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Welcome to GCSE Edexcel Maths revision.

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Topic M 27: Angles, polygons and shape properties.

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This video covers Foundation and Higher tiers.

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Parallel lines never meet in a plane; perpendicular lines meet at 90 degrees.

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A vertex is a corner, an edge joins vertices, and a plane is a flat surface.

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Use conventional labels, for example angle A B C has its vertex at B.

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Angles on a straight line sum to 180 degrees, angles around a point sum to 360 degrees, and vertically opposite angles are equal.

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A right angle is 90 degrees.

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Parallel lines and a transversal

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With parallel lines crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180 degrees.

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State the parallel-line reason when finding an unknown angle.

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A triangle's interior angles sum to 180 degrees.

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An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three equal sides and three 60 degrees angles.

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A triangle's exterior angle equals the sum of its two opposite interior angles.

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If those are 43 degrees and 61 degrees, the exterior angle is 104 degrees.

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An n-sided polygon has interior-angle sum open bracket n minus 2 close bracket multiplied by 180 degrees.

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Divide a polygon into triangles from one vertex to see why this works.

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The exterior angles of a convex polygon, one at each vertex in the same turning direction, sum to 360 degrees.

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For a regular n-gon,

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each exterior angle is the fraction with numerator open bracket 360 degrees close bracket and denominator open bracket n close bracket ,

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end fraction and each interior angle is 180 degrees minus that value.

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Worked example: A regular pentagon has interior-angle sum 3 multiplied by 180 degrees equals 540 degrees.

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Each interior angle is 108 degrees and each exterior angle is 72 degrees.

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Any quadrilateral has interior-angle sum 360 degrees.

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A parallelogram has parallel opposite sides, equal opposite sides and equal opposite angles; adjacent angles sum to 180 degrees.

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A rectangle is a parallelogram with four right angles and equal diagonals.

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A rhombus is a parallelogram with all four sides equal; its diagonals meet at right angles and bisect each other.

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The diagonals of every parallelogram bisect each other.

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A square's diagonals are equal, perpendicular and angle-bisecting; a rectangle's diagonals are not generally perpendicular.

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State properties that actually follow from the named shape.

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A square is both a rectangle and a rhombus.

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Its four sides are equal and its four angles are right angles.

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A kite has two pairs of adjacent equal sides and perpendicular diagonals, with one diagonal bisecting the other.

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A trapezium has a pair of parallel sides.

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An isosceles trapezium has equal non-parallel sides and equal base angles.

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Use the definition stated in a question if conventions differ.

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A line of symmetry reflects a shape onto itself.

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Rotational symmetry order counts how many times it fits itself during a full turn; a non-square rectangle has order 2.

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Worked example: In an isosceles triangle with vertex angle 40 degrees, the base angles are equal and total 140 degrees.

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Each is 70 degrees.

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Name the equal sides or angle property that justifies equality.

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When proving a geometric statement, link each equality to a reason: parallel lines maps to alternate angles equal maps to triangle angle sum maps to required angle.

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A diagram is not assumed to scale unless the question says so.

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That completes Angles, polygons and shape properties.

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Revisit the notes and test yourself on the revision website.