Since all rigid motions of the plane preserve angles, angle $C''DE$ must map to angle $FDE$. Postulate 14 (SAS Postulate): If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle. Angle $FDE$ is congruent to angle $C''DE$ because angle $C''DE$ is the image under two reflections of angle $CAB$ which is congruent to angle $FDE$ by hypothesis. So we have to check that reflection about line $DE$ maps $C''$ to $F$.
The only reflection that leaves $D$ and $E$ fixed is the one about line $DE$. In this last step we must move $C''$ to $F$ via a reflection while leaving $D$ and $E$ fixed. The result of the second reflection is pictured below: The reason we know that $D$ is on the perpendicular bisector of $B'E$ is that it is equidistant from $E$ and $B'$ by the hypothesis that $AB$ is congruent to $DE$ and the perpendicular bisector of a line segment $xy$ consists of all points in the plane equidistant from $x$ and $y$. Note that it is important that this perpendicular bisector contains $D$ so that our second reflection preserves what we accomplished in the first step. In this step we wish to move $B'$ to $E$ and so we must reflect again, this time about the perpendicular bisector of $B'E$. Also pictured below is the new triangle $DB^\prime C^\prime$ obtained by reflecting triangle $ABC$. Learn vocabulary, terms, and more with flashcards, games, and other study tools.
So we must reflect about the perpendicular bisector of $AD$ which is pictured below. In the first part of this problem, we wish to send $A$ to $D$ via a reflection. If two triangles are congruent, if I say that. triangle congruence side side side side angle side. SSS and SAS are important shortcuts to know when solving proofs.
Another shortcut is side-angle-side (SAS), where two pairs of sides and the angle between them are known to be congruent. Download scientific diagram SAS geometry in the upper divertor of DIII-D.
For every testing method, you are checking the three parts identified between the two triangles. That side is out there, all alone, not between the angles. So Side Angle Side (SAS) means one side, the angle next to that side, and then the side next to that angle. Reflection about line $L$ sends point $P$ in the plane to point $Q$ exactly when $L$ is the perpendicular bisector of $PQ$ This congruence shortcut is known as side-side-side (SSS). An included angle or side is physically between the others in the triangle.