Geometry

Geometry studies shapes, angles and their relationships. An angle describes how far one direction turns towards another.

Mark equal angles and seek similar triangles before introducing variables. Ratios of corresponding sides square when converted to area ratios.

Two angles in a triangle are 50° and 60°. Find the third angle. 1. The interior angles of a triangle on a flat plane total 180°. 2. The known angles total 50° + 60° = 110°. 3. The remaining angle is 180° − 110° = 70°.

The three labels are the triangle’s interior angles. Their sum on a flat plane is 180°, so the unknown is 180 − 50 − 60 = 70°. The sketch is schematic: do not measure its apparent angles or side lengths to answer a question. Use only stated properties and valid geometric relationships.

In triangle ABC, D lies on AB and E on AC, with DE parallel to BC and AD:DB=2:3. AD/AB=2/5, so triangle ADE is similar to ABC with length scale 2/5. Its area is 4/25 of the whole. If ABC has area 100, ADE has area 16 and trapezium DBCE has area 84. DE/BC remains 2/5, not 4/25.

A drawing may not be to scale. Use the stated measurements and properties. A ratio of part to remaining part is not a ratio of part to whole. Draw the full segment before using similarity.

A triangle has angles of 50 degrees and 60 degrees. Together they make 110 degrees. The missing angle is 180 - 110 = 70 degrees. In similar triangles, matching angles are equal and matching sides have the same ratio.

Angles on a straight line total 180 degrees. Parallel-line angle equalities require the lines to actually be parallel.

Two adjacent angles on a straight line are 3x and 2x. Then 5x = 180, so x = 36 and the angles are 108 and 72 degrees.

Interior angles of a quadrilateral total 360 degrees. Apply special properties only after the shape is identified.

A parallelogram has one angle of 70 degrees. Adjacent angles sum to 180, so the next is 110. Opposite angles are equal, giving 70,110,70,110.

An n-sided polygon can be divided into n-2 triangles from a vertex, so its interior-angle sum is (n-2)180 degrees.

A regular hexagon has total interior angle 720 degrees and each angle is 720/6 = 120. Its exterior angles each equal 60 and total 360.

A radius is perpendicular to a tangent at the point of contact. An angle at the centre is twice an angle on the same arc at the circumference.

Two tangents from one external point have equal lengths. If one is 8 cm, the other is also 8 cm; equality follows from the tangent theorem, not the drawing.

Mark equal angles and seek similar triangles before introducing variables. Ratios of corresponding sides square when converted to area ratios.

In triangle ABC, D lies on AB and E on AC, with DE parallel to BC and AD:DB=2:3. AD/AB=2/5, so triangle ADE is similar to ABC with length scale 2/5. Its area is 4/25 of the whole. If ABC has area 100, ADE has area 16 and trapezium DBCE has area 84. DE/BC remains 2/5, not 4/25.

A ratio of part to remaining part is not a ratio of part to whole. Draw the full segment before using similarity.

Angles of a triangle sum to 180°. An exterior angle equals the sum of the two opposite interior angles. Any two side lengths must add to more than the third for a nondegenerate triangle. Parallel-line angle equalities require parallel lines.

Similar triangles have proportional corresponding sides and squared area ratios. A line parallel to one side divides the other two sides proportionally. For a triangle with base b and perpendicular height h, area Δ=bh/2. With sides a,b,c and semiperimeter s=(a+b+c)/2, Heron's formula is Δ=√[s(s−a)(s−b)(s−c)]. Also Δ=ab sin(C)/2=rs=abc/(4R), where r and R are the inradius and circumradius and C is the angle between a and b.

A 3-4-5 right triangle has area 3×4/2=6 and s=6. Thus inradius r=Δ/s=1 and circumradius R=abc/(4Δ)=2.5. Its altitude from the right angle to the hypotenuse is 2Δ/5=2.4.

The centroid is the intersection of medians and divides each median 2:1 from the vertex. The incenter meets angle bisectors, the circumcenter meets perpendicular bisectors, and the orthocenter meets altitudes. In a right triangle with right angle A and altitude AD to BC: AD²=BD×DC, AB²=BD×BC and AC²=DC×BC. The cosine rule c²=a²+b²−2ab cos(C) applies to acute, right and obtuse triangles.

Height must be perpendicular to the chosen base. A drawing alone does not establish equal lengths or a right angle.

A simple quadrilateral has four sides and interior-angle sum 360°. Diagonals join opposite vertices. Apply a special shape's properties only when its defining conditions are established.

A parallelogram has both pairs of opposite sides parallel, equal opposite sides and angles, and diagonals that bisect each other. A rectangle additionally has four right angles and equal diagonals. A rhombus has four equal sides and perpendicular diagonals; its diagonals need not be equal. A square has both rectangle and rhombus properties. A kite has two pairs of equal adjacent sides; the diagonal joining the equal-pair vertices is its symmetry diagonal in the usual convex case. A trapezium has a pair of parallel opposite sides; some conventions require exactly one pair, so follow the question's definition. An isosceles trapezium has equal nonparallel sides, equal base angles and equal diagonals under the usual exclusive convention.

A trapezium has parallel sides 8 cm and 14 cm and perpendicular height 5 cm. Area = (8+14)×5/2 = 55 cm². A rhombus with diagonals 6 and 8 cm has area 6×8/2=24 cm²; each side is √(3²+4²)=5 cm.

Rectangle diagonals are equal but usually not perpendicular. Rhombus diagonals are perpendicular but usually not equal.

A simple polygon with n sides has interior-angle sum (n−2)×180°. A regular polygon has equal sides and equal angles. Equal sides alone do not establish regularity.

For a regular convex polygon, each interior angle is (n−2)×180°/n and each exterior turning angle is 360°/n. The exterior turning angles of any convex polygon sum to 360° when taken consistently. The number of diagonals is n(n−3)/2: each vertex connects to n−3 non-neighbouring vertices, and each diagonal is counted twice.

A regular octagon has interior-angle sum 6×180°=1080°, each interior angle 135° and each exterior angle 45°. Its diagonals number 8×5/2=20. A regular hexagon of side a consists of six equilateral triangles, so area is 3√3 a²/2.

The formula 180°−360°/n gives each angle only for a regular or equiangular polygon, not an arbitrary convex polygon.

A circle is the set of points at a fixed distance r from its centre. Its boundary has length 2πr; the enclosed disk has area πr². A chord joins two points on the circle, and a diameter is a chord through the centre.

Equal chords in the same circle subtend equal central angles and are equally distant from the centre. A perpendicular from the centre to a chord bisects the chord. An inscribed angle is half the central angle on the same arc; an angle in a semicircle is 90°. Opposite angles of a cyclic quadrilateral sum to 180°. A tangent is perpendicular to the radius at contact. Tangents from the same external point have equal lengths. Intersecting chords satisfy PA×PB=PC×PD; for an external point with tangent PT and secant PAB, PT²=PA×PB, with PA the external part and PB the whole secant.

A circle of radius 5 cm has a chord 3 cm from its centre. Half the chord is √(25−9)=4 cm, so its length is 8 cm. A 72° sector of radius 10 cm has arc length (72/360)×20π=4π cm and area (72/360)×100π=20π cm².

A chord is not an arc. An arc subtending a central angle θ in degrees has length θ×2πr/360; using radians instead requires rθ. A sector includes the centre, while a segment is bounded by a chord and an arc.