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ASSIGNMENT 7 – CONGRUENT TRIANGLES AND PASCH’S AXIOM
Definition 1: Two triangles are congruent if and only if there is some way to match the
vertices of one triangle to that of the other so that the corresponding sides are congruent
and the corresponding angles are congruent (We will use the symbol ‘  ‘ to denote
congruence).
Definition 2: An isosceles triangle is a triangle that has two sides congruent. The
congruent sides are called the ‘legs’ of the triangle and the third side is called the ‘base’.
The ‘base angles’ are the those angles sharing the base as a side.
Pasch’s axiom: Let A, B, C be three non-collinear points and let l be a line that does not
pass through A, B, or C. If l passes through the side AB , it must pass through either a
point on AC or a point on BC , but not both.
Definition 3: Let l be a line and A, B two points not on l. If A = B, or if segment AB
contains no point on l, we say that A, and B are on the same side of l. Otherwise, they are
on opposite sides. We say that the set of points all on one side of l are in the same halfplane.
Definition 4: Define a point D to be in the interior of an angle ABC if D is on the
same side of BC as A and also D is on the same side of AB as C. (The interior is the
intersection of two half planes).
Exercises:
State:
1. SAS
2. ASA
3. AAS
4. SSS
(NOTE: The congruence results above are independent of the parallel postulate)
5. State and prove the Isosceles Triangle Theorem (ITT) for planes. If we assumed
Euclid’s development of axioms and theorems to prove this result, we would be using
circular reasoning – Explain why.
6. State the converse of the ITT.
7. Use the ITT to show that the bisector of the top angle of an isosceles triangle is also
the perpendicular bisector of the base of that triangle.
8. Is it possible for a line to intersect all three sides of a triangle. If so, does this
contradict Pasch’s axiom?
9. Use Pasch’s Axiom to prove Hilbert’s Plane Separation Property (First state the pair
of results which Hilbert termed the Plane Separation Property).
10. Define precisely what is meant by the interior of a triangle.
11. Use congruent triangles to prove that the angle bisector construction is valid.
12. Show that for two right triangles, if the hypotenuse and leg of one triangle are
congruent to the hypotenuse and leg of the other, then the two triangles are congruent
[Try a proof by contradiction].
13. State and prove SASAS congruence result for quadrilaterals.
14. Do Exercises 2.2.7, 2.2.9, 2.2.10.
15. Give counterexamples to show that generally SAS is not true on spheres. Comment
on ASA, AAS, and SSS on spheres.