By John Burnett & Peter Orlowski. All the Geometry you need to know for the SAT/PSAT comes down to...

SAT/PSATGEOMETRY

PREPARATIONBy John Burnett & Peter Orlowski

SAT/PSAT GEOMETRY 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.

ITEM #1: ANGLE TERMINOLOGY

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.

Acute Right

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).

f. Angle B = 20° d. Angle E = 90° g. Angle H = 140° k. Angle L

= 91°

g. 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 F p. Angle M – Angle H q. Angle J – Angle E

ITEM #2: ADJACENT ANGLE SUMS

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

ITEM #3: VERTICAL ANGLES

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 = ____________

ITEM #4: PARALLEL LINES

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.

ITEM #5: KINDS OF TRIANGLES BY SIDES

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..

ITEM #6: TRIANGLE SIDE LENGTHS

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


ITEM #7: TRIANGLE INEQUALITY

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


ITEM #8: ANGLE SUMS IN TRIANGLES

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


ITEM #9: KINDS OF TRIANGLES BY ANGLES

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)


ITEM #10: RIGHT TRIANGLES

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 a2 + b2 = c2

c. These two facts are true of only a right triangle.

PRACTICE RIGHT TRIANGLES


ITEM #11: PYTHAGOREAN TRIPLETS

a. Pythagorean Triplets are a triad of three numbers, a, b, and c, where a2 + b2 = c2

b. Example: 32 + 42 = 52

c. E.g.: 62 + 82 = 102

d. E.g.: 92 + 122 = 152

e. E.g.: 302 + 402 = 502

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


ITEM #12: SPECIAL RIGHT TRANGLES

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


ITEM #13: TRIANGLE AREA

a. Area= ½ b∙hb. 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


ITEM #14: SIMILAR POLYGONS

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 k2.

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


ITEM #15: INTERIOR ANGLE SUM OF A CONVEX POLYGON

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


ITEM #16: EXTERIOR ANGLE SUM OF A CONVEX POLYGON

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


ITEM #17: PROPERTIES OF PARALLELOGRAMS

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


ITEM #18: AREA FORMULAE FOR SOME QUADRILATERALSClick icon to add picture

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


ITEM #18: PERIMETER FORMULAE FOR QUADRILATERALS

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


ITEM #19: CIRCLE AND SECTOR FORMULAE

a. The perimeter of a circle of radius r = 2π∙r.

b. The area of a circle of radius r = π∙ r2.

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∙ r2 ∙ A/360º.

PRACTICE FORMULAE FOR CIRCLES AND SECTORS


By John Burnett & Peter Orlowski. All the Geometry you need to know for the SAT/PSAT comes down to...

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