Question

1. an incomplete proof of the theorem that the sum of the interior angles of a triangle is 180° is shown. given ( adparallel bc ) prove ( mangle1 + mangle2 + mangle3 = 180^{circ} ). a. what is reason 1? b. what is reason 4? c. what is reason 5?

Explanation:

a. Reason 1

When two parallel lines are cut by a transversal, alternate - interior angles are congruent. Here, \(AD\parallel BC\) and \(AC\) is a transversal for \(\angle1\) and \(\angle4\), and \(AB\) is a transversal for \(\angle2\) and \(\angle5\).

b. Reason 4

The sum of the measures of adjacent angles that form a linear - pair (or in this case, angles that make up a straight - line angle) is given by the Angle Addition Postulate. The Angle Addition Postulate states that if \(D\) is in the interior of \(\angle ABC\), then \(m\angle ABD+m\angle DBC = m\angle ABC\). Here, \(\angle4\), \(\angle3\), and \(\angle5\) are adjacent angles that form \(\angle EAD\).

c. Reason 5

\(\angle EAD\) is a straight angle. By the definition of a straight angle, the measure of a straight angle is \(180^{\circ}\).

Answer:

a. Alternate - interior angles theorem.
b. Angle Addition Postulate.
c. Definition of a straight angle.