Montana Math MT.CORE.GEOM.TRANS.2

MathGrades 9–12Transformations

Standard Cluster

Define the congruence of two and show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other by: Using the definition of congruence in terms of rigid motions to show that two triangles are congruent if, and only if, corresponding pairs of sides and corresponding pairs of angles are congruent, and Verifying that two triangles are congruent if, but not only if, the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS).

2026 Montana Mathematics Content Standards · Geometric Reasoning

Cluster contents

Standards in This Cluster

MT.CORE.GEOM.TRANS.2 is a cluster heading. These are the individual standards under it.

MT.CORE.GEOM.TRANS.2.b
Verifying that two triangles are congruent if, but not only if, the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-si...

Teacher's field guide

What This Cluster Means

What Students Need to Do: Students match corresponding sides and angles in two triangles. They use slides, turns, and flips to show that the triangles line up and that matching lengths and angles are preserved. They also use ASA, SAS, or SSS to prove congruence from fewer marked parts.
What Mastery Looks Like: A student identifies the correct matching vertices and uses ASA, SAS, or SSS when the given parts support it. They describe motions that make the triangles overlap and explain why matching all sides and angles means the triangles are congruent.
Common Misconceptions: Students may match vertices by their position on the page instead of by the marked parts. Some think AAA proves congruence, even though triangles can have the same angles and different sizes. Others use SSA as a test or think a flipped triangle cannot be congruent.
How to Assess It: Give students A(0,0), B(3,0), C(0,4) and D(6,2), E(3,2), F(6,6). Ask them to name the motions that map ABC onto DEF and identify one congruence test that applies.

Lesson moves

Ways to Teach It

Give students two identical scalene paper triangles; have them slide, turn, or flip one on tracing paper and label matching vertices.
Show two triangles with the same three angles but different side lengths; ask students to explain why AAA does not prove congruence.
Hand out triangle cards with marked parts; partners match congruent pairs, name ASA, SAS, or SSS, and flag SSA.
Sketch a 3-4-5-foot triangular roof brace and two replacements, 3-4-5 and 3-4-6; students use SSS to choose the match.

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