SOLUTION: A triangle with interior angles labeled 1, 2, and 3. The exterior angle at 3 is labeled angle 4. Given: Symbol for angle 4. is an exterior angle of the triangle Prove: Symbols fo

Algebra ->  Geometry-proofs -> SOLUTION: A triangle with interior angles labeled 1, 2, and 3. The exterior angle at 3 is labeled angle 4. Given: Symbol for angle 4. is an exterior angle of the triangle Prove: Symbols fo      Log On


   

Question 389290: A triangle with interior angles labeled 1, 2, and 3. The exterior angle at 3 is labeled angle 4.
Given: Symbol for angle 4. is an exterior angle of the triangle
Prove: Symbols for the measure of angle 4 = measure of angle 1 plus the measure of angle 2.
Two column proof: Statement 1 says “angle 4 is an exterior angle of the triangle.” Reason 1 says “given”. Statement 2 says “the measures of angles 1, 2 and 3 added together equal 180 degrees.” Reason 2 says “a” indicating this is the first place where a student should provide an answer. Statement 3 says “angles 3 and 4 are a linear pair.” Reason 3 says “b” indicating this is the next place where a student should provide an answer. Statement 4 says “angles 3 and 4 are supplementary angles.” Reason 4 says “If two angles form a linear pair, then they are supplementary.” Statement 5 says “measure of angle 3 plus the measure of angle 4 equals 180 degrees.” Reason 5 says “c” indicating this is the next place where a student should provide an answer. Statement 6 says “ the sum of angles 1, 2 and 3 equal the sum of angle 3 and 4.” Reason 6 says “d” indicating this is the next place where a student should provide an answer. Statement 7 says “the sum of the measure of angles 1 and 2 equal the measure of angle 4.” Reason 7 says “e” indicating this is the next place where a student should provide an answer.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
a: The angles of a triangle always add up to 180 degrees.
b: 4 is an exterior angle.
c: Angles that are supplementary add up to 180 degrees.
d: 180 = 180
e: Subtract the measure of angle 3 from both sides of the equality.

The way I would prove this uses the same steps, but is much shorter and just as concise. Suppose that angle 1 and angle 2 measure alpha and beta respectively. Since angle 1 + angle 2 + angle 3 = 180, then angle 3 = 180 - angle 1 - angle 2 = 180+-+alpha+-+beta.

Since angle 3 + angle 4 = 180, angle 4 = 180 - angle 3 = 180+-+%28180+-+alpha+-+beta%29+=+alpha+%2B+beta which is the same as angle 1 + angle 2.