About this topic
Before you begin
Build geometric arguments from angle, circle and similarity properties, and distinguish length, area and volume scale factors.
Learning goals
- Classify angles by their size
- Apply the sum of angles around a point
- Deduce all unknown angles at two intersecting lines
- Deduce unknown angles from geometric relationships
- Produce complete parallel-line angle relations and values from numerical or affine-expression givens
- Give the number of lines of symmetry and the order of rotational symmetry of a plane figure
- Classify shapes by their properties
- Connect triangle, rectangle and square facts to deduce angles in a drawn composite figure
- Connect parallelogram and connected shape facts to deduce angles using existing lines
- Connect trapezium and connected shape facts to deduce angles using existing lines
- Use interior and exterior angle sums of convex and regular polygons to find angles or the number of sides
- Name the most specific special quadrilateral from its stated side, angle and diagonal properties
- Construct geometric figures from stated conditions
- Use the equal-distance properties of perpendicular bisectors and angle bisectors
- Reason about congruence, similarity and scale factors
- Find a scale factor and unknown lengths from matched sides of similar triangles
- Recover a similarity length scale and apply its square or cube to lengths, areas, volumes and masses
- Find an angle in a stated circle configuration and name the circle property used
- Find a length in a stated circle configuration and name the circle symmetry property used
Syllabus statements covered
- Right, acute, obtuse and reflex angles
- Vertically opposite angles, angles on a straight line and angles at a point
- Angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles
- Properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon), including symmetry properties
- Classifying special quadrilaterals on the basis of their properties
- Angle sum of interior and exterior angles of any convex polygon
- Construction of simple geometrical figures from given data using compasses, ruler, set squares and protractors, where appropriate
- Congruent figures and similar figures
- Properties of similar triangles and polygons: - corresponding angles are equal - corresponding sides are proportional
- Enlargement and reduction of a plane figure
- Scale drawings
- Properties and construction of perpendicular bisectors of line segments and angle bisectors
- Determining whether two triangles are - congruent - similar
- Ratio of areas of similar plane figures
- Ratio of volumes of similar solids
- Solving simple problems involving similarity and congruence
- Symmetry properties of circles: - equal chords are equidistant from the centre - the perpendicular bisector of a chord passes through the centre - tangents from an external point are equal in length - the line joining an external point to the centre of the circle bisects the angle between the tangents
- Angle properties of circles: - angle in a semicircle is a right angle - angle between tangent and radius of a circle is a right angle - angle at the centre is twice the angle at the circumference - angles in the same segment are equal - angles in opposite segments are supplementary
- Right, acute, obtuse and reflex angles
- Vertically opposite angles, angles on a straight line, angles at a point
- Angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles
- Properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon), including symmetry properties
- Classifying special quadrilaterals on the basis of their properties
- Angle sum of interior and exterior angles of any convex polygon
- Construction of simple geometrical figures from given data using compasses, ruler, set squares and protractors, where appropriate
- Congruent figures
- Similar figures
- Properties of similar triangles and polygons:
- Corresponding angles are equal
- Corresponding sides are proportional
- Enlargement and reduction of a plane figure
- Solving simple problems involving congruence and similarity
- Scale drawings
- Properties and construction of perpendicular bisectors of line segments and angle bisectors
- Determining whether two triangles are:
- Congruent
- Similar
- Ratio of areas of similar plane figures
- Ratio of volumes of similar solids
- Symmetry properties of circles:
- Equal chords are equidistant from the centre
- The perpendicular bisector of a chord passes through the centre
- Tangents from an external point are equal in length
- The line joining an external point to the centre of the circle bisects the angle between the tangents
- Angle properties of circles:
- Angle in a semicircle is a right angle
- Angle between tangent and radius of a circle is a right angle
- Angle at the centre is twice the angle at the circumference
- Angles in the same segment are equal
- Angles in opposite segments are supplementary
Choose a lesson
Work through them in order.
- Not startedParallel lines and angle reasonsNextName the geometric condition before using an angle equality or sum.
- Not startedAngle relations and equationsNextTrace every angle pair, choose equality or a sum, and check algebraic results against the geometry.
- Not startedLine and rotational symmetryNextTest the whole figure under a reflection or rotation and count matching positions in a full turn.
- Not startedTriangle angle propertiesNextUse the angle sum and equal-side markings to justify unknown triangle angles.
- Not startedParallel quadrilaterals and their anglesNextUse marked parallel sides and named quadrilateral properties to justify unknown angles.
- Not startedPolygon angle sumsNextDistinguish totals for a convex polygon from equal individual angles in a regular polygon.
- Not startedTriangle construction from dataNextConstruct a triangle from stated lengths or angles and check that the data determine a possible figure.
- Not startedBisectors and geometric constructionNextUse equal distances to justify compass constructions and check the given dimensions.
- Not startedCongruence and similarityNextMatch corresponding parts before comparing lengths, angles and scale.
- Not startedSimilar triangles and unknown lengthsNextMatch corresponding vertices, find the scale factor and calculate unknown sides in separate, nested and crossed triangles.
- Not startedEnlargement, reduction and scale drawingsNextUse one scale factor for every length, keep angles unchanged, and move between real lengths and a scale drawing.
- Not startedTriangle tests and similar-solid scalesNextUse sufficient corresponding data and distinguish length, area and volume scale factors.
- Not startedCircle angle and tangent propertiesNextIdentify the chord, arc, radius and tangent that justify each circle relationship.
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