Question

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which is a true statement about the diagram?
$m\angle5 + m\angle6 = m\angle1$
$m\angle3 + m\angle4 + m\angle5 = 180^{circ}$
$m\angle1 + m\angle2 = 180^{circ}$
$m\angle2 + m\angle3 = m\angle5$

Explanation:

Step1: Analyze the angle - sum property of a triangle

In a triangle, the sum of the interior angles is \(180^{\circ}\). But for the second option \(m\angle3 + m\angle4 + m\angle5=180^{\circ}\), \(\angle3,\angle4,\angle5\) are not the interior angles of a single triangle.

Step2: Analyze the linear - pair property

A linear pair of angles is supplementary (sum to \(180^{\circ}\)). For the third option \(m\angle1 + m\angle2 = 180^{\circ}\), \(\angle1\) and \(\angle2\) are not a linear pair.

Step3: Analyze the exterior - angle property

The exterior - angle property of a triangle states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
For the first option, by the exterior - angle property of a triangle, if we consider the triangle with angles \(\angle2,\angle3\) and the non - adjacent exterior angle \(\angle5\) is not related to \(\angle1\) in the way \(m\angle5 + m\angle6=m\angle1\).
For the fourth option, if we consider the triangle where \(\angle5\) is an exterior angle and \(\angle2\) and \(\angle3\) are non - adjacent interior angles. According to the exterior - angle theorem of a triangle: \(m\angle2 + m\angle3=m\angle5\)

Answer:

\(m\angle2 + m\angle3 = m\angle5\)