Proof of the Exterior Angle Theorem.The exterior angle theorem for triangles states that the sum...

Baladdaa9

Baladdaa9

Answered

2022-07-22

Proof of the Exterior Angle Theorem.
The exterior angle theorem for triangles states that the sum of “The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.” How can we prove this theorem?

So, for the triangle above, we need to prove why C B D = B A C + B C A

Answer & Explanation

Brienueentismvh

Brienueentismvh

Expert

2022-07-23Added 11 answers

Step 1
In the A B C, we know that ABD is a straight line. So A B C = 180 C B D. From the angle sum property of triangles we can infer that B A C + A B C + B C A = 180 or A B C = 180 ( B A C + B C A ).
Step 2
Therefore: A B C = 180 C B D = 180 ( B A C + B C A ) C B D = ( B A C + B C A ) C B D × 1 = ( B A C + B C A ) × 1 C B D = B A C + B C A
Ciara Rose

Ciara Rose

Expert

2022-07-24Added 4 answers

Step 1
Euclid gives a visually satisfying proof of the exterior angle theorem by drawing BE parallel to AC, and observing that C B E = A C B (alternate interior angles) and E B D = C A B (corresponding angles), making C B D = A C B + C A B. This theorem includes the further important result that the three angles of a triangle sum to 180 o , or "two right angles" as Euclid says.

Step 2
But if, as I suspect, the true intent of OP's question is, assuming the truth of the exterior angle theorem, prove that the sum of the three exterior angles of a triangle is 360 o , then we can argue as follows.

Since by the exterior angle theorem C B D = B C A + B A C and A C E = C B A + B A C and B A F = C B A + B C A then by addition C B D + A C E + B A F = 2 B C A + 2 C B A + 2 B A C = 2 180 o = 360 o .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?