Advanced Mathematics

Determine the excluded values of a rational expression.

Simplify a rational expression by factoring and cancelling common factors.

Add and subtract rational expressions using a least common denominator.

Simplify a complex rational expression containing multiple fractions.

Solve a rational equation and reject solutions excluded from its domain.

Simplify radical expressions using product and quotient properties.

Rationalize a denominator containing one or two terms.

Solve a radical equation and check for extraneous solutions.

Add, multiply, and divide complex numbers in standard form.

Simplify integer powers of the imaginary unit.

Solve a quadratic equation whose solutions are nonreal complex numbers.

Factor a real polynomial when one nonreal complex zero is known.

Solve an exponential equation by applying logarithms.

Solve a logarithmic equation while enforcing domain restrictions.

Simplify the difference quotient of a polynomial function.

Use the Binomial Theorem to expand a binomial to a specified power.

Distinguish definitions, postulates, and theorems in a deductive argument.

Write the converse, inverse, and contrapositive of a conditional statement.

Disprove a universal mathematical claim by constructing a counterexample.

Translate marked geometric diagrams into symbolic statements.

Identify the givens and required conclusion of a geometric proof problem.

Justify an algebraic chain of statements using properties of equality.

Construct a two-column proof using segment or angle addition.

Prove a trigonometric identity by transforming one side into the other.

Identify linear pairs, vertical angles, and complementary or supplementary angle pairs.

Prove the Vertical Angles Theorem from linear-pair relationships.

Calculate angle measures formed by parallel lines and a transversal.

Prove that two lines are parallel using converses of transversal angle theorems.

Prove the Triangle Angle-Sum Theorem using a parallel auxiliary line.

Apply the Exterior Angle Theorem to determine unknown angle measures.

Write a paragraph proof involving parallel lines and triangle angle relationships.

Describe why rigid transformations preserve lengths and angle measures.

Match corresponding vertices and sides in a triangle-congruence statement.

Determine whether SSS, SAS, ASA, AAS, or HL proves two triangles congruent.

Explain why SSA and AAA do not establish triangle congruence.

Construct a proof that two triangles are congruent from diagrammed givens.

Use corresponding parts of congruent triangles to prove an additional relationship.

Prove the Isosceles Triangle Theorem using congruent triangles.

Prove the converse of the Isosceles Triangle Theorem.

Determine the scale factor and corresponding measurements under a dilation.

Determine whether AA, SAS, or SSS establishes triangle similarity.

Construct a proof that two triangles are similar.

Use corresponding sides of similar triangles to solve a proportion.

Apply the Triangle Proportionality Theorem to divided sides.

Derive a right-triangle geometric-mean relationship from similarity.

Prove the Pythagorean Theorem using similar triangles.

Connect trigonometric ratios to side relationships in similar right triangles.

Derive the interior-angle sum formula for a polygon by triangulation.

Derive the exterior-angle sum of a convex polygon.

Prove that opposite sides and angles of a parallelogram are congruent.

Apply a converse test to prove that a quadrilateral is a parallelogram.

Distinguish parallelograms, rectangles, rhombi, and squares by their defining properties.

Prove that a quadrilateral is a rectangle, rhombus, or square from sufficient givens.

Prove the Triangle Midsegment Theorem.

Prove the trapezoid midsegment relationship.

Calculate distance, midpoint, and slope from coordinates.

Use slopes to establish that lines are parallel or perpendicular.

Write the equation of a perpendicular bisector of a segment.

Partition a segment in a specified ratio using coordinates.

Classify a triangle from coordinate data and justify the classification.

Classify a quadrilateral from coordinate data and justify the classification.

Write a coordinate proof of a geometric claim.

Represent translations, reflections, and rotations using coordinate rules.

Relate central angles to intercepted arcs.

Apply the Inscribed Angle Theorem to determine angle or arc measures.

Use relationships among congruent chords, arcs, and distances from the center.

Prove that a radius to a point of tangency is perpendicular to the tangent.

Apply the theorem that tangent segments from one external point are congruent.

Determine angles formed by intersecting chords, secants, or tangents.

Apply power-of-a-point relationships to solve chord, secant, and tangent problems.

Write and interpret the standard equation of a circle in the coordinate plane.

Derive the area formula for a triangle from the area of a parallelogram.

Calculate the area of a regular polygon using its apothem.

Calculate arc length and sector area from a central angle.

Calculate the surface area of prisms and cylinders from their nets or dimensions.

Apply cross-sectional reasoning to calculate prism and cylinder volumes.

Explain the one-third volume relationship for pyramids and cones.

Use Cavalieri’s Principle to compare the volumes of solids.

Calculate the surface area and volume of a sphere.

Construct a segment’s perpendicular bisector with compass and straightedge and justify the construction.

Construct an angle bisector with compass and straightedge and justify the construction.

Construct a line parallel or perpendicular to a given line through a specified point.

Construct a triangle from specified side and angle data and determine whether the construction is unique.

Describe a geometric locus defined by an equal-distance condition.

Choose between synthetic and coordinate methods for a given proof problem.

Introduce an auxiliary line that exposes congruent or similar triangles in a proof.

Complete a multi-step proof that combines results about triangles, quadrilaterals, or circles.