Montana Math MT.CORE.GEOM.ARG.1.c

MathGrades 9–12Geometric Arguments, Reasoning, and Proof

The Standard

MT.CORE.GEOM.ARG.1
Investigate, conjecture, prove theorems, and communicate the proofs in a variety of ways by: Proving theorems about lines and angles. Theorems include: vertical angles are congruent, when a transversal crosses parallel lines alternate interior angles are congruent and corresponding angles are congruent, and the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints, Proving theorems about triangles. Theorems include: the sum of the measures of the interior angles of a triangle is 180˚, the Pythagorean Theorem, the base angles of isosceles triangles are congruent, and a line parallel to one side of a triangle divides the other two sides proportionally, Proving theorems about parallelograms and other quadrilaterals. Theorems include: necessary and sufficient conditions for rectangles, parallelograms, rhombi, and kites, and Proving theorems about circles. Theorems include: the relationship between central, inscribed, and circumscribed angles, inscribed angles on a diameter are right angles, and the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Investigate, conjecture, prove theorems, and communicate the proofs in a variety of ways by: Proving theorems about parallelograms and other quadrilaterals. Theorems include: necessary and sufficient conditions for rectangles, parallelograms, rhombi, and kites, and

2026 Montana Mathematics Content Standards · Geometric Reasoning

Teacher's field guide

What This Standard Means

What Students Need to Do

Students draw diagonals and mark equal lengths and angles to test claims about quadrilaterals. They prove when a condition guarantees a parallelogram, rectangle, rhombus, or kite, and use counterexamples to reject false claims.

What Mastery Looks Like

A student can prove both directions of a claim, such as why a quadrilateral is a parallelogram if and only if its diagonals bisect each other. They can use added conditions to prove a figure is a rectangle, rhombus, or kite.

Common Misconceptions

Students may think equal diagonals alone make a rectangle, but an isosceles trapezoid has equal diagonals too. They may think perpendicular diagonals always make a rhombus, or prove only one direction of an if-and-only-if claim.

How to Assess It

Exit ticket: A quadrilateral’s diagonals bisect each other and meet at right angles. Name the figure, justify your answer in two steps, and say whether it must be a square.

Lesson moves

Ways to Teach It

  1. Cut out a rectangle, rhombus, kite, and slanted parallelogram; draw their diagonals, measure the segments and angles, then record one conjecture.

  2. Ask: Can equal diagonals alone prove a quadrilateral is a rectangle? Draw an isosceles trapezoid to support your answer.

  3. Deal cards labeled 'parallelogram,' 'equal diagonals,' and 'perpendicular diagonals'; partners combine cards to name a figure and defend each match.

  4. Measure both pairs of opposite sides and the diagonals of a picture frame, then use the results to argue whether it is rectangular.

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