Question

1. why is ( mangle4 + mangle5 + mangle6 = 360^{circ} )? addition property subtraction property triangle sum theorem exterior angle theorem

Explanation:

The Addition Property is not relevant as it just states if \(a = b\) and \(c = d\), then \(a + c=b + d\). The Subtraction Property is also not relevant. The Triangle Sum Theorem states that the sum of the interior angles of a triangle is \(180^{\circ}\). The Exterior Angle Theorem 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. But when we consider the sum of the exterior angles of a triangle, \(\angle4\), \(\angle5\), and \(\angle6\) are the exterior angles of a triangle. The sum of the exterior angles of any polygon (a triangle is a 3 - sided polygon) is \(360^{\circ}\). This is a well - known geometric property.

Answer:

None of the options provided are correct. The sum of the exterior angles of a triangle (or any polygon) is \(360^{\circ}\) which is a general geometric exterior - angle - sum property. If we assume there is a mis - labeling and we consider the fact that when we look at the angles around a point (by extending the sides of the triangle), using the concept of a full - circle (a full - circle has a measure of \(360^{\circ}\)) and angle addition (since \(\angle4\), \(\angle5\), and \(\angle6\) can be thought of as angles around a 'virtual' point formed by extending the sides of the triangle), we can also say that by the property of angles around a point (a full - rotation is \(360^{\circ}\)) and angle addition (the sum of non - overlapping angles around a point). But among the given options, if we consider the basic idea of adding angles (\(\angle4+\angle5+\angle6\)) to get a total, the closest (though not a perfect match) is the Addition Property. However, in a strict geometric sense, the sum of exterior angles of a triangle is \(360^{\circ}\) is a standard result. If forced to choose from the given options (assuming some context where it's about adding angles around a point - like structure), the Addition Property (but this is a stretch as the Addition Property is more of an algebraic property \(a + b=c + d\) if \(a = c\) and \(b = d\)). But if we consider the operation of adding the measures of the three angles \(\angle4\), \(\angle5\), \(\angle6\) together to get \(360^{\circ}\) (the act of addition), the answer is Addition Property.