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Ishikawa Entrance Mathematics study materials
Ishikawa Entrance Mathematics study resources

Ishikawa Entrance Mathematics: Geometry: Congruence, Similarity and Measurement

Ishikawa Entrance Mathematics · Work through your exam, one topic at a time.

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Geometry: Congruence, Similarity and Measurement

Congruence and Similarity is proof-and-ratio territory. You need to select a valid congruence or similarity condition, match corresponding vertices correctly, and use side, area, and parallel-line ratios without mixing them up.

Core rules

  • Congruence conditions: SSS, SAS, ASA, AAS, and RHS for right triangles. In SAS, the equal angle must lie between the two pairs of equal sides.
  • Similarity conditions: AA, SAS with two sides in proportion and the included angle equal, and SSS with all three side ratios equal.
  • If the similarity ratio is k, corresponding lengths are in ratio k, areas in k², and volumes in k³.
  • A line parallel to one side of a triangle divides the other two sides proportionally. The midpoint theorem gives a segment half the length of the third side.

Worked example

In triangle ABC, D lies on AB, E lies on AC, and DE is parallel to BC. Given AD:DB=2:3 and BC=15, find DE. Because DE is parallel to BC, triangles ADE and ABC are similar by AA: they share angle A, and corresponding angles are equal. The ratio AD:AB=2:(2+3)=2:5. Therefore DE:BC=2:5, so DE=15×2/5=6.

Traps

  • Writing a similarity statement with vertices in the wrong order.
  • Using k for an area ratio instead of k².
  • Treating SSA as a congruence condition.
  • Using AD:DB directly as the similarity ratio instead of AD:AB.
  • Forgetting that a shared angle can supply one of the AA pairs.

Measuring lengths, areas and volumes

For a right triangle, the hypotenuse c satisfies c²=a²+b². Identify the side opposite the right angle before substituting; the longest side is the hypotenuse. A rectangle has area length times width, and a triangle has half the product of its perpendicular base and height. A prism has volume base area times perpendicular height. A cylinder uses the circular base area πr²; a cone with the same base and height has one-third that volume. Surface area counts all exposed faces and uses squared units, while volume uses cubed units. These formulas describe different quantities, so write the requested unit before choosing a calculation.