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Geometry, grade 9

Ninth-grade geometry.Subjects

The topics, in order:

  • the isosceles triangle: the converse theorems;
  • how a triangle’s sides and angles are related;
  • the right triangle with a 30∘\displaystyle 30^{\circ} angle;
  • parallel lines: the converse theorems;
  • the kite and the trapezoid;
  • the parallelogram, the rectangle, the rhombus and the square;
  • the median to the hypotenuse of a right triangle;
  • the ratio of the areas of similar triangles.

Two kinds of lessons:

  • learning lessons – an explanation, solved proofs and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The isosceles triangle: the converse theoremLearningA triangle with two equal angles is isosceles: the proof with an angle bisector and congruence, and its uses – finding sides and proving.
  2. The isosceles triangle: bisector, median and altitudeLearningIf two of the angle bisector, the median and the altitude from one vertex coincide, the triangle is isosceles: two proofs, calculations and proofs to try.
  3. Practice: the isosceles trianglePracticeAngles and sides in an isosceles triangle, an isosceles triangle on a coordinate system, and proofs with the converse theorems.
  4. Sides and angles of a triangleLearningIn a triangle the larger side faces the larger angle, and the larger angle faces the larger side: the proof, ordering sides and angles, and proofs with inequalities.
  5. A right triangle with a 30° angleLearningIn a right triangle with a 30° angle, the leg facing it is half the hypotenuse: the proof inside an equilateral triangle, the converse, calculations and proofs.
  6. Parallel lines: the converse theoremsLearningEqual corresponding angles, equal alternate angles or co-interior angles adding up to 180° – and the lines are parallel: why it holds, calculations and proofs.
  7. Practice: triangles and parallel linesPracticeSides and angles in a triangle, the isosceles triangle, the triangle with a 30° angle and parallel lines: calculations and proofs.
  8. The kiteLearningA kite – a quadrilateral with two pairs of equal adjacent sides: its angles, the main diagonal and its proof by congruence, and a kite in a coordinate system.
  9. The trapezoidLearningThe trapezoid: its bases and legs, the angles beside a leg, its height, area and perimeter, and a trapezoid in a coordinate system.
  10. The isosceles trapezoidLearningThe isosceles trapezoid: equal base angles and equal diagonals, the converse theorems, and proving that a quadrilateral is a trapezoid.
  11. Practice: the kite and the trapezoidPracticeQuestions on the kite, the trapezoid and the isosceles trapezoid: angles, lengths, area and perimeter, the coordinate system, and proofs.
  12. The parallelogram: definition and propertiesLearningA parallelogram is a quadrilateral whose opposite sides are parallel: its opposite sides and angles are equal, adjacent angles add up to 180°, and the diagonals bisect each other. Proofs, calculations and a parallelogram in a coordinate system.
  13. Proving that a quadrilateral is a parallelogramLearningThe converse theorems: a quadrilateral with equal opposite sides, or a pair of sides parallel and equal, or diagonals that bisect each other, or equal opposite angles, is a parallelogram. Proofs and uses.
  14. Practice: the parallelogramPracticeAngles, sides and diagonals of a parallelogram, a parallelogram in a coordinate system, and proofs: the parallelogram's properties and the converse theorems.
  15. The rectangleLearningA rectangle is a parallelogram with a right angle: all its angles are right and its diagonals are equal. The converse theorems – how to prove that a quadrilateral is a rectangle.
  16. The rhombusLearningA rhombus is a parallelogram with two equal adjacent sides: all its sides are equal, and its diagonals are perpendicular and bisect its angles. The converse theorems – how to prove that a quadrilateral is a rhombus.
  17. The squareLearningA square is both a rectangle and a rhombus: it has all the properties of both. The converse theorems – how to prove that a quadrilateral is a square – and the family of quadrilaterals.
  18. Practice: the rectangle, the rhombus and the squarePracticeThe family of quadrilaterals, calculations in a rectangle and a rhombus, a rhombus in a coordinate system, and proofs: the properties of the rectangle, the rhombus and the square, and the converse theorems.
  19. The median to the hypotenuse of a right triangleLearningIn a right triangle the median to the hypotenuse is half the hypotenuse: the proof with a rectangle, the converse theorem, and its uses – calculations and proofs in quadrilaterals.
  20. The ratio of the areas of similar trianglesLearningThe ratio of the areas of similar triangles is the square of the ratio of similarity: why, and calculations both ways – from the ratio of similarity to the areas, and from the areas to the ratio and a side.
  21. Review: the whole coursePracticeEverything together: the isosceles triangle, sides and angles, the triangle with a 30° angle, parallel lines, the kite, the trapezoid, the parallelogram, the rhombus, the median to the hypotenuse and the areas of similar triangles – calculations and proofs.
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