Geometry Review To begin this section, it is helpful to become familiar with the vocabulary used in geometry. The list below defines some of the main geometrical terms. It is followed by an overview of geometrical equations and figures. arc part of a circumference area the space inside a two-dimensional figure bisect cut in two equal parts circumference the distance around a circle chord a line segment that goes through a circle, with its endpoint on the circle congruent identical in shape and size. The geometric notation of “congruent” is ≅ . diameter a chord that goes directly through the center of a circle—the longest line you can draw in a circle equidistant exactly in the middle hypotenuse the longest leg of a right triangle, always opposite the right angle line a straight path that continues infinitely in two directions. The geometric notation for a line is AB ឈ . line segment the part of a line between (and including) two points. The geometric notation for a line segment is PQ . parallel lines in the same plane that will never intersect perimeter the distance around a figure perpendicular two lines that intersect to form 90-degree angles quadrilateral any four-sided figure radius a line from the center of a circle to a point on the circle (half of the diameter) ray a line with an endpoint that continues infinitely in one direction. The geometric notation for a ray is AB . tangent line a line meeting a smooth curve (such as a circle) at a single point without cutting across the curve. Note that a line tangent to a circle at point P will always be perpendicular to the radius drawn to point P. volume the space inside a three-dimensional figure –THE SAT MATH SECTION– 127 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 127 –THE SAT MATH SECTION– 128 Angles An angle is formed by an endpoint, or vertex, and two rays. Naming Angles There are three ways to name an angle. 1. An angle can be named by the vertex when no other angles share the same vertex: ∠A. 2. An angle can be represented by a number written across from the vertex: ∠1. 3. When more than one angle has the same vertex, three letters are used, with the vertex always being the middle letter: ∠1 can be written as ∠BAD or as ∠DAB, ∠2 can be written as ∠DAC or as ∠CAD. Classifying Angles Angles can be classified into the following categories: acute, right, obtuse, and straight. ■ An acute angle is an angle that measures less than 90°. ■ A right angle is an angle that measures 90°. A right angle is symbolized by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90°, but less than 180°. ■ A straight angle is an angle that measures 180°. Thus, both of its sides form a line. Complementary Angles Two angles are complementary if the sum of their measures is equal to 90°. ∠1 + ∠2 = 90° 1 2 Complementary Angles Straight Angle 180° Obtuse Angle Right Angle Symbol A cute Angle 1 2 A C D B Endpoint or Vertex ray ray 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 128 Supplementary Angles Two angles are supplementary if the sum of their measures is equal to 180 degrees. Adjacent Angles Adjacent angles have the same vertex, share a side, and do not overlap. ∠1 and ∠2 are adjacent. The sum of all adjacent angles around the same vertex is equal to 360°. Angles of Intersecting Lines When two lines intersect, vertical angles are formed. Vertical angles have equal measures and are supple- mentary to adjacent angles. ■ m∠1 = m∠3 and m∠2 = m∠4 ■ m∠1 = m∠4 and m∠3 = m∠2 ■ m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180° ■ m∠3 + m∠4 = 180° and m∠1 + m∠4 = 180° Bisecting Angles and Line Segments Both angles and lines are said to be bisected when divided into two parts with equal measures. Example: Therefore, line segment AB is bisected at point C. According to the figure, ∠A is bisected by ray AC. 35° 35° 1 2 3 4 1 2 3 4 ∠1 + ∠2 + ∠3 + ∠4 = 360° 1 2 Adjacent Angles 1 2 ∠1 + ∠2 = 180° Supplementary Angles –THE SAT MATH SECTION– 129 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 129 Angles Formed by Parallel Lines When two parallel lines are intersected by a third line, vertical angles are formed. ■ Of these vertical angles, four will be equal and