Question

1. explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is 180°.

Explanation:

Step1: Draw a triangle

Draw a triangle $\triangle ABC$.

Step2: Use parallel - line property

Draw a line $l$ parallel to side $BC$ passing through point $A$. By the alternate - interior angles theorem, $\angle B=\angle BAX$ and $\angle C = \angle CAY$ (where $X$ and $Y$ are points on the line $l$). Since $\angle BAX+\angle BAC+\angle CAY = 180^{\circ}$ (a straight - line angle), and $\angle B=\angle BAX$, $\angle C=\angle CAY$, then $\angle A+\angle B+\angle C = 180^{\circ}$. This works for any triangle because the properties of parallel lines and alternate - interior angles are universal. The construction of the parallel line can be done for any triangle, and the angle - relationships hold true regardless of the size or shape of the triangle.

Answer:

The sum of the angle measures in any triangle is $180^{\circ}$ because by constructing a parallel line to one side of the triangle through the opposite vertex, we can use the properties of parallel lines (alternate - interior angles) to show that the sum of the three interior angles of the triangle is equal to a straight - line angle, which is $180^{\circ}$. This construction and the application of angle - relationships are valid for all triangles.