Triangle Congruence Criteria and Proofs

Identify and calculate unknown angles using complementary, supplementary, vertical, and parallel line transversal relationships with algebraic equations.

Apply interior and exterior angle theorems, side-angle relationships, and the triangle inequality to classify and solve triangles.

Prove triangles congruent using SSS, SAS, ASA, AAS, and HL criteria and deduce corresponding parts.

Prove, calculate, and apply the Pythagorean theorem and its converse to 2D and 3D real-world problems.

Classify quadrilaterals, prove polygon properties, and calculate interior and exterior angle sums of n-sided polygons.

Construct direct proofs, converse statements, counterexamples, and proofs by contradiction to establish geometric truths.

Define pi, calculate circumference and arc length, and analyze chords, tangents, and central and inscribed angles.

Derive circle area, calculate sector and segment areas, and solve perimeter and area for composite plane figures.

Execute precise geometric constructions using compass, straightedge, protractor, and dynamic geometry software.

Perform and describe translations, reflections, and rotations on and off the coordinate plane and identify symmetries.

Apply dilations, establish triangle similarity criteria (AA, SAS, SSS), and solve scale drawing and perimeter/area ratio problems.

Define sine, cosine, and tangent ratios, compute exact values for 30°, 45°, 60°, and solve right triangles.

Represent 3D objects with front/top/side orthographic projections, nets, isometric drawings, and plane cross-sections.

Calculate lateral surface area, total surface area, and volume of prisms, pyramids, cylinders, and composite solids, modeling volume relationships.