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

- 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.

- 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:

Equilateral:

- The angle sum of a triangle is 180Â°:

*a + b + c*=180Â°

- 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*

- 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*.

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

*c*^{2}=*a*^{2}+*b*^{2}

- 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:

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

- 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*.

- 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:

*e*=*a*+*b*and*e*>*a*and*e*>*b*

- 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

- 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*= 2*w*+ 2*l*:

*A*=*l*Ã—*w*

*P*= 2*w*+ 2*l*

- 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*= 4*s*, where*s*is the length of a side:

*A*=*s*^{2}

*P*= 4*s*

- 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:

*A*= $\left(\frac{{b}_{1}+{b}_{2}}{2}\right)$*h*

- 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*= 2*wl*+ 2*hl*+ 2*wh*

- 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*=*x*^{3}

*S*= 6*x*^{2}

- The volume of a cylinder is
*V*= Ï€*r*^{2}*h*, and the lateral surface (excluding the top and bottom) is

*S*= 2*Ï€rh*, where*r*is the radius and*h*is the height:

*V*= Ï€*r*^{2}*h*

*S*= 2Ï€*rh*+ 2Ï€*r*^{2}

- 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*.

- 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:

*AB*â‰…*AC*

- 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 2*Ï€r*, where*r*is the radius:

*A*= Ï€*r*^{2}

*C*= 2Ï€*r*

- 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.