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High school geometry

Lesson 2: Triangle congruence from transformations

Justify triangle congruence

Below are triangle, A, B, C and triangle, D, E, F. We assume that A, B, equals, D, E, B, C, equals, E, F, and m, angle, B, equals, m, angle, E.

Here is a rough outline of a proof that triangle, A, B, C, \cong, triangle, D, E, F:

We can map triangle, A, B, C using a sequence of rigid transformations so that A, prime, equals, D and B, prime, equals, E.
[Show drawing.]
If C, prime and F are on the same side of D, E, with, \overleftrightarrow, on top, then C, prime, equals, F.
[Show drawing.]
If C, prime and F are on opposite sides of D, E, with, \overleftrightarrow, on top, then we reflect triangle, A, prime, B, prime, C, prime across D, E, with, \overleftrightarrow, on top and then C, start superscript, prime, prime, end superscript, equals, F, A, start superscript, prime, prime, end superscript, equals, D and B, start superscript, prime, prime, end superscript, equals, E.
[Show drawing.]

What is the justification that C, prime, equals, F in step 2?

(Choice A)
C, prime and F are the same distance from E along the same ray.
(Choice B)
Both C, prime and F lie on intersection points of circles centered at D and E with radii D, F and E, F, respectively. There are two such possible points, one on each side of D, E, with, \overleftrightarrow, on top.
(Choice C)
C, prime and F are at the intersection of the same pair of rays.

Stuck?