1. x˚ 40˚ y˚ The figure above shows three intersecting lines. What is the value of x 1 y ? (A) 50 (B) 80 (C) 130 (D) 140 (E) 150 The correct answer is (D). The angle vertical to the one indicated as being 40° must also measure 40°. That 40° angle, together with the angles whose measures are x° and y°, combine to form a straight (180°) line. In other words, 40 1 x 1 y 5 180. Thus, x 1 y 5 140. A slightly tougher “wheel-spoke” question might focus on overlapping angles and require you to apply rule 1 (about vertical angles) to determine the amount of the overlap. Look at this “wheel-spoke” figure: A GRE question about this figure might test your ability to recognize one of the following relationships: Chapter 11: Math Review: Geometry 263 NOTE Degree measures of the individual central angles of a wheel-spoke figure total 360°, which is the number of degrees in a circle. You’ll explore this concept again later in the chapter. www.petersons.com x° 1 y° 2 z° 5 180° x° 1 y° exceeds 180° by the amount of the overlap, which equals z°, the angle vertical to the overlapping angle. x° 1 y° 1 v° 1 w° 5 360° The sum of the measures of all angles, excluding z°, is 360°; z is excluded because it is already accounted for by the overlap of x and y. y° 2 w° 5 z° w° equals its vertical angle, so y 2 w equals the portion of y vertical to angle z. Parallel Lines and Transversals GRE problems involving parallel lines also involve at least one transversal, which is a line that intersects each of two (or more) lines. Look at this next figure, in which l 1 i l 2 and l 3 i l 4 : The upper-left “cluster” of angles (numbered 1, 2, 3, and 4) matches each of the three other clusters. In other words: • All the odd-numbered angles are congruent (equal in size) to one another. • All the even-numbered angles are congruent (equal in size) to one another. If you know the size of just one angle, you can determine the size of all 16 angles. PART IV: Quantitative Reasoning264 www.petersons.com 2. In the figure above, lines P and Q are parallel to each other. If m∠x 5 75°, what is the measure of ∠y? (A) 75° (B) 85° (C) 95° (D) 105° (E) 115° The correct answer is (D). The angle “cluster” where lines P and R intersect corresponds to the cluster where lines Q and R intersect. Thus, ∠x and ∠y are supplementary (their measures add up to 180°). Given that ∠x measures 75°, ∠y must measure 105°. TRIANGLES The triangle (a three-sided polygon) is the test makers’ favorite geometric figure. You’ll need to understand triangles not only to solve “pure” triangle problems but also to solve certain problems involving four-sided figures, three-dimensional figures, and even circles. After a brief review of the properties of any triangle, we’ll focus on right triangles (which include one right, or 90°, angle), isosceles triangles (in which two sides are congruent), and equilateral triangles (in which all sides and angles are congruent). Properties of All Triangles Here are four properties that all triangles share: Length of the sides: Each side is shorter than the sum of the lengths of the other two sides. (Otherwise, the triangle would collapse into a line.) Angle measures: The measures of the three angles total 180°. Angles and opposite sides: Comparative angle sizes correspond to the com- parative lengths of the sides opposite those angles. For example, a triangle’s largest angle is opposite its longest side. (The sides opposite two congruent angles are also congruent.) Chapter 11: Math Review: Geometry 265 ALERT! The ratio of angle sizes need not be identical to the ratio of lengths of sides. For example, if a certain triangle has angle measures of 30°, 60°, and 90°, the ratio of the angles is 1:2:3. But this doesn’t mean that the ratio of the opposite sides is also 1:2:3. It’s not! www.petersons.com Area: The area of any triangle is equal to one-half the product of its base and its height (or altitude): Area 5 1 2 3 base 3 height. You can use any side as the base to calculate area. Right Triangles and the Pythagorean Theorem In a right triangle, one angle measures 90° (and, of course, each of the other two angles measures less than 90°). The Pythagorean theorem expresses the relationship among the sides of any right triangle. This theorem is usually expressed as a 2 + b 2 = c 2 , where a and b are the two legs (the two shortest sides) that form the right angle, and c is the hypotenuse—the longest side, opposite the right angle: For any right triangle, if you know the length of two sides, you can determine the length of the third side by applying the theorem. If the two shortest sides (the legs) of a right triangle are 2 and 3 units long, then the length of the triangle’s third side (the hypotenuse) is = 13 units: 2 2 1 3 2 5 13 5 c 2 ; c 5 = 13 If a right triangle’s longest side (hypotenuse) is 10 units long and another side (one of the legs) is 5 units long, then the third side is 5 = 3 units long: a 2 1 5 2 5 10 2 ; a 2 5 75; a 5 = 75 5 = ~25!