SOLUTION: Hello. I am having difficulty doing this paragraph proof. Can you guide me through the proof and help me solve it? thanks. Angle ABC and Angle DBF are vertical. Prove that Angle

Algebra ->  Geometry-proofs -> SOLUTION: Hello. I am having difficulty doing this paragraph proof. Can you guide me through the proof and help me solve it? thanks. Angle ABC and Angle DBF are vertical. Prove that Angle       Log On


   

Question 725785: Hello. I am having difficulty doing this paragraph proof. Can you guide me through the proof and help me solve it? thanks.
Angle ABC and Angle DBF are vertical. Prove that Angle ABD is congruent to Angle FBC
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
A pair of angles whose sides form two lines is called vertical angles.
Vertical angles are congruent and it is easy to prove. We just use the fact that a linear pair of angles are supplementary; that is their measures add up to 180°.

so, first draw lines l and m intersecting at point P
angles 1 (left of the point P) and 3 (right of the point P) are vertical angles since their sides form lines l and m, similarly, angles 2 (above of the point P) and 4 (below of the point P) are vertical angles for the same reason

Proof
We show that < 1 congruent < 3.
By definition, m < 1 + m < 2 = 180° because linear pair of angles are supplementary.
Then m< 2 + m < 3 = 180° for same reason: linear pair of angles are supplementary.


using substitution property of equality, we have
m < 1 + m < 2 = m < 2 + m < 3 ; that is 180° = 180°.
Subtracting m < 2 from both sides, we have
m+< 1 =+m+< 3.
Therefore, vertical angles are congruent.