By John Burnett & Peter Orlowski

 All the Geometry you need to know for the SAT/PSAT comes down to 20 key facts, and can be covered in 20 slides.  30% of the SAT questions use geometry. So cramming these 20 facts into your head can make a considerable difference in test scores.  Practicing using these facts will help even more!  These 20 facts will be presented in this slide show without any proof. Take them on faith for now, because all we care about for now is getting a high score on the SAT. We will care for our souls, about really learning Geometry, when we prove these facts in class at another time.

a. An angle is formed by any two rays with a common endpoint. b. An acute angle has between 0º and 90º. c. A right angle has a measure of exactly 90º. d. An obtuse angle has between 90º and 180º. e. A straight “angle” has exactly 180º. f. A reflex “angle” has between 180º and 360º. AcuteRight Obtuse

PRACTICE ANGLE TERMINOLOGY Angle R is a right angle, angle A is an acute angle, angle S is a straight angle, angle O is obtuse, and angle X is a reflex angle. Which statements are possibly true? a.Angle S = 50° f. Angle S = 90° l. Angle S = 120° r. Angle S = 180° b.Angle R = 50°g. Angle R = 90°m. Angle R = 120°s. Angle R = 180° c.Angle A = 50°h. Angle A = 90°n. Angle A = 120° t. Angle A = 180° d.Angle O = 50°j. Angle O = 90° p. Angle O = 120° u. Angle O = 180° e.Angle X = 50° k. Angle X = 90° q. Angle X = 120° v. Angle X = 180° Identify the angle as straight (s), right (R), acute (A), obtuse (O), or reflex (X). a.Angle B = 20°d. Angle E = 90°g. Angle H = 140°k. Angle L = 91° b.Angle C = 190°e. Angle F = 13°h. Angle J = 180°l. Angle M = 200° c.Angle D = 90.1°f. Angle G = 89.9°j. Angle K = 179.9°m. Angle N = 180.1° n. Angle B + Angle Fp. Angle M – Angle Hq. Angle J – Angle E

a. Two angles with a common vertex & side between them are called “adjacent”. b. If together they form a right angle, they add up to 90º. The converse is also true. c. If together they form a straight line, they add up to 180º. The converse is also true. d. The sum of all angles around a point is 360º. Vocabulary: Complementary Angles, Supplementary Angles, Explementary Angles 180

PRACTICE ADJACENT ANGLE SUMS

a. A pair of crossing lines make two sets of vertical angles. b. Vertical angles have equal degree measurements.

PRACTICE VERTICAL ANGLE FACTS 1.What two angles are vertical? __________ 2.What is the sum of angle measures a, b, and c? _______ 3.What is the sum of angle measures d, e, and f? ________ 4.What is the sum of angle measures d, and f? _______ 5.What two angles are complementary? ________ 6.Express angle d in an equation in terms of angles a, and b. ___________ 7.If angle d = x and angle c = 2x, what is the measure of angle f? ____________ 8.If angle f = 6x + 4° and c = 70°, what is x? What is d? x = _____ ; d = ______ 9.Let a = 60°, and b = 50°. Find the remaining angles. c = ____________ d= ______________e = ________________f = ____________

a. When a pair of parallel lines are cut by a transversal, angles facing the same way or exactly opposite ways have equal measures. b. Angles not facing the same or opposite ways are supplementary. c. If the transversal is known to be perpen- dicular to either parallel line, all angles are 90º. Cheaters trick: One can assume that all angles that look acute are equal, and all that look obtuse are equal. The SAT beats this fact to death!

PRACTICE PARALLEL LINE FACTS See handouts distributed in class.

a. All triangles are either equilateral, isosceles or scalene. b. Only equilateral triangles have all angles equal. c. Only scalene triangles have no angles equal. d. Only isosceles triangles have just two angles equal, namely the pair opposite the equal legs.

PRACTICE KINDS OF TRIANGLES (BY SIDES) See handouts distributed in class..

a. The greater side of a triangle is always & only opposite the greater angle. b. The lesser side of a triangle is always & only opposite the lesser angle. c. Equal sides of a triangle are always & only opposite equal angles.

PRACTICE TRIANGLE SIDE LENGTHS See handouts distributed in class..

a. The sum of any two sides of a triangle is greater than the third side. (That’s why we walk straight to a place, instead of walking off to one side and then straight at it!) b. The difference between any two sides of a triangle is less than the third side.

PRACTICE TRIANGLE INEQUALITY See handouts distributed in class..

a. The exterior angle of a triangle exceeds either remote interior angle. b. The exterior angle of a triangle has the same degree measure as the sum of the two remote interior. c. The sum of all interior angles in a triangle equal exactly 180º Vocabulary: Interior angle, Exterior angle, Adjacent Interior angle, Remote Interior angle

PRACTICE ANGLE SUMS IN TRIANGLES See handouts distributed in class..

a. If all the angles in a triangle are acute, the triangle is called acute. b. If just one angle in a triangle is right, the triangle is called right. c. If just one angle in a triangle is obtuse, the triangle is called obtuse.

