1 · Angle pairs
Complementary angles add to 90°; supplementary angles add to 180°. Angles that sit side by side on a straight line are supplementary.
When two lines cross, the angles across from each other are vertical angles, and they are equal. If one is 65°, the one across is 65° and the two beside it are 115°.
2 · Parallel lines and a transversal
A line crossing two parallel lines makes only two sizes of angle: a small one and a big one, and they add to 180°.
Corresponding angles (same position at each crossing) are equal. Alternate interior angles (between the parallel lines, on opposite sides of the transversal) are equal. Same-side interior angles add to 180°.
3 · Angles in triangles and polygons
The angles of a triangle add to 180°. An outside (exterior) angle equals the two far inside angles added together.
A polygon with n sides splits into n − 2 triangles, so its angles add to (n − 2) × 180°. A hexagon: 4 × 180 = 720°, and each angle of a regular hexagon is 720 ÷ 6 = 120°.
4 · The Pythagorean theorem
In a right triangle, the two legs a and b and the longest side c (across from the right angle) follow a² + b² = c².
A 10-foot ladder with its foot 6 feet from a wall: 6² + h² = 10², so h² = 64 and the ladder reaches 8 feet up. Common whole-number sets: 3-4-5, 5-12-13, 8-15-17, and their multiples like 6-8-10.
5 · Circles
Circumference = π × diameter = 2πr. Area = πr². A circle with diameter 10 has radius 5, circumference 10π, and area 25π.
Many ISEE answers are left "in terms of π". Using the diameter where the radius belongs is the most common mistake: it makes the area four times too big.
6 · Parts of circles and shaded regions
A half circle has half the area and half the curved edge. A quarter circle has a quarter.
For a shaded region, subtract: a circle of radius 3 inside a 6 by 6 square leaves 36 − 9π shaded.
7 · Similar figures
Similar figures have the same shape; matching sides are in the same ratio. If a 6-foot person casts a 4-foot shadow when a tree casts a 30-foot shadow, then 6/4 = h/30 and the tree is 45 feet tall.
If every length is multiplied by k, the area is multiplied by k² and the volume by k³. Doubling the sides of a square makes its area 4 times as big, not 2.
8 · Distance and midpoint
The distance between (1, 2) and (7, 10): the run is 6 and the rise is 8, so the distance is √(36 + 64) = 10. It is the Pythagorean theorem on a grid.
The midpoint averages the coordinates: between (−4, 3) and (6, 9) it is ((−4 + 6) ÷ 2, (3 + 9) ÷ 2) = (1, 6).
9 · Reflections and translations
Reflecting across the x-axis changes the sign of y: (3, −5) becomes (3, 5). Reflecting across the y-axis changes the sign of x: (3, −5) becomes (−3, −5).
A translation slides a point: 4 left and 2 up turns (1, 1) into (−3, 3). Left and right change x; up and down change y.
10 · On the ISEE
Figures are not always drawn to scale, so trust the numbers and the marks (right-angle boxes, tick marks, arrows for parallel lines) more than how the picture looks.
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