Russia 7 класс Geometry (advanced)
Chapters: 7
1. Beginnings of geometry
Basic concepts and axioms
- Euclid's Geometry: Definitions, Axioms and the Five Postulates – Geometry began as practical measuring of land and altars in Egypt, India and Mesopotamia. Indian Sulbasutras (like Baudhayana's) gave rope rules for making squares, doubling a square and the diagonal rule. Around 300 BCE, Euclid of Alexandria organised geometry as a chain of reasoning: start from a few definitions, common-sense axioms and five geometry postulates, and prove everything else. The fifth postulate is about when two lines meet, and it leads to the idea of parallel lines.
2. Triangles
Congruence tests
- Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
3. Parallel lines and polygon angles
Parallel lines · Angle sums
- Lines and Angles: Linear Pair, Vertically Opposite and Parallel Lines – An angle is the turn between two rays that start from the same point. Angles on a straight line add up to 180° (linear pair). When two lines cross, the opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to 180°. The reverse is also true, and we can prove such facts by contradiction: assume the opposite and show it leads to something impossible.
4. Right triangles
Right triangles
- Right Triangles – A right triangle has one 90° angle. The side across it is the hypotenuse; the other two sides are legs. Its two acute angles add up to 90°. The median to the hypotenuse is half the hypotenuse. If an angle is 30°, the leg across from it is half the hypotenuse. Two right triangles are congruent by HL, LL, LA or HA.
5. Circle
Circle
- The Circle: Chords, Tangents and Triangle Circles – A circle is all points at one fixed distance (the radius) from a centre. A chord joins two points on it; the longest chord is the diameter. A line can cut a circle at two points (secant), touch it at one (tangent) or miss it. A tangent is perpendicular to the radius. Every triangle has an incircle inside and a circumcircle around it.
6. Locus
Locus method
- Locus: The Path of a Point That Follows a Rule – A locus is the set of ALL points that obey one rule, and no other points. Fixed distance from one point gives a circle. Equal distance from two points gives the perpendicular bisector. Equal distance from two crossing lines gives the angle bisector. Where two loci meet, you find points that obey both rules, like the centre of the circumcircle. Points that see a segment at a fixed angle lie on an arc, called the capable arc.
7. Constructions
Construction problems
- Geometric Constructions: Drawing Exactly with Compass and Straightedge – A geometric construction draws a figure exactly using only a compass and a straightedge (a ruler used for straight lines). Key constructions: the perpendicular bisector of a segment, the bisector of an angle, a perpendicular from a point to a line, angles of 60°, 30°, 90° and 45°, a triangle from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and regular polygons such as the hexagon. Each works because equal compass arcs make equal lengths, which give congruent triangles.