Question

line m is constructed as the first step to prove the triangle angle - sum theorem. which of the following must be true in order to complete the proof? o a m || n o b bc = ac o c m∠1 + m∠2 = 90 o d m∠1 = m∠2

Explanation:

Step1: Recall triangle - angle sum proof concept

To prove the Triangle Angle - Sum Theorem, we usually construct a line parallel to one of the sides of the triangle. When we construct a line \(m\) parallel to side \(AB\) (assuming \(n\) is the line containing \(AB\)), we use the properties of parallel lines and transversals.
If \(m\parallel n\), then alternate - interior angles are equal. This helps in relating the angles of the triangle to a straight - line angle (which is 180°).

Step2: Analyze each option

• Option A: If \(m\parallel n\), we can use the properties of parallel lines (alternate - interior and corresponding angles) to show that the sum of the interior angles of the triangle is 180°. This is a key step in the proof of the Triangle Angle - Sum Theorem.
• Option B: \(BC = AC\) makes the triangle isosceles, but this has no direct relation to the general proof of the Triangle Angle - Sum Theorem for all triangles.
• Option C: \(m\angle1 + m\angle2=90^{\circ}\) is not relevant to the proof of the Triangle Angle - Sum Theorem. The sum of the three interior angles of a triangle is 180°, not the sum of just two non - supplementary angles in a non - right triangle context.
• Option D: \(m\angle1 = m\angle2\) is not a necessary condition for the proof of the Triangle Angle - Sum Theorem.

Answer:

A. \(m\parallel n\)