Question 162701This question is from textbook Geometry for enjoyment and challenge
: Given: angle AFE is congruent to angle DEF
ray FC bisects angle AFE
ray EB bisects angle DEF
Prove: angle one is congruent to angle two
This question is from textbook Geometry for enjoyment and challenge
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! ------------------------------------
given:
angle AFE is congruent to angle DEF
ray FC bisects angle AFE at F
ray EB bisects angle DEF at E
prove:
angle 1 is congruent to angle two
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proof:
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statement 1:
let angle 1 = angle AFC
let angle 2 = angle CFE
let angle 3 = angle DEB
let angle 4 = angle BEF
reason 1:
by naming convention, any angle formed by the endpoint of a ray can be called by either the letter at the vertex of the ray, the three letters of the rays with the angle formed by the ray as the middle letter, or a number designated to represent the angle formed by the intersection of the rays.
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statement 2:
angle 1 equals angle 2.
angle 3 equals angle 4.
reason 2:
if a ray bisects an angle, it creates two equal angles with the bisecting ray as the common side between them.
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statement 3:
angle 1 plus angle 2 equals angle AFE.
angle 3 plus angle 4 equals angle DEF
reason:
if a ray divides an angle, it creates two angles, the sum of which equals the divided angle.
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statement 4:
2 times angle 1 equals angle AFE.
2 times angle 2 equals angle AFE.
2 times angle 3 equals angle DEF.
2 times angle 4 equals angle DEF.
reason 4:
substitution of equal elements.
angle 1 equals angle 2 so you can substitute angle 1 for angle 2. same for the others.
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statement 5:
2 times angle 1 equals 2 times angle 3.
2 times angle 2 equals 2 times angle 4.
reason 5:
transitive property of equality. 2*1=AFE, 2*3=DEF, AFE=DEF, 2*1=2*3. same for 2*2=2*4.
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statement 6:
angle 1 equals angle 3
angle 2 equals angle 4
reason 6:
algebraic property of equations. 2*a=2*b; divide both sides of equation by 2 to get a = b.
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statement 7:
angle 1 is congruent to angle 2
angle 3 is congruent to angle 4
angle 1 is congruent to angle 3
angle 2 is congruent to angle 4
reason 7:
two angles are congruent if they have the same measure.
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note:
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not sure which angles you called 1 and 2, but this proof works for any of those situations and all you have to do is rename.
draw a picture and then change as you see fit.
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the wording here is in all probability not the exact wording in the text. if you don't use the exact wording in the text, then be as precise in your own wording as you can to remove any ambiguity and to make what you are trying to say as clear as possible.
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use terms, definitions, postulates, previously proven theorems, for reasons of statements in your proof.
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