Hard multi-step angle problems: parallel lines with algebra, zig-zag shapes between parallel lines, isosceles triangle chains, exterior angles, regular polygons, two-unknown angle systems and angle ratios.

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Angle Pairs: Complementary, Supplementary & Vertical

Use complementary angles, linear pairs and vertical angles to find angle measures and solve for x.

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Arc Length & Area of a Sector

Find arc lengths and sector areas from the central angle and radius, and work backwards to find the angle or radius.

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Area of Regular Polygons (Apothem)

Find central angles and areas of regular polygons using the apothem formula A = 1/2 · a · P.

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Tangents, Chords & Secant Segments

Use tangent-radius perpendicularity, congruent tangent segments and the segment products for chords, secants and tangents to find lengths.

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Angles Formed by Chords, Secants & Tangents

Find angles and arcs when chords cross inside a circle, when secants or tangents meet outside it, and when a tangent meets a chord.

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Equations of Circles

Write the standard-form equation of a circle and find the center and radius from an equation, including by completing the square.

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Expert Angle Problems: Triangle Centers, Circles & Polygons

Olympiad-style angle chasing: incenter, excenter, orthocenter and circumcenter angles, cyclic quadrilaterals, tangents and secants, arcs in a ratio, regular polygons and star polygons, isosceles staircases, polygons with angles in an arithmetic sequence, and light reflected between two mirrors.

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Isosceles & Equilateral Triangles

Use the Base Angles Theorem and properties of equilateral triangles to find angles, sides and x.

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Parallel Lines Cut by a Transversal

Use corresponding, alternate interior, alternate exterior and same-side interior angles to find angle measures and solve for x.

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Partitioning a Directed Line Segment

Find the point that divides a segment in a given ratio m:n on a number line and on the coordinate plane, and work backwards to an endpoint or to the ratio.

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Centroid, Circumcenter & Incenter

Use the centroid (2:1 on each median), circumcenter (equidistant from vertices) and incenter (equidistant from sides) to find lengths and solve for x.

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Pyramids & Prisms: Volume and Surface Area

Find volumes of pyramids and triangular prisms, and lateral and total surface areas of square pyramids using the slant height.

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Segment Addition & Midpoints

Use the Segment Addition Postulate and the definition of a midpoint to find lengths, coordinates and x.

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Properties of Parallelograms, Rectangles, Rhombuses & Kites

Use the side and diagonal properties of parallelograms, rectangles, rhombuses and kites to find lengths, angles and x.

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Special Right Triangles: 45-45-90 & 30-60-90

Use the side ratios of 45°-45°-90° and 30°-60°-90° triangles to find missing sides in simplest radical form.

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Trapezoids: Midsegments, Angles & Diagonals

Use the trapezoid midsegment theorem and the properties of isosceles trapezoids (equal legs, equal base angles, equal diagonals) to find lengths, angles and x.

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