Prove Euclid's Fifth Postulate using Triangle Sum?

Solutions

Expert Solution

Euclides fifth postulates States that " if a line segment intersect two lines forming two interior angle on the same side that sum to less than two right angle , then the two lines , if extended indefinitely , meet on that side on which the angles sum to less than two right angle".

We can prove it by contradiction method.

Let AB and CD be two straight lines and another straight line L meet with AB at E and With CD at F .

Where <BEF+<DFE is less than 180°.

We have to prove that if AB and CD are extended indefinitely then AB and CD wil meet on the side in which <BEF and <DFE occurs.

Assume that the two lines AB and CD will meet on the side in which <AEF and <CFE occurs , and the meeting point is M.

Now <BEF+<DFE is less than 180° implies that <AEF+<CFE is greater than 180°.

Then from triangle EMF we have <EMF=180°-(<MEF+<MFE) =180°- (angle greater than 180°) = negative value. But any Angle can not be negative. So,our assumption was wrong . So by contradiction method two lines meet on the side in which <BEF and <DFE occurs. And if AB and CD meet at N then<ENF=180°-(<NEF+<NFE).

This completes the proff.

milcah answered 1 year ago

Next > < Previous

Related Solutions

Prove that the sum of angles of a hyperbolic triangle is less than 180. Detailed Proof

Prove the following using the triangle inequality: Given a convex quadrilateral, prove that the point determined...

Prove the following using the triangle inequality: Given a convex quadrilateral, prove that the point determined by the intersection of the diagonals is the minimum distance point for the quadrilateral - that is, the point from which the sum of the distances of the vertices is minimal.

Prove that the angle bisectors of a triangle are concurrent using the Pointwise Characterization of an...

Prove that the angle bisectors of a triangle are concurrent using the Pointwise Characterization of an Angle Bisector.

PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that...

PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that side BC is of length 2, the circumradius is of length 3/2 and the median AA' is of length 2 and is the midpoint of BC.

3. State thoroughly and precisely Euclids fifth postulate. According to Playfair, "Given a line l and...

3. State thoroughly and precisely Euclids fifth postulate. According to Playfair, "Given a line l and a point P not on l, there is exactly one line through P that is parallel to l" Are Euclids fifth postulate and Playfair's claim logically equivalent? If yes, justify your stance. If no, produce a counter example.

what is the maximum sum of the angles for a triangle on a sphere

Prove that the following statements are equivalent. A. The Euclidean parallel postulate holds. B. Given any...

Prove that the following statements are equivalent. A. The Euclidean parallel postulate holds. B. Given any triangle △ABC and given any segment DE, there exists a triangle △DEF having DE as one of its sides such that △ABC ∼ △DEF (Wallis’ postulate on the existance of similar triangles). (you cannot use measures)

Prove that the SMSG axiomatic set is not independent. SMSG Axioms: Postulate 1. Given any two...

Prove that the SMSG axiomatic set is not independent. SMSG Axioms: Postulate 1. Given any two distinct points there is exactly one line that contains them. Postulate 2. Distance Postulate. To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points. Postulate 3. Ruler Postulate. The points of a line can be placed in a correspondence with the real numbers such that: To every point of the line there...

Find a point within a triangle such that the sum of the square of its distances...

Find a point within a triangle such that the sum of the square of its distances from the three angular points is a minimum.

Prove Proposition 3.22(SSS Criterion for Congruence). Given triangle ABC and triangle DEF. If AB is congruent...

Prove Proposition 3.22(SSS Criterion for Congruence). Given triangle ABC and triangle DEF. If AB is congruent to DE, BC is congruent to EF, and AC is congruent to DF, then triangle ABC is congruent to triangle DEF. (Hint:Use three congruence axioms to reduce to the case where A=D, C=F, and points B and E are on opposite sides of line AC.)

Subjects