acute, and four will be equal and obtuse. ■ Any combination of an acute and obtuse angle will be supplementary. In the above figure: ■ ∠b, ∠c, ∠f, and ∠g are all acute and equal. ■ ∠a, ∠d, ∠e, and ∠h are all obtuse and equal. ■ Also, any acute angle added to an any obtuse angle will be supplementary. Examples: m∠b + m∠d = 180° m∠c + m∠e = 180° m∠f + m∠h = 180° m∠g + m∠a = 180° Example: In the figure below, if m || n, what is the value of x? Because ∠x is acute, you know that it can be added to x + 10 to equal 180. The equation is thus x + x + 10 = 180. Solve for x:2x + 10 = 180 –10 –10 ᎏ 2 2 x ᎏ = ᎏ 17 2 0 ᎏ x =85 Therefore, m∠x = 85 and the obtuse angle is equal to 180 – 85 = 95. Angles of a Triangle The measures of the three angles in a triangle always equal 180°. Exterior Angles An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles: ■ An exterior angle and interior angle that share the same vertex are supplementary. ■ An exterior angle is equal to the sum of the non- adjacent interior angles. + = 180° and = + + + = 180° x° (x + 10)° b m n a –THE SAT MATH SECTION– 130 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 130 Example: m∠1 = m∠3 + m∠5 m∠4 = m∠2 + m∠5 m∠6 = m∠3 + m∠2 ■ The sum of the exterior angles of a triangle equals 360°. Triangles It is possible to classify triangles into three categories based on the number of equal sides: Scalene Isosceles Equilateral Triangle Triangle Triangle (no equal sides) (two equal sides) (all sides equal) It is also possible to classify triangles into three categories based on the measure of the greatest angle: Acute Triangle Right Triangle Obtuse Triangle greatest angle greatest angle greatest angle is acute is 90° is obtuse Angle-Side Relationships Knowing the angle-side relationships in isosceles, equi- lateral, and right triangles is useful knowledge to have in taking the SAT. ■ In isosceles triangles, equal angles are opposite equal sides. AB ba C m∠a = m∠b 50° 60° 70° 150° Obtuse Right Acute Equilateral Isosceles Scalene 3 4 6 5 1 2 –THE SAT MATH SECTION– 131 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 131 ■ In equilateral triangles, all sides are equal and all angles are equal. ■ In a right triangle, the side opposite the right angle is called the hypotenuse. Pythagorean Theorem The Pythagorean theorem is an important tool for working with right triangles. It states: a 2 + b 2 = c 2 ,where a and b represent the legs and c represents the hypotenuse. This theorem allows you to find the length of any side as long as you know the measure of the other two. a 2 + b 2 = c 2 1 2 + 2 2 = c 2 1 + 4 = c 2 5 = c 2 ͙5 ෆ = c Pythagorean Triples In a Pythagorean triple, the square of the largest num- ber equals the sum of the squares of the other two numbers. Example: As demonstrated above: 1 2 + 2 2 = (͙5 ෆ ) 2 1, 2, and ͙5 ෆ are also a Pythagorean triple because: 1 2 + 2 2 = 1 + 4 = 5 and (͙5 ෆ ) 2 = 5. Pythagorean triples are useful for helping you identify right triangles. Some common Pythagorean triples are: 3:4:5 8:15:17 and 5:12:13 Multiples of Pythagorean Triples Any multiple of a Pythagorean triple is also a Pythagorean triple. Therefore, if given 3:4:5, then 9:12:15 is also a Pythagorean triple. Example: If given a right triangle with sides measuring 6, x, and 10, what is the value of x? Because it is a right triangle, use the Pythagorean theorem. Therefore, 10 2 – 6 2 = x 2 100 – 36 = x 2 64 = x 2 8= x 2 1 √ ¯¯¯ 5 Hypotenuse Right Equilateral 60º 60º60º 55 5 –THE SAT MATH SECTION– 132 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 132 . 10 to equal 180 . The equation is thus x + x + 10 = 180 . Solve for x:2x + 10 = 180 –10 –10 ᎏ 2 2 x ᎏ = ᎏ 17 2 0 ᎏ x =85 Therefore, m∠x = 85 and the obtuse angle is equal to 180 – 85 = 95. Angles. obtuse angle will be supplementary. Examples: m∠b + m∠d = 180 ° m∠c + m∠e = 180 ° m∠f + m∠h = 180 ° m∠g + m∠a = 180 ° Example: In the figure below, if m || n, what is the value of x? Because ∠x is acute,. Vertex ray ray 56 58 SAT2 006[04](fin).qx 11/21/05 6:44 PM Page 1 28 Supplementary Angles Two angles are supplementary if the sum of their measures is equal to 180 degrees. Adjacent Angles Adjacent angles have the