Exploring Congruent Triangles - Interactive Activity

Two figures are congruent when they have the same shape and the same size. For triangles, that one sentence has a precise meaning: every side of one triangle matches a side of the other triangle, and every angle of one triangle matches an angle of the other triangle. The skill of this lesson is keeping track of which part matches which part. That matching is called the correspondence, and once it is known, every measure in one triangle can be read from the other triangle. Throughout this page the same pair of triangles is used, with the correspondence A corresponds to D, B corresponds to E, and C corresponds to F, so the congruence statement is △ABC ≅ △DEF.

Definitions and Main Ideas

Congruent figures: figures that have the same shape and the same size. Congruence is not affected by translating, rotating, or reflecting a figure, because none of those moves changes the figure’s shape or size.
Congruent triangles: two triangles are congruent if and only if their corresponding parts are congruent — all three pairs of corresponding sides are congruent (equal in length) and all three pairs of corresponding angles are congruent (equal in measure).
Corresponding parts: the sides and angles of two figures that occupy the same relative position. In a congruence statement, corresponding vertices are named in the same position in both names.
Congruence statement: a statement such as △ABC ≅ △DEF that names two congruent figures with their vertices listed in corresponding order: A corresponds to D, B corresponds to E, and C corresponds to F.
CPCTC: Corresponding Parts of Congruent Triangles are Congruent. Once two triangles are known to be congruent, any pair of corresponding sides or corresponding angles must be congruent.

Main ideas

Congruent means same shape and same size: every corresponding side and every corresponding angle matches exactly.
Always write congruence statements in corresponding order. The first triangle can be named starting at any vertex, but the second triangle must follow the same correspondence.
Read parts from a statement by position: in △ABC ≅ △DEF, side BC joins the second and third vertices, so it corresponds to side EF, and angle B corresponds to angle E.
CPCTC turns one congruence statement into six facts: three pairs of congruent sides (equal in length) and three pairs of congruent angles (equal in measure).
Moving a figure by a translation, a rotation, or a reflection never changes its size or shape, so the image is always congruent to the original.

Correspondence Explorer

What to do: The two triangles below are congruent. The triangle on the right is a rotation of the triangle on the left, so it has the same shape and the same size, even though it points in a different direction. Press the button for any angle or any side of △ABC. That part is highlighted in gold on both triangles, and the readout names the corresponding pair.

What to notice: corresponding parts always carry the same measure. Corresponding angles are congruent (equal in measure), and corresponding sides are congruent (equal in length). Every measure shown is computed from the drawing itself.

CPCTC Explorer

What to do: This is the same pair of triangles, △ABC ≅ △DEF, but now only △ABC has its measures labeled. Press a button to ask for a part of △DEF. The part is highlighted in △DEF, its measure appears on △DEF in the diagram, and the readout finds that measure by CPCTC: it first names the part of △ABC that sits in the same position in the congruence statement, and that part has the measure you need. Press a different button and the highlight and the measure on △DEF move to the new part.

What to notice: you never measure △DEF directly. Position in the statement does all the work — the first vertex matches the first vertex, the second matches the second, and the third matches the third.

Write the Congruence Statement

What to do: Each challenge shows two congruent triangles with congruence marks: matching tick marks on sides that are congruent (equal in length), and matching arcs on angles that are congruent (equal in measure). One matching pair of vertices is given to start you off, and the first triangle is named in a stated order. Choose the one congruence statement that names the second triangle in true corresponding order.

The 1-2-3 strategy: number the congruent angle pairs 1, 2, and 3 — in these diagrams the pair with one arc is pair 1, the pair with two arcs is pair 2, and the pair with three arcs is pair 3. Then read the vertices of the first triangle in its stated order, note the number pattern, and name the second triangle by following the same number pattern. The side tick marks must agree as well, so use them as a check.

Your Turn

Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.

Exploring Congruent Triangles — Mr. J Geometry. All activities on this page run entirely in your browser.