Write out the proof proving that vertical angles are congruent.?

Question

Can only use postulates, thereoms, definitons. Thanx. I don't get it at alll. Stuck!

Answer 1

Whenever I come across questions like this one, I realize two things: (1) geometry students often feel burdened by all the strangely-named 'rules' (axioms, postulates, etc.); and (2) proofs can be intimidating!

I'd like to try making this one simple, citing as few theorems as possible. First, I will just say it "in English" for you.

When two lines cross to form an X, you easily see at least 4 angles. The up-down pair of angles (I'll call them v and ^) are 'vertical angles' and so are the right-left pair (I'll call them < and >). Someone could have come up with a better name than 'vertical' ... especially since 'vertical' makes you think of "standing up straight." Here, 'vertical' means "of the vertex." Since the lines cross and form a vertex, the angles get called vertical.

Now, why would the angles v and ^ be the same measure (congruent)? The easiest answer is "Because they both have the same supplement."

Which supplement do they have in common? You could say either < or >, it doesn't matter.

Of course, I'm being a bit lazy by not using geometry-standard terms, so now we must label everything correctly and formally:

Given two intersecting lines, AB and CD, whose intersection is at the point X. Prove that angle AXC is congruent to DXB. (Hopefully your drawing will match my setup!)

PROOF:

(1) Angle AXC is supplementary to angle CXB.
Reason: Angles AXC and CXB form a straight angle.

(2) Angle DXB is also supplementary to angle CXB.
Reason: (Same as Step 1)

(3) Angle AXC is congruent to angle DXB.
Reason: Two angles that are supplementary to the same third angle, are congruent.

Hope this sets you on your way!

Answer 2

Vertical Angles Theorem Proof

Answer 3

Vertical angles are angles that form when two lines cross, like the opposite sides of an "X". Suppose you have an "X", and the angle at the top is numbered 1, the angle at the right is numbered 2, the bottom is numbered 3, and the left is numbered 4. Then angle 1 and angle 3 are vertical angles (and so are angles 2 and 4).

To do the proof ...
The measure of angle 1 + the measure of angle 2 = 180
(because they make a linear pair, and the angles in a linear pair are supplementary)
The measure of angle 2 + the measure of angle 3 = 180
(because they also make a linear pair and are supplementary)
The measure of angle 1 + the measure of angle 2 = the measure of angle 2 + the measure of angle 3
(by substitution or the symmetric and transitive properties of equality)
The measure of angle 1 = the measure of angle 3
(by the addition property of equality ... adding negative measure of angle 2 to both sides)
Angle 1 is congruent to angle 3
(by the definition of congruet)

Answer 4

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RE:
Write out the proof proving that vertical angles are congruent.?
Can only use postulates, thereoms, definitons. Thanx. I don't get it at alll. Stuck!

Answer 5

I don't either.