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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 supplementary 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 to the right, a = b, and c = d.


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


    Alternate interior angles are equal.

    Corresponding angles are equal.


    Interior angles on the same side of the transversal are supplementary.


  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°:

    a + b + c=180°

  13. The area of a triangle is A = 12 bh , where b is the base and h is the height.


    A = 12 bh

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


    50° is larger than 30°, so side b longer than side a.

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

    c2 = a2 + b2

  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:

    ad = be = cf

  18. If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar.


    In the figure to the right, 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

  22. In a 30°–60°–90° triangle, the sides have the following relationships:

    In general


  23. In a 45°–45°–90° triangle, the sides have the following relationships:


  24. Opposite sides of a parallelogram are both parallel and congruent:


  25. The diagonals of a parallelogram bisect each other:


  26. 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:


    A = l × w
    P = 2w + 2l


  27. 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:


    A = s2
    P = 4s


  28. The diagonals of a square bisect each other and are perpendicular to each other:



  29. 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:



  30. The area of a trapezoid is the average of the bases times the height:


    A = b1+b22 h

  31. 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:

    V = l × w × h
    S = 2wl + 2hl + 2wh


  32. 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:

    V = x3
    S = 6x2


  33. The volume of a cylinder is V = πr2h, and the lateral surface (excluding the top and bottom) is
    S = 2πrh, where r is the radius and h is the height:


    V = πr2h
    S = 2πrh + 2πr2


  34. 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.


  35. 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:



  36. Two tangents to a circle from a common exterior point of the circle are congruent:




  37. An angle inscribed in a semicircle is a right angle:
  38. A central angle has by definition the same measure as its intercepted arc.


  39. An inscribed angle has one-half the measure of its intercepted arc.



  40. The area of a circle is πr2, and its circumference (perimeter) is 2πr, where r is the radius:

    A = πr2
    C = 2πr


  41. 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.


  42. When drawing geometric figures, don’t forget extreme cases.

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