PRACTICE KINDS OF TRIANGLES (BY ANGLES) See handouts distributed in class..

a. The two smallest angles of a right triangle add up to 90º. b. If the shortest sides are a and b, and the longest side is c, then a 2 + b 2 = c 2 c. These two facts are true of only a right triangle.

PRACTICE RIGHT TRIANGLES See handouts distributed in class..

a. Pythagorean Triplets are a triad of three numbers, a, b, and c, where a 2 + b 2 = c 2 b. Example: = 5 2 c. E.g.: = 10 2 d. E.g.: = 15 2 e. E.g.: = 50 2 f. Or: (3n) 2 + (4n) 2 = (5n) 2 Pythagorean Triplets make right triangles with easy to express sides (but the acute angles cannot be expressed with integers.)

PRACTICE PYTHAGOREAN TRIPLETS See handouts distributed in class..

a. Fold an equilateral triangle in half and you get a scalene right triangle with angles 30º, 60º, 90º. If its hypotenuse is 2x, its legs must be x, and x√3. b. Fold a square along its diagonal and you get an isosceles right triangle w/ angles 45º, 45º, 90º. If its legs are x, its hypotenuse must be x√2. These “special” right triangles have easy to express angles, but their sides are irrational. One can use the Pythagorean Theorem to calculate them.

PRACTICE SPECIAL RIGHT TRIANGLES See handouts distributed in class..

a. Area= ½ b∙h b. If a is the leg of a triangle, and C is the angle at the base of the leg, then the height of the triangle is a∙sin(C) c. So if you don’t know the height, Area = ½ b∙a∙sin(C)

PRACTICE TRIANGLE AREA See handouts distributed in class..

a. Similar polygons have equal matching angles but all matching lengths (e.g. sides, diagonals, heights) are proportional. b. The number that the lines of one polygon need to be multiplied to get the lines of the other is the “scale”. c. If the scale is k, then the ratio of matching areas is k 2. Make sure you match up the parts of one polygon correctly with the parts of the other. Prime notation can help here.

PRACTICE SIMILAR POLYGONS See handouts distributed in class..

An n-sided convex polygon can be divided into n triangles, the sum of whose angles is 180ºn. Subtract 360º from the center and you’re left with only the polygon’s angles. a. A convex polygon is a polygon where a straight line can be drawn from any point on the perimeter to any point in the interior without crossing the perimeter. b. The sum of all the interior angles in a convex polygon of n sides is 180ºn —360º.

PRACTICE CONVEX POLYGON INTERIOR ANGLE SUMS See handouts distributed in class..

You can easily convince yourself that this must be true by drawing all the exterior angles out in a clockwise direction, then sliding a copy of each angle to one point in the middle of the polygon. a. If you add up one exterior angle from each corner of a convex polygon, the sum is always, exactly 360º. It doesn’t matter how many sides it has or how irregular its shape!

PRACTICE CONVEX POLYGON EXTERIOR ANGLE SUMS See handouts distributed in class..

a. By definition, opposite sides are parallel. b. Opposite sides have equal length. c. Opposite angles have equal degree measures. d. Consecutive angles are supplementary. e. Diagonals bisect each other. f. None of these properties are found in any other quadrilateral!

PRACTICE PROPERTIES OF PARALLELOGRAMS See handouts distributed in class..

a. Area of square = side squared. b. Area of parallelogram = base times height. c. Area of trapezoid = half top side times height plus half bottom side times height. d. Area of “diamond”(i.e. quadrilateral with perpendicular diagonals) = half the product of the diagonals The diamond formula works for any square or rhombus, since they are special cases of diamonds.

PRACTICE AREA FORMULAE FOR QUADRILATERALS See handouts distributed in class..

a. The perimeter of any quadrilateral is just the sum of the four sides. This formula simplifies for some quadrilaterals. b. For squares or rhombi, it is four times any side. c. For parallelograms or rectangles, it is twice the sum of two consecutive sides.

PRACTICE PERIMETER FORMULAE FOR QUADRILATERALS See handouts distributed in class..

a. The perimeter of a circle of radius r = 2π∙r. b. The area of a circle of radius r = π∙ r 2. c. The arc of a sector of radius r and angle A = 2π∙r ∙ A/360º. d. The area of a sector of radius r and angle A = π∙ r 2 ∙ A/360º.

PRACTICE FORMULAE FOR CIRCLES AND SECTORS See handouts distributed in class..