### Geometry

1. There are four major types of angle measures:

An acute angle has measure less than 90Ëš:

A right angle has measure 90Ëš:

An obtuse angle has measure greater than 90Ëš:

A straight angle has measure 180Â°:

2. Two angles are supplementary if their angle sum is 180Ëš:

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

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 60Â°:

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

Isosceles:

Equilateral:

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

a + b + c=180Â°

13. The area of a triangle is A = $\frac{1}{2}$ bh , where b is the base and h is the height.

A = $\frac{1}{2}$ bh

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

50Â° is larger than 30Â°, 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:

$\frac{a}{d}$ = $\frac{b}{e}$ = $\frac{c}{f}$

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 âˆ 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 30Â°â€“60Â°â€“90Â° triangle, the sides have the following relationships:

In general

23. In a 45Â°â€“45Â°â€“90Â° 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 Ã— 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 = $\left(\frac{{b}_{1}+{b}_{2}}{2}\right)$ 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 Ã— w Ã— 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 = Ï€r2h, and the lateral surface (excluding the top and bottom) is
S = 2Ï€rh, where r is the radius and h is the height:

V = Ï€r2h
S = 2Ï€rh + 2Ï€r2

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 diameterâ€™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:

AB â‰… AC

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 Ï€r2, and its circumference (perimeter) is 2Ï€r, where r is the radius:

A = Ï€r2
C = 2Ï€r

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, donâ€™t forget extreme cases.