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Geometry, grade 8

Eighth-grade geometry.Subjects

The topics, in order:

  • angles between parallel lines, and the exterior angle of a triangle;
  • Pythagoras’ theorem;
  • transformations and congruent shapes;
  • the congruence theorems for triangles, and proofs that use them;
  • the triangle inequality, the median, the altitude and the angle bisector;
  • the isosceles triangle and its converse theorems;
  • similar triangles.

There are two kinds of lessons:

  • Learning lessons – an explanation, solved examples and a few questions to try;
  • Practice lessons – questions only.

How the page works: Using a page.

  1. Angles between parallel linesLearningAdjacent and vertical angles, and a line that cuts two parallel lines: corresponding, alternate and co-interior angles, and angle calculations.
  2. The exterior angle of a triangleLearningWhy the angles of a triangle add up to 180°, the exterior angle equals the sum of the two interior angles not adjacent to it, and angle calculations.
  3. Practice: anglesPracticeAngles between parallel lines, the sum of the angles of a triangle and the exterior angle: calculations, equations and explanations.
  4. Pythagoras' theoremLearningA right triangle: its legs and hypotenuse, the squares on its sides, Pythagoras' theorem with a proof by areas, and finding the hypotenuse.
  5. Using Pythagoras' theoremLearningFinding a leg, problems from life – a ladder, a screen, the diagonal of a rectangle – the converse: is the triangle right-angled, and Pythagorean triples.
  6. Practice: Pythagoras' theoremPracticeFinding the hypotenuse and a leg, rectangles, trapezoids and coordinates, problems from life and the converse.
  7. Transformations: reflection, translation and rotationLearningThree ways to move a shape – a reflection in a line, a translation and a rotation about a point – and what they keep: the lengths of the sides and the sizes of the angles.
  8. Congruent shapesLearningTwo shapes are congruent when one can be moved exactly onto the other by reflections, translations and rotations: corresponding vertices, sides and angles, and writing the congruence in the right order.
  9. The congruence theorems SAS and ASALearningThree pairs of equal parts are enough for two triangles to be congruent: two sides and the angle between them, or a side and the two angles beside it.
  10. The congruence theorems SSS and SSALearningThree equal sides are enough for congruence; two sides and an angle are enough only when the angle is opposite the longer of the two sides – and what happens when it is opposite the shorter one.
  11. Proving with congruent trianglesLearningA table of claims and reasons: proving that two triangles are congruent, and from the congruence, that sides or angles are equal.
  12. Practice: transformations and congruent trianglesPracticeReflections, translations and rotations on coordinate axes, corresponding parts, the four congruence theorems, and proofs with congruent triangles.
  13. The triangle inequalityLearningTwo sides of a triangle add up to more than the third: when three lengths make a triangle, and between which numbers the third side's length lies.
  14. The median, the altitude and the angle bisectorLearningThree segments from a vertex of a triangle to the side opposite it: the median, the altitude and the angle bisector; where the medians meet, and the median halving the triangle's area.
  15. The isosceles triangleLearningThe legs, the base and the apex angle; the base angles are equal – the proof by congruence; the bisector of the apex angle is also a median and an altitude; the equilateral triangle.
  16. The isosceles triangle: the converse theoremsLearningA triangle with two equal angles is isosceles, and so is a triangle where two of the angle bisector, the median and the altitude coincide: proofs by congruence, calculations and proofs to try.
  17. Practice: the sides of a triangle, the median and the isosceles trianglePracticeThe triangle inequality, the median and area, angles in an isosceles triangle, the converse theorems, and proofs.
  18. Similar trianglesLearningSimilar triangles: the same shape in another size – corresponding angles equal, corresponding sides in the same ratio. The ratio of similarity, enlarging and reducing, and finding a missing side with a proportion.
  19. The similarity theoremsLearningThree theorems that show two triangles are similar without checking every angle and side: AA, SAS and SSS. Uses: a segment parallel to a side of a triangle, and the height of a tree from its shadow.
  20. Practice: similar trianglesPracticeThe ratio of similarity and missing sides, proofs by the similarity theorems, a segment parallel to a side, the shadow of a street lamp, and which triangles are always similar.
  21. Review: geometry, grade 8PracticeQuestions from the whole course: angles between parallel lines, the exterior angle, Pythagoras' theorem, reflection and congruence, the triangle inequality, the median, the isosceles triangle and similar triangles.
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