~3!55 = 3 PYTHAGOREAN SIDE TRIPLETS A Pythagorean side triplet is a specific ratio among the sides of a triangle that satisfies the Pythagorean theorem. In each of the following triplets, the first two numbers represent the comparative lengths of the two legs, whereas the third—and greatest—number represents the comparative length of the hypotenuse (on the GRE, the first four triplets appear far more frequently than the last two): 1:1: = 2 ~1 2 1 1 2 5~ = 2! 2 ! 1: = 3:2 ~1 2 1~ = 3! 2 5 2 2 ! 3:4:5 (3 2 1 4 2 5 5 2 ) 5:12:13 (5 2 1 12 2 5 13 2 ) 8:15:17 (8 2 1 15 2 5 17 2 ) 7:24:25 (7 2 1 24 2 5 25 2 ) Each triplet above is expressed as a ratio because it represents a proportion among the triangle’s sides. All right triangles with sides having the same proportion, or ratio, have the same shape. For example, a right triangle with sides of 5, 12, and 13 is PART IV: Quantitative Reasoning266 www.petersons.com smaller but exactly the same shape (proportion) as a triangle with sides of 15, 36, and 39. 3. Two boats leave the same dock at the same time, one traveling due east at 10 miles per hour and the other due north at 24 miles per hour. How many miles apart are the boats after 3 hours? (A) 68 (B) 72 (C) 78 (D) 98 (E) 110 The correct answer is (C). The distance between the two boats after 3 hours forms the hypotenuse of a triangle in which the legs are the two boats’ respective paths. The ratio of one leg to the other is 10:24, or 5:12, so you know you’re dealing with a 5:12:13 triangle. The slower boat traveled 30 miles (10 mph 3 3 hours). Thirty corresponds to the number 5 in the 5:12:13 ratio, so the multiple is 6(53 6 5 30). 5:12:13 5 30:72:78. PYTHAGOREAN ANGLE TRIPLETS In two (and only two) of the unique triangles identified in the preceding section as Pythagorean side triplets, all degree measures are integers: The corresponding angles opposite the sides of a 1:1: = 2 triangle are 45°, 45°, and 90°. The corresponding angles opposite the sides of a 1: = 3:2 triangle are 30°, 60°, and 90°. If you know that the triangle is a right triangle (one angle measures 90°) and that one of the other angles is 45°, then given the length of any side, you can determine the unknown lengths. For example: • If one leg is 5 units long, then the other leg must also be 5 units long, while the hypotenuse must be 5 = 2 units long. • If the hypotenuse (the longest side) is 10 units long, then each leg must be 5 = 2 units long. Divide the hypotenuse by = 2: 10 = 2 5 10 = 2 2 5 5 = 2 Chapter 11: Math Review: Geometry 267 TIP To save valuable time on GRE right-triangle problems, learn to recognize given numbers (lengths of triangle sides) as multiples of Pythagorean triplets. www.petersons.com Similarly, if you know that the triangle is a right triangle (one angle measures 90°) and that one of the other angles is either 30° or 60°, then given the length of any side you can determine the unknown lengths. For example: • If the shortest leg (opposite the 30° angle) is 3 units long, then the other leg (opposite the 60° angle) must be 3 = 3 units long, and the hypotenuse must be 6 units long (3 3 2). • If the longer leg (opposite the 60° angle) is 4 units long, then the shorter leg (opposite the 30° angle) must be 4 = 3 3 units long (divide by = 3: 4 = 3 5 4 = 3 3 ), while the hypotenuse must be 8 = 3 3 (twice as long as the shorter leg). • If the hypotenuse is 10 units long, then the shorter leg (opposite the 30° angle) must be 5 units long, while the longer leg (opposite the 60° angle) must be 5 = 3 units long (the length of the shorter leg multiplied by = 3). 4. In the figure below, AC is 5 units long, m∠ABD 5 45°, and m∠DAC 5 60°. How many units long is BD ? 60˚ 5 45 ˚ A BD C (A) 7 3 (B) 2 = 2 (C) 5 2 (D) 3 = 3 2 (E) 7 2 The correct answer is (C). To find the length of BD, you first need to find AD. Notice that DADC is a 30°-60°-90° triangle. The ratio among its sides is 1: = 3:2. Given that AC is 5 units long, AD must be 5 2 units long. (The ratio 1:2 is equivalent to the ratio 5 2 : 5. Next, notice that DABD is a 45°-45°-90° triangle. The ratio among its sides is 1:1: = 2. You know that AD is 5 2 units long. Thus, BD must also be 5 2 units long. Isosceles Triangles An isosceles triangle has the following two special properties: Two of the sides are congruent (equal in length). The two angles opposite the two congruent sides are congruent (equal in size, or degree measure). PART IV: Quantitative Reasoning268 www.petersons.com If you know any two angle measures of a triangle, you can determine whether the triangle is isosceles. 