1.If two parallel lines are cut by a third line, the alternate interior angles are congruent. In the figure above,

*c* = *x* and *w* = *d*

2. If two parallel lines are cut by a third line, the corresponding angles are congruent. In the figure,

*a* = *w*, *b* = *x*, *c* = *y*, and *d* = *z*

3. If two parallel lines are cut by a third line, the sum of the measures of the interior angles on the same side of the transversal is 180��. In the figure,

*c* + *w* = 180 and *d* + *x* = 180

1. The sum of the measures of the interior angles of a triangle is 180°. In the figure above,

*x* = 70 because 60 + 50 + *x* = 180

2. When two lines intersect, vertical angles are congruent. In the figure,

*y* = 50

3. A straight angle measures 180°. In the figure,

*z* = 130 because *z* + 50 = 180

4. The sum of the measures of the interior angles of a polygon can be found by drawing all diagonals of the polygon from one vertex and multiplying the number of triangles formed by 180°.

Since this polygon is divided into 3 triangles, the sum of the measures of its angles is 3 x 180°, or 540°.

Unless otherwise noted in the SAT, the term "polygon" will be used to mean a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180°.

A polygon is "regular" if all its sides are congruent and all its angles are congruent.

1. Pythagorean theorem: In any right triangle, *a*^{2} + *b*^{2} = *c*^{2}, where *c* is the length of the longest side and *a* and *b* are the lengths of the two shorter sides.

To find the value of *x*, use the Pythagorean Theorem.

*x*^{2} = 3^{2} + 4^{2}

*x*^{2} = 9 + 16

2. In any equilateral triangle, all sides are congruent and all angles are congruent.

Because the measure of the unmarked angle is 60°, the measures of all angles of the triangle are equal; and, therefore, the lengths of all sides of the triangle are equal: *x* = *y* = 10.

3. In an isosceles triangle, the angles opposite congruent sides are congruent. Also, the sides opposite congruent angles are congruent. In the figures below, *a* = *b* and *x* = *y*.

4. In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. In the figure below, *a* < *b* < *c*.

5. Two polygons are similar if and only if the lengths of their corresponding sides are in the same ratio and the measures of their corresponding angles are equal.

If polygons *ABCDEF* and *GHIJKL* are similar, then and are corresponding sides, so that . Therefore, *x* = 9 = *HI*.

**Note:** means the line segment with endpoints *A* and *F*, and *AF* means the length of .

#### Rectangles

Area of a rectangle = length × width = × *w*

Perimeter of a rectangle = 2( + *w*) = 2 + 2*w*

#### Circles

Area of a circle = *r*^{2} (where r is the radius)

Circumference of a circle = 2*r* = *d* (where *d* is the diameter)

#### Triangles

Area of a triangle = (base × altitude)

Perimeter of a triangle = the sum of the lengths of the three sides

Triangle inequality: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

#### Volume

Volume of a rectangular solid (or cube) = × *w* × *h*

(*l* is the length, *w* is the width, and *h* is the height)

Volume of a right circular cylinder = *r*^{2}h

(*r* is the radius of the base, and *h* is the height)

**Be familiar with the formulas that are provided in the Reference Information included with the test directions. Refer to the test directions in the sample test.(**pdf/1.1M)

1. In questions that involve the *x-* and *y-*axes, *x-*values to the right of the *y*-axis are positive and x-values to the left of the *y-*axis are negative. Similarly, *y-*values above the *x-*axis are positive and *y-*values below the *x-*axis are negative. In an ordered pair (*x*, *y*), the *x-*coordinate is written first. Point *P* in the figure above appears to lie at the intersection of gridlines. From the figure, you can conclude that the *x-*coordinate of *P* is –2 and the *y-*coordinate of *P* is 3. Therefore, the coordinates of point *P* are (–2, 3). Similarly, you can conclude that the line shown in the figure passes through the point with coordinates (–2, –1) and the point (2, 2).

2. Slope of a line =

A line that slopes upward as you go from left to right has a *positive* slope. A line that slopes downward as you go from left to right has a *negative* slope. A horizontal line has a slope of zero. The slope of a vertical line is undefined.

Parallel lines have the same slope. The product of the slopes of two perpendicular lines is –1, provided the slope of each of the lines is defined. For example, any line perpendicular to line above has a slope of .

The equation of a line can be expressed as *y* = *mx *+ *b*, where *m* is the slope and *b* is the *y-*intercept. Since the slope of line is , the equation of line can be expressed as , since the point (–2, 1) is on the line, *x* = –2 and *y* = 1 must satisfy the equation. Hence, so and the equation of line is .

3. A quadratic function can be expressed as *y* = *a* (*x* – *h*)^{2} + *k*, where the vertex of the parabola is at the point (*h*, *k*) and a $\ne $0. If a > 0, the parabola opens upward; and if a < 0, the parabola opens downward.

The parabola above has its vertex at (–2, 4). Therefore, *h* = –2 and *k* = 4. The equation can be represented by *y* = *a* (*x* + 2)^{2} + 4. Since the parabola opens downward, we know that a < 0. To find the value of *a*, we also need to know another point on the parabola. Since we know the parabola passes through the point (1, 1), *x* = 1 and *y* = 1 must satisfy the equation.

Hence, 1 = *a*(1 + 2)^{2} + 4,

so .

Therefore, an equation for the parabola is .