Mia checks five rectangles and notices each one has diagonals that bisect each other, so she conjectures all rectangles have this property. What type of reasoning is she using?

A proof uses the theorem that vertical angles are congruent. Since angle 1 and angle 2 are vertical angles, it concludes m<1 = m<2. What type of reasoning is this?

Carlos measures several triangles and finds each angle sum is 180 degrees, then conjectures every triangle has angle sum 180 degrees. What type of reasoning is he using?

Nina knows a midpoint divides a segment into two congruent segments. Since M is the midpoint of AB, she concludes AM = MB. What type of reasoning is this?

A regular polygon pattern has perimeters 9 cm, 12 cm, and 15 cm. Jalen predicts the next perimeter is 18 cm because the pattern adds 3 cm each time. What reasoning is this?

A postulate says exactly one line passes through any two distinct points. Points A and B are distinct, so Priya concludes exactly one line passes through A and B. What reasoning is this?

After drawing several parallelograms, Leo notices opposite angles are congruent and conjectures all parallelograms have congruent opposite angles. What type of reasoning is he using?

A theorem says perpendicular lines form right angles. Since line l is perpendicular to line m, Aisha concludes the angle where they meet measures 90 degrees. What reasoning is this?

Conjecture: Every quadrilateral with one pair of parallel sides is a parallelogram. Among a rectangle, square, isosceles trapezoid, and rhombus, what is a counterexample?

Conjecture: Every triangle with two congruent sides is equilateral. Among triangles with side lengths 5 cm, 5 cm, 5 cm; 5 cm, 5 cm, 8 cm; 5 cm, 6 cm, 7 cm; and 4 cm, 4 cm, 4 cm, what is a counterexample?

Conjecture: If a quadrilateral has four right angles, then it is a square. Among a square, a 6 cm by 10 cm rectangle, a 60-degree rhombus, and a kite with one right angle, what is a counterexample?

Conjecture: Every quadrilateral with four congruent sides is a square. Among a square, a rectangle measuring 5 cm by 8 cm, a 60-degree rhombus, and a trapezoid, what is a counterexample?

Conjecture: Every solid with two congruent parallel bases is a rectangular prism. Among a cube, rectangular prism, triangular prism, and square pyramid, what is a counterexample?

Conjecture: If two angles are congruent, then they are vertical angles. Among vertical 50-degree angles, base angles of an isosceles triangle each 50 degrees, 40-degree and 140-degree angles, and 50-degree and 60-degree angles, what is a counterexample?

Conjecture: Every chord of a circle is a diameter. Among a diameter through the center, a radius, a tangent line, and a chord not through the center, what is a counterexample?

Conjecture: Every pair of lines that never meet is parallel. Among opposite sides of a rectangle, skew edges of a rectangular prism, perpendicular lines, and the same line named twice, what is a counterexample?

A pattern of angle measures is 20 degrees, 35 degrees, 50 degrees, and 65 degrees. If the pattern continues, what is the next measure?

A polygon pattern has figures with 3 sides, 4 sides, 5 sides, and 6 sides. If the pattern continues, how many sides should the next polygon have?

Drawing all diagonals from one vertex divides polygons into 2, 3, 4, and 5 triangles for a quadrilateral through a heptagon. If the pattern continues, how many triangles are formed in an octagon?

The number of diagonals in polygons with 3, 4, 5, and 6 sides is 0, 2, 5, and 9. The increases are 2, 3, and 4. If this pattern continues, how many diagonals does a 7-sided polygon have?