Consider the diagram below. \triangle{ABC} \cong \triangle{DEF}.

What sequence of rigid transformations could map \triangle{ABC} to \triangle{DEF}?

What segment is the image of \overline{AC} after performing rigid transformations on the pre-image? How do you know?

What angle is the image of \angle B after performing motions on the pre-image? How do you know?

Each segment and each angle in \triangle{ABC} maps to another segment or angle in \triangle{DEF}. What can you say about the relationship between each mapped pair?

When a figure can be mapped to an image using translations, reflections, and rotations, we can state that the triangles are congruent by definition of **rigid transformations**. Once we've established congruency between two triangles by mapping transformations, we can then justify the congruence of any pair of corresponding parts.

We can reference this theorem with its acronym, CPCTC.

When the figures are oriented in the same direction, it is easier to identify the corresponding parts. If the figures have been reflected or rotated try to find a reference point (such as a labeled pair, a shared side, or a right angle) to help us identify the corresponding parts.

Likewise, if we know all corresponding parts of a figure are congruent, then we know there is a way to map one figure to the other. This means two figures are **congruent** if all corresponding sides and all corresponding angles are congruent.

Prove corresponding segments of congruent triangles are congruent.

Worked Solution

Consider the congruent triangles shown below:

a

Show that the triangles are congruent using rigid transformations.

Worked Solution

b

Identify the congruent parts of the triangles and write a congruency statement.

Worked Solution

Find the values of x and y that would prove these triangles are congruent.

Worked Solution

Hamida is working on a building for her architecture class final exam. A tower in her building is constructed from congruent triangles that meet at the top of the tower's roof.

If \angle BCD \cong \angle BDC, and m \angle BCD=67 \degree, determine m \angle DBE.

Worked Solution

Idea summary

If two triangles are congruent, all corresponding segment and angle pairs will be congruent.

If all corresponding sides and all corresponding angles between a pair of figures are congruent, then the two figures will be congruent.