5. A B C 40º 70º In the figure above, BC is 6 units long, m∠A 5 70°, and m∠B 5 40°. How many units long is AB ? (A) 5 (B) 6 (D) 7 (C) 8 (E) 9 The correct answer is (B). Since m∠A and m∠B add up to 110°, m∠C 5 70° (70 1 110 5 180), and you know the triangle is isosceles. What’s more, since m∠A 5 m∠C, AB ≅ BC. Given that BC is 6 units long, AB must also be 6 units long. A line bisecting the angle connecting the two congruent sides divides an isosceles triangle into two congruent right triangles. So if you know the lengths of all three sides of an isosceles triangle, you can determine the area of these two right triangles by applying the Pythagorean theorem, as in the next example. 6. Two sides of a triangle are each 8 units long, and the third side is 6 units long. What is the area of the triangle, expressed in square units? (A) 14 (B) 12 = 3 (C) 18 (D) 22 (E) 3 = 55 The correct answer is (E). Bisect the angle connecting the two congruent sides (as in DABC in the next figure). The bisecting line is the triangle’s height (h), and the triangle’s base is 6 units long. Chapter 11: Math Review: Geometry 269 www.petersons.com C B A You can determine the triangle’s height (h) by applying the Pythagorean theorem: 3 2 1 h 2 5 8 2 h 2 5 64 2 9 h 2 5 55 h 5 = 55 A triangle’s area is half the product of its base and height. Thus, the area of DABC 5 1 2 ~6! = 55 5 3 = 55 Equilateral Triangles An equilateral triangle has the following three properties: All three sides are congruent (equal in length). The measure of each angle is 60°. Area 5 s 2 = 3 4 (s 5 any side) Any line bisecting one of the 60° angles divides an equilateral triangle into two right triangles with angle measures of 30°, 60°, and 90°; in other words, into two 1: = 3:2 triangles, as shown in the right-hand triangle in the next figure. (Remember that Pythagorean angle triplet?) C AB 60º 60º 60º In the left-hand triangle, if s 5 6, the area of the triangle 5 9 = 3. To confirm this formula, bisect the triangle into two 30°-60°-90° ~1: = 3:2! triangles (as in the right-hand triangle in the preceding figure). The area of this equilateral triangle is 1 2 ~2! = 3, or = 3. The area of each smaller right triangle is = 3 2 . PART IV: Quantitative Reasoning270 www.petersons.com QUADRILATERALS A quadrilateral is a four-sided geometric figure. The GRE emphasizes four specific types of quadrilateral: the square, the rectangle, the parallelogram (including the rhombus), and the trapezoid. In this section, you’ll learn the unique properties of each type, how to distinguish among them, and how the GRE test makers design questions about them. Rectangles, Squares, and Parallelograms Here are five characteristics that apply to all rectangles, squares, and parallelograms: The sum of the measures of all four interior angles is 360°. Opposite sides are parallel. Opposite sides are congruent (equal in length). Opposite angles are congruent (the same size, or equal in degree measure). Adjacent angles are supplementary (their measures total 180°). A rectangle is a special type of parallelogram in which all four angles are right angles (90°). A square is a special type of rectangle in which all four sides are congruent (equal in length). For the GRE, you should know how to determine the perimeter and area of each of these three types of quadrilaterals. Referring to the next three figures, here are the formulas (l 5 length and w 5 width): Rectangle Perimeter 5 2l 1 2w Area 5 l 3 w Chapter 11: Math Review: Geometry 271 www.petersons.com Square Perimeter 5 4s [s 5 side] Area 5 s 2 Area 5 S 1 2 D (diagonal) 2 Parallelogram Perimeter 5 2l 1 2w Area 5 base (b) 3 altitude (a) GRE questions involving squares come in many varieties. For example, you might need to determine area, given the length of any side or either diagonal, or perimeter. Or, you might need to do just the opposite—find a length or perimeter given the area. For example: The area of a square with a perimeter of 8 is 4. (s 5 8 4 4 5 2, s 2 5 4) The perimeter of a square with an area of 8 is 8 = 2. (s 5 = 8 5 2 = 2,4s 5 4 3 2 = 2) The area of a square with a diagonal of 6 is 18. (A 5 S 1 2 D 6 2 5 S 1 2 D ~36!518) Or, you might need to determine a change in area resulting from a change in perimeter (or vice versa). PART IV: Quantitative Reasoning272 www.petersons.com . comparative lengths of the two legs, whereas the third—and greatest—number represents the comparative length of the hypotenuse (on the GRE, the first four triplets. triangles share: Length of the sides: Each side is shorter than the sum of the lengths of the other two sides. (Otherwise, the triangle would collapse