Question

points a, b, and c form a triangle. complete the statements to prove that the sum of the interior angles of △abc is 180°. statement reason points a, b, and c form a triangle given let (overline{de}) be a line passing through b and parallel to (overline{ac}) definition of parallel lines (angle3congangle5) and (angle1congangle4) (mangle1 = mangle4) and (mangle3 = mangle5) alternate interior angles theorem (mangle4 + mangle2 + mangle5 = 180^{circ}) alternate exterior angles theorem (mangle1 + mangle2 + mangle3 = 180^{circ}) corresponding angles theorem congruent angles have equal measures

Explanation:

Step1: Identify angle - congruence reason

Since $\angle3\cong\angle5$ and $\angle1\cong\angle4$, the reason for $m\angle1 = m\angle4$ and $m\angle3 = m\angle5$ is that congruent angles have equal measures.

Step2: Analyze the straight - line angle sum

$\angle4,\angle2,\angle5$ form a straight - line at point $B$ on line $DE$. By the definition of a straight - line, the sum of the measures of angles on a straight - line is $180^{\circ}$, so $m\angle4 + m\angle2 + m\angle5=180^{\circ}$.

Step3: Substitute equal - measure angles

Since $m\angle1 = m\angle4$ and $m\angle3 = m\angle5$, we can substitute $\angle4$ with $\angle1$ and $\angle5$ with $\angle3$ in the equation $m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$, getting $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$.

Answer:

The reason for $m\angle1 = m\angle4$ and $m\angle3 = m\angle5$ is "Congruent angles have equal measures"; the reason for $m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$ is "Definition of a straight - line (sum of angles on a straight - line is 180°)"; the reason for $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$ is "Substitution property of equality (substituting $m\angle4$ with $m\angle1$ and $m\angle5$ with $m\angle3$)".