Congruent Triangles Explained: Geometry Rules & Examples

In 7th grade math, students learn about congruent figures. Two triangles are congruent if they have the same size and shape, meaning all their sides and angles match—even if one triangle is flipped, turned, or slid on the plane. A common mistake is to confuse “same shape” with “same position.” Another error is not matching the correct corresponding parts. Recognizing and avoiding these mistakes is key to mastering congruent triangles in Common Core Grade 7 geometry.

What Does “Congruent” Mean?

In math, the word congruent means exactly the same in size and shape. Two figures are congruent if one can be transformed into the other by flipping, turning, or sliding, without resizing.

What Are Congruent Triangles?

Congruent triangles have the same size and shape. Their corresponding sides and angles are equal.

This idea is often written as:

This is sometimes called CPCTC, which stands for: Corresponding Parts of Congruent Triangles are Congruent.

Example Problems: Congruent Triangles

Example 1

Problem: Given that \( \triangle ABC \cong \triangle XYZ, \; m{\large \angle}A = 40^\circ, \; m{\large \angle}B = 70^\circ \). What is the measure of \({\large \angle}Z \, ?\)

Solution:

\(∵ \; \triangle ABC \cong \triangle XYZ, \) all corresponding angles are equal

\[∴ \; m{\large \angle}C = m{\large \angle}Z\]

\[∵ \; m{\large \angle}C = 180^\circ – \big(m{\large \angle}A + m{\large \angle}B\big) = 180^\circ – (40^\circ + 70^\circ) = 70^\circ\]

\[∴ \; m{\large \angle}Z = m{\large \angle}C = 70^\circ\]

Example 2

Problem: Given that △PQR ≅ △MNO, PQ = 5 cm, QR = 7 cm, perimeter of △PQR = 20 cm. What is the length of side MO?

Solution:

∵△PQR ≅ △MNO

∴ MO = PR = 20 cm − 5 cm − 7 cm = 8 cm

Additional Math Topics for Triangles – with Free Worksheets

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