Question

1. ∠efg is formed by an extended ray and the adjacent side of △def.

a. what type of angle is ∠efg?
b. what are ∠d and ∠e called with respect to the exterior angle ∠efg?
c. what is ∠efd called with respect to ∠efg?

1. write a conjecture relating the measures of an exterior angle of a triangle and its remote interior angles. call this conjecture the exterior angle conjecture.

Explanation:

Step1: Determine the type of ∠EFG

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. So, ∠EFG is an exterior angle of △DEF.

Step2: Identify ∠D and ∠E with respect to ∠EFG

The remote - interior angles of an exterior angle of a triangle are the two non - adjacent interior angles. So, ∠D and ∠E are the remote - interior angles of ∠EFG.

Step3: Identify ∠EFD with respect to ∠EFG

∠EFD and ∠EFG form a linear pair (they are adjacent angles and their non - common sides form a straight line). So, ∠EFD is the adjacent interior angle of ∠EFG.

Step4: Formulate the Exterior Angle Conjecture

Let \(m\angle EFG\) be the measure of the exterior angle, and \(m\angle D\) and \(m\angle E\) be the measures of the remote - interior angles. By the property of triangle angles (the sum of the interior angles of a triangle is \(180^{\circ}\), i.e., \(m\angle D+m\angle E + m\angle EFD=180^{\circ}\), and since \(m\angle EFD + m\angle EFG=180^{\circ}\) (linear pair)), we can subtract the two equations: \((m\angle EFD + m\angle EFG)-(m\angle D+m\angle E + m\angle EFD)=180 - 180\). Simplifying gives \(m\angle EFG=m\angle D + m\angle E\).

Answer:

a. ∠EFG is an exterior angle of △DEF.
b. ∠D and ∠E are the remote - interior angles of ∠EFG.
c. ∠EFD is the adjacent interior angle of ∠EFG.

1. The Exterior Angle Conjecture: The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote - interior angles.