Question

which statements regarding the diagram of δebc are true? select three options. ∠bec is an exterior angle. ∠dec is an exterior angle. ∠abe and ∠ebc are supplementary angles. ∠bcf and ∠dec are supplementary angles. ∠bec is a remote interior angle to exterior ∠bcf.

Explanation:

Step 1: Recall definitions of exterior, supplementary, and remote interior angles

• Exterior angle: An angle formed by one side of a triangle and the extension of another side.
• Supplementary angles: Two angles whose sum is \(180^\circ\) (they form a linear pair or are adjacent and form a straight line).
• Remote interior angle: An interior angle of a triangle that is not adjacent to a given exterior angle.

Step 2: Analyze each option

1. \(\angle BEC\) is an exterior angle:

\(\angle BEC\) is an interior angle of \(\triangle EBC\) (it's formed by sides \(EB\) and \(EC\) within the triangle), so this is false.

1. \(\angle DEC\) is an exterior angle:

\(\angle DEC\) is formed by side \(EC\) of \(\triangle EBC\) and the extension of side \(EB\) (from \(B\) to \(A\) to \(E\) extended? Wait, actually, \(D - E - C\) is a line? Wait, looking at the diagram, \(D\), \(E\), and then \(C\)? Wait, no, the triangle is \(EBC\). The angle \(\angle DEC\): \(E\) is a vertex, \(D - E - B\) is a line? Wait, maybe \(D\), \(E\), \(B\) are colinear? Wait, the diagram has \(A\) on a line with \(B\), and \(D\), \(E\) on another line. Wait, maybe \(D - E - B\) is a straight line? Wait, no, let's re-examine. For \(\triangle EBC\), the exterior angle at \(E\) would be formed by extending \(EB\) or \(EC\). If \(D - E - B\) is a straight line, then \(\angle DEC\) is formed by extending \(EB\) (from \(B\) through \(E\) to \(D\)) and side \(EC\). So \(\angle DEC\) is an exterior angle of \(\triangle EBC\) (since it's outside the triangle, formed by one side \(EC\) and the extension of \(EB\)). So this is true.

1. \(\angle ABE\) and \(\angle EBC\) are supplementary angles:

\(A - B - E\) (or \(A - B - D\)? Wait, the diagram shows \(A\) on a line with \(B\), and \(E\) connected to \(B\). If \(A\), \(B\), and the line through \(E\) (maybe \(D - E - B\) is straight? Wait, if \(A\), \(B\), and \(E\) are colinear? Wait, no, the diagram: \(A\) is a point above \(B\), \(B\) is connected to \(E\) and \(C\). Wait, maybe \(A - B - E\) is a straight line? If \(A\), \(B\), \(E\) are colinear, then \(\angle ABE\) and \(\angle EBC\) form a linear pair (they are adjacent and their non - common sides form a straight line), so their sum is \(180^\circ\), meaning they are supplementary. So this is true.

1. \(\angle BCF\) and \(\angle DEC\) are supplementary angles:

\(\angle BCF\) is an exterior angle at \(C\) (formed by extending \(BC\) to \(F\)), and \(\angle DEC\) is an exterior angle at \(E\). There's no reason for their sum to be \(180^\circ\) (they are not adjacent, not forming a linear pair). So this is false.

1. \(\angle BEC\) is a remote interior angle to exterior \(\angle BCF\):

For exterior angle \(\angle BCF\) (at vertex \(C\) of \(\triangle EBC\)), the remote interior angles are the interior angles not adjacent to \(\angle BCF\). \(\angle BCF\) is adjacent to \(\angle ECB\) (since \(E - C - F\) is a straight line, \(\angle ECB\) and \(\angle BCF\) are supplementary). So the other interior angles of \(\triangle EBC\) are \(\angle BEC\) and \(\angle EBC\). \(\angle BEC\) is not adjacent to \(\angle BCF\), so it is a remote interior angle to \(\angle BCF\). So this is true.

Answer:

• \(\boldsymbol{\angle DEC}\) is an exterior angle.
• \(\boldsymbol{\angle ABE}\) and \(\boldsymbol{\angle EBC}\) are supplementary angles.
• \(\boldsymbol{\angle BEC}\) is a remote interior angle to exterior \(\boldsymbol{\angle BCF}\).