Question

fill in the blanks to complete the proof that ( mangle1 + mangle3 + mangle2 = 180^{circ} ) given the diagram.
(2 points)
given the diagram shown, ( anglesquarecongangle5;angle2congangle4 ) because alternate interior angles are congruent when lines are parallel.
then, i know that ( mangle1 = mangle5;mangle2 = mangle4 ) because congruent angles have equal measures. ( manglesquare)( +mangle3 + mangle4 = 180^{circ} ) by the definition of a straight angle. finally, ( mangle1 + mangle3 + manglesquare=180^{circ} ) by substitution.

Explanation:

Step1: Identify congruent angles

From the diagram and the property of alternate - interior angles (when lines are parallel), \(\angle1\cong\angle5\).

Step2: Use the angle - sum property of a straight angle

We know that for a straight angle, \(m\angle5 + m\angle3+m\angle4=180^{\circ}\) (by the definition of a straight angle, which is \(180^{\circ}\)).

Step3: Substitute congruent angles

Since \(m\angle1 = m\angle5\) and \(m\angle2 = m\angle4\), substituting these into \(m\angle5 + m\angle3+m\angle4=180^{\circ}\) gives \(m\angle1 + m\angle3+m\angle2=180^{\circ}\).

Answer:

The first blank is \(1\), the second blank is \(5\), and the third blank is \(2\).