How can you prove that Vertical Angles are congruent?

Question

Can you please explain the reason, and also write the chart [Statement, Reason,] if possible.

Answer 1

Given: <1 and <2 are vertical angles.
Prove: <1 = <2
Plan: Draw a figure and label the vertical angles 1 and 2. Number
a third angle 3.

PROOF:
1) m<1 + m<3 = 180; m<2 + m<3 = 180 (Angle Addition Postulate)
2) m<1 + m<3 = m<2 + m<3 (Substitution Property)
3) m<3 = m<3 (Reflexive Property)
4) m<1 = m<2 (Subtraction Property)

Answer 2

Let A, B be vertical angles and C be an angle adjacent to both A and B. Then m∠A+m∠C=180° (since they together form a straight line), and by the same, m∠B+m∠C=180°. But by simple algebra, m∠A=180°-m∠C, and m∠B=180°-m∠C, thus by transitivity of equality, m∠A=m∠B, and A≡B. Q.E.D.

Answer 3

You should be able to do it using supplementary angles.

\2/
1X3
/4\

Line those up, with the numbers representing the angles.

1S2 (angle one is supplementary to angle 2)
2S3
Therefore 1=3, since the measure of the supplement is fixed.