Congruent shapes

Two shapes are congruent when one of them can be moved exactly onto the other: by a reflection, a translation, a rotation, or several of them one after another.

• Every shape is congruent to its image in a reflection, a translation or a rotation: they change no length and no angle.
• Congruent shapes have the same size and the same shape; only their places differ.
• The sign for “is congruent to” is $\displaystyle \cong$.

When a triangle is laid on a triangle congruent to it, every vertex falls on a vertex: these are corresponding vertices.
In the drawing, $\displaystyle A$ falls on $\displaystyle D$, $\displaystyle B$ on $\displaystyle E$, and $\displaystyle C$ on $\displaystyle F$.

• Corresponding sides are equal: $\displaystyle {AB}={DE}$, $\displaystyle {BC}={EF}$, $\displaystyle {CA}={FD}$.
• Corresponding angles are equal: $\displaystyle {\angle A}={\angle D}$, $\displaystyle {\angle B}={\angle E}$, $\displaystyle {\angle C}={\angle F}$.
• Six pairs of equal parts in all: three pairs of sides and three pairs of angles.
• In the drawing, the same number of small strokes marks equal sides, and the same Greek letter equal angles.

In $\displaystyle \triangle ABC\cong\triangle DEF$ the order of the letters is the order of the correspondence: the first letter corresponds to the first, the second to the second, the third to the third.

• Every pair can be read from it: $\displaystyle AB$ (the first two letters) corresponds to $\displaystyle DE$; $\displaystyle \angle B$ (the middle letter) to $\displaystyle \angle E$.
• $\displaystyle \triangle ABC\cong\triangle EDF$ is wrong: it says that $\displaystyle A$ corresponds to $\displaystyle E$.
• The letters of both triangles may be reordered together: $\displaystyle \triangle BCA\cong\triangle EFD$ is right too.

Given: $\displaystyle \triangle KLM\cong\triangle PQR$.
$\displaystyle {KL}={6}$, $\displaystyle {QR}={8}$, $\displaystyle {\angle M}={50^{\circ}}$.
We find sides and angles of triangle $\displaystyle PQR$:

• $\displaystyle {PQ}={KL}={6}$: the first two letters of each triangle.
• $\displaystyle {LM}={QR}={8}$: the last two letters.
• $\displaystyle {\angle R}={\angle M}={50^{\circ}}$: the third letter.

Given: $\displaystyle \triangle ABC\cong\triangle XYZ$.
Fill in.

$\displaystyle {AC}={\ ?}$

$\displaystyle {\angle Y}={\ ?}$

$\displaystyle {YZ}={\ ?}$

The triangles in the drawing are congruent.
Write the congruence in the right order, $\displaystyle \triangle ABC\cong\triangle\ ...$, and give the reason from the marks on the sides.

Given: $\displaystyle \triangle DEF\cong\triangle RST$.
$\displaystyle {DE}={5}$, $\displaystyle {EF}={7}$, $\displaystyle {FD}={9}$.
$\displaystyle {\angle D}={40^{\circ}}$, $\displaystyle {\angle E}={60^{\circ}}$.
Find all the sides and all the angles of triangle $\displaystyle RST$.

Are the two shapes always congruent? Give a reason.

Two squares, each with a side of 3 cm.

Two rectangles, each with an area of 12 cm².

Two triangles, each angle of which is $\displaystyle 60^{\circ}$.