Geometry
- 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°:
- Two angles are supplementar if their angle sum is 180Ëš:
- Two angles are complementary if their angle sum is 90Ëš:
- Perpendicular lines meet at right angles:
- 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
- When parallel lines are cut by a transversal, three important angle relationships exist:
- The shortest distance from a point not on a line to the line is along a perpendicular line.
- A triangle containing a right angle is called a right triangle. The right angle is denoted by a small square:
- A triangle with two equal sides is called isosceles. The angles opposite the equal sides are called the base angles:
- In an equilateral triangle, all three sides are equal and each angle is 60Â°:
- The altitude to the base of an isosceles or equilateral triangle bisects the base and bisects the vertex angle:
Isosceles:;
- The angle sum of a triangle is 180Â°:
- The area of a triangle is , where b is the base and h is the height.
- In a triangle, the longer side is opposite the larger angle, and vice versa:
- Pythagorean Theorem (right triangles only): The square of the hypotenuse is equal to the sum of the squares of the legs.
- A Pythagorean triple: the numbers 3, 4, and 5 can always represent the sides of a right triangle and they appear very often: 5^{2} = 3^{2} + 4^{2}.
- 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:
- If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar.
- Two triangles are congruent (identical) if they have the same size and shape.
- In a triangle, an exterior angle is equal to the sum of its remote interior angles and is therefore greater than either of them:
- 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 |
- In a 30Â°â€“60Â°â€“90Â° triangle, the sides have the following relationships:
In general for x
- In a 45Â°â€“45Â°â€“90Â° triangle, the sides have the following relationships:
- Opposite sides of a parallelogram are both parallel and congruent:
- The diagonals of a parallelogram bisect each other:
- 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:
- If the opposite sides of a rectangle are equal, it is a square and its area is A = s^{2} and its perimeter is P = 4s, where s is the length of a side:
- The diagonals of a square bisect each other and are perpendicular to each other:
- 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:
- The area of a trapezoid is the average of the bases times the height:
- 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:
- 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:
- The volume of a cylinder is , and the lateral surface (excluding the top and bottom) is S = 2Ï€rh, where r is the radius and h is the height:
- A line segment form the circle to its center is a radius.
- 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:
- Two tangents to a circle from a common exterior point of the circle are congruent:
- An angle inscribed in a semicircle is a right angle:
- A central angle has by definition the same measure as its intercepted arc.
- An inscribed angle has one-half the measure of its intercepted arc.
- The area of a circle is Ï€r^{2}, and its circumference (perimeter) is 2Ï€r, where r is the radius:
- 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.
- When drawing geometric figures, donâ€™t forget extreme cases.