Download Plane Geometry - Answer Explanations

Plane Geometry
Angles
Complimentary Angles: Two angles that add to 90˚
Supplementary Angles: Two angles that add to 180˚
Acute Angle: Any angle greater than 0˚ and less than 90˚
Obtuse Angle: Any angle greater than 90˚ and less than 180˚
Right Angle: A 90˚ angle
Parallel Line Properties: When two parallel lines are cut by a transversal, two types of angles
are formed: big angles and little angles. All the big angles are equal and all the little angles are
equal, and the big angles are supplementary to the little angles. In school, you likely learned
this in terms of each of the individual properties (alternating interior angles, corresponding
angles, etc.), but it is much easier to think of parallel line properties as one property, as
described above.
Vertical Angles: When two lines intersect, four angles are formed. Each pair of opposite, or
“vertical,” angles are equal.
Polygons
Number of Diagonals in an n-sided figure
n • (n-3)/2
Sum of Interior Angles in an n-sided figure
(n-2) • 180
Triangles
Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the
length of the third side. In other words, if a triangle has sides of length a, b, and c, a + b c, a +
c b, and b + c a. Therefore, if a triangle has one side of 5 inches and another side of 9
inches, the third side would have to be less than 13 inches (the sum of 9 and 5) and greater
than 4 inches (the difference of 9 and 5). Note that 4 and 13 themselves are not possible values
for the third side, as the inequality is non-inclusive.
Sum of Angles: The sum of the angles in any triangle is 180˚.
Area: A = (1/2)hb where h is the triangle’s height and b is the triangle’s base. Note that the
height, or altitude, of a triangle is by definition perpendicular to its base. Sometimes, the
altitude is outside the triangle itself.
Right Triangle: A triangle with one right angle
Equilateral Triangle: A triangle in which all three sides are of equal length. Equilateral triangles
are also, by definition, equiangular: each angle of an equilateral triangle is equal to 60˚.
Isosceles Triangle: A triangle with two sides of equal length. The base angles (the angles
opposite the equal sides) of an isosceles triangle are also equal.
Quadrilaterals
Sum of Angles: The sum of the angles in any convex quadrilateral is 360˚.
Square: A quadrilateral with four sides of equal length and four angles of 90˚ each.
A = s2 P = 4s, where s = side length.
Rectangle: A quadrilateral with four angles of 90˚ each.
A = lw P = 2l + 2w, where l = length and w = width
Rhombus: A quadrilateral with four sides of equal length.
A = bh, where b = base and h = height. The height is by definition perpendicular to the base.
P = 4s, where s = side length.
Opposite sides of a rhombus are parallel. Opposite angles of a rhombus are equal, and
consecutive angles are supplementary.
Parallelogram: A quadrilateral with two sets of parallel sides.
A = bh where b = base and h = height. The height is by definition perpendicular to the base.
P = 2S1 + 2S2, where S1 and S2 are adjacent sides.
Opposite sides of a parallelogram are equal in length. Opposite angles of a parallelogram are
equal, and consecutive angles are supplementary.
Trapezoid: A quadrilateral with one set of parallel sides.
A = .5h(b1 + b2), where h = height and b1 and b2 are the two bases. The area of a trapezoid
can also be thought of as A = height • (the average of the bases). P = S1 + S2 + S3 + S4, where
S1, S2, S3, and S4 are the four sides.
Congruent: Same size and shape
Two angles are congruent if and only if they have the same measure.
Two line segments are congruent if and only if they have the same length.
Two shapes are congruent if and only if they are the same size and the same shape.
Triangle Congruence
Triangle congruence can be proven in the following ways:
SSS (Side-Side-Side): Two triangles are congruent if they have three pairs of congruent sides.
SAS (Side-Angle-Side): Two triangles are congruent if they have two pairs of congruent sides
and the angle in between the two congruent sides is also congruent.
AAS (Angle-Angle-Side), ASA (Angle-Side-Angle): Two triangles are congruent if they have two
pairs of congruent angles and one congruent side.
HL (Hypotenuse Leg): Two right triangles are congruent if they have congruent hypotenuses and
one pair of congruent legs.
Similar: Same shape, but not necessarily the same size.
Triangle Similarity
AAA (Angle-Angle-Angle): Two triangles are similar if they have three pairs of congruent angles.
In reality, only two pairs of congruent angles are needed to establish the congruence of the
triangles, since if two pairs of angles are congruent, the third pair of angles is necessarily
congruent.