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  1. There are four major types of angle measures:
    An acute angle has measure less than 90˚:
    A right angle has measure 90˚:
    An obtuse angle has measure greater than 90˚:
    A straight angle has measure 180°:

  2. Two angles are supplementar if their angle sum is 180˚:

  3. Two angles are complementary if their angle sum is 90˚:

  4. Perpendicular lines meet at right angles:


  5. When two straight lines meet at a point, they form four angles. The angles opposite each other are called vertical angles, and they are congruent (equal). In the figure, a = b, and c = d.
    a = b and c=d

  6. When parallel lines are cut by a transversal, three important angle relationships exist:

  7. The shortest distance from a point not on a line to the line is along a perpendicular line.

  8. A triangle containing a right angle is called a right triangle. The right angle is denoted by a small square:



  9. A triangle with two equal sides is called isosceles. The angles opposite the equal sides are called the base angles:

  10. In an equilateral triangle, all three sides are equal and each angle is 60°:

  11. The altitude to the base of an isosceles or equilateral triangle bisects the base and bisects the vertex angle:

  12. The angle sum of a triangle is 180°:

  13. The area of a triangle is , where b is the base and h is the height.


  14. In a triangle, the longer side is opposite the larger angle, and vice versa:

  15. Pythagorean Theorem (right triangles only): The square of the hypotenuse is equal to the sum of the squares of the legs.

  16. A Pythagorean triple: the numbers 3, 4, and 5 can always represent the sides of a right triangle and they appear very often: 52 = 32 + 42.
  17. Two triangles are similar (same shape and usually different size) if their corresponding angles are equal. If two triangles are similar, their corresponding sides are proportional:

  18. If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar.
    In the figure, the large and small triangles are similar because both contain a right angle and they share A.

  19. Two triangles are congruent (identical) if they have the same size and shape.
  20. In a triangle, an exterior angle is equal to the sum of its remote interior angles and is therefore greater than either of them:
    e = a + b and e > a and e > b
  21. In a triangle, the sum of the lengths of any two sides is greater than the length of the remaining side:
x + y > z
y + z > x
x + z > y
  1. In a 30°–60°–90° triangle, the sides have the following relationships:
    In general for x
  1. In a 45°–45°–90° triangle, the sides have the following relationships:
  2. Opposite sides of a parallelogram are both parallel and congruent:
  3. The diagonals of a parallelogram bisect each other:

  4. A parallelogram with four right angles is a rectangle. If w is the width and l is the length of a rectangle, then its area is A = lw and its perimeter is P = 2w + 2l:

  5. If the opposite sides of a rectangle are equal, it is a square and its area is A = s2 and its perimeter is P = 4s, where s is the length of a side:

  6. The diagonals of a square bisect each other and are perpendicular to each other:
  7. A quadrilateral with only one pair of parallel sides is a trapezoid. The parallel sides are called bases, and the non-parallel sides are called legs:
  8. The area of a trapezoid is the average of the bases times the height:
  9. The volume of a rectangular solid (a box) is the product of the length, width, and height. The surface area is the sum of the area of the six faces:
  10. If the length, width, and height of a rectangular solid (a box) are the same, it is a cube. Its volume is the cube of one of its sides, and its surface area is the sum of the areas of the six faces:
  11. The volume of a cylinder is , and the lateral surface (excluding the top and bottom) is S = 2πrh, where r is the radius and h is the height:
  12. A line segment form the circle to its center is a radius.
    A line segment with both end points on a circle is a chord.
    A chord passing though the center of a circle is a diameter.
    A diameter can be viewed as two radii, and hence a diameter’s length is twice that of a radius.
    A line passing through two points on a circle is a secant.
    A piece of the circumference is an arc.
    The area bounded by the circumference and an angle with vertex at the center of the circle is a sector.
  13. A tangent line to a circle intersects the circle at only one point. The radius of the circle is perpendicular to the tangent line at the point of tangency:
  14. Two tangents to a circle from a common exterior point of the circle are congruent:
  15. An angle inscribed in a semicircle is a right angle:
  16. A central angle has by definition the same measure as its intercepted arc.
  17. An inscribed angle has one-half the measure of its intercepted arc.
  18. The area of a circle is πr2, and its circumference (perimeter) is 2πr, where r is the radius:
  19. To find the area of the shaded region of a figure, subtract the area of the unshaded region from the area of the entire figure.
  20. When drawing geometric figures, don’t forget extreme cases.

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