Lesson 5: Solving for Unknown Measures in Congruent Figures

Property

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. If you know two triangles are congruent, you can set the measures of their corresponding sides or corresponding angles equal to each other to solve for unknowns.

Examples

• Finding lengths: Given $\Delta ABC \cong \Delta XYZ$, $m\angle A = 50^\circ$, and $YZ = 12$ cm. To find side $BC$, identify the corresponding part. Side $BC$ corresponds to side $YZ$, so $BC = 12$ cm.
• Solving with Algebra: If $\Delta PQR \cong \Delta TUV$, side $QR = 2x + 5$, and side $UV = 15$. Since corresponding sides are equal, set up the equation: $2x + 5 = 15$. Solving gives $2x = 10$, so $x = 5$.
• Perimeters: If two pentagons are congruent and the perimeter of the first is 65 meters, the perimeter of the second must also be 65 meters.

Explanation

CPCTC is the primary reason we prove triangles congruent in the first place.

Property

If two figures are congruent, then the measures of their corresponding parts are equal.
If $\triangle ABC \cong \triangle DEF$, then the lengths of corresponding sides are equal ($AB = DE$) and the measures of corresponding angles are equal ($m\angle A = m\angle D$).

Examples