.

Since, radii of a circle are perpendicular to the tangents at the point of contacts.

`\therefore GE \perp CB\\\\</p><p>\& \:GF \perp CA\\\\</p><p>m\angle GEC =m\angle GFC = 90\degree \\\\`

In quadrilateral GECF, by interior angle sum property:

`m\angle C + m\angle E + m\angle F + m\angle EGF = 360\degree \\\\</p><p>\therefore 72\degree + 90\degree + 90\degree + m\angle EGF = 360\degree \\\\</p><p>\therefore 252\degree + m\angle EGF = 360\degree \\\\</p><p>\therefore m\angle EGF = 360\degree - 252\degree \\\\</p><p>\therefore m\angle EGF = 108\degree \\`

Answer 2

• 108°

Explanation:

• Since GE ⊥ BC and GF ⊥ AC,
• ∠GEC = ∠GFC = 90°

Sum of 4 angles is 360°, therefore:

• ∠ECF + ∠EGF = 360° - 180°
• ∠EGF = 180° - 72°
• ∠EGF = 108°