Download Key Geometric Ideas from Courses 1 and 2

Key Geometric Ideas from Courses 1 and 2
This unit builds on important geometric concepts and relationships developed in the previous Core-Plus
Mathematics geometry units. Specifically:
Definitions
Isosceles triangle A triangle with at least two sides the same length
Median of a triangle The line segment joining a vertex to the midpoint of the opposite side.
Parallelogram A quadrilateral with two pairs of opposite sides the same length
Rectangle A quadrilateral with four right angles
Kite A convex quadrilateral with two distinct pairs of consecutive sides the same length
Rhombus A quadrilateral with all four sides the same length
Square A rhombus with one right angle
Congruent figures Figures that have the same shape and size, regardless of position or orientation
Complementary angles Two angles whose measures sum to 90˚
Midpoint of a segment The point on the segment that is the same distance from each endpoint
Perpendicular bisector of a segment A line or a line segment that is perpendicular to a segment and contains its
midpoint
Slope of a segment The slope of a segment that contains two points with coordinates (x1, y1) and (x2, y2) is
y2 – y1
(x1 ≠ x2).
x2 – x1
Line reflection A motion determined by a “mirror line” (or line of reflection) that is the perpendicular bisector of the
segment connecting a point and its reflected image; a point on the line reflection is its own image.
Translation A sliding motion that is determined by a distance and direction; the coordinate rule for a translation
h units horizontally and k units vertically is (x, y) → (x + h, y + k).
Rotation A turning motion determined by a point called the center of rotation and a directed angle of rotation
Size transformation of magnitude k A size transformation of magnitude k centered at the origin is defined by the
rule (x, y) → (kx, ky).
Trigonometric functions; 0˚ ≤ θ ≤ 360˚
y
tangent of θ = tan θ = x (x ≠ 0)
y
sine of θ = sin θ = r
cosine of θ = cos θ = xr
Trigonometric ratios for right triangles
length of side opposite !A
tangent of ∠A = tan A = a
= length of side adjacent to !A
b
length of side opposite !A
sine of ∠A = sin A = a
=
length of hypotenuse
c
length of side adjacent to !A
cosine of ∠A = cos A = b
=
length of hypotenuse
c
Relationships
Pythagorean Theorem If the lengths of the sides of a right triangle are a, b, c, with the side of length c opposite the
right angle, then a2 + b2 = c2.
Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the
square of the length of the third side, then the triangle is a right triangle.
Triangle Inequality The sum of the lengths of any two sides of a triangle is always greater than the length of the
third side.
Triangle Angle Sum Property The sum of the measures of the angles in a triangle is 180˚.
Quadrilateral Angle Sum Property The sum of the measures of the angles in a quadrilateral is 360˚. The sum of the
measures of the angles in a quadrilateral is 360˚.
Polygon Angle Sum Property The sum of the measures of the interior angles of a polygon with n sides is (n – 2)180˚.
Base Angles of Isosceles Triangle A triangle is an isosceles triangle if and only if it has two congruent angles.
Side-Side-Side (SSS) congruence condition If three sides of a triangle are congruent to the corresponding sides of
another triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) congruence condition If two sides and the angle between the sides of one triangle are
congruent to the corresponding parts of another triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA) congruence condition If two angles and the side between the angles of one triangle are
congruent to the corresponding parts of another triangle, then the two triangles are congruent.
Opposite Angles Property of Parallelograms Opposite angles in a parallelogram are congruent.
Conditions ensuring a parallelogram (1) If opposite sides are parallel or (2) the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a parallelogram.
Condition ensuring a rectangle If the diagonals of a parallelogram are the same length, then the parallelogram is
a rectangle.
Condition ensuring a square If a rectangle has two consecutive sides the same length, then the rectangle is
a square.
30˚-60˚ right triangle relationship For a right triangle with acute angles of measures 30˚ and 60˚, the length of the
side opposite the 30˚ angle is half the length of the hypotenuse. The length of the side opposite the 60˚ angle is 3
times the length of the side opposite the 30˚ angle.
45˚-45˚ right triangle relationship For a right triangle with acute angles of measures 45˚, the length of the
hypotenuse is 2 times the length of either of the equal legs of the right triangle.
Distance formula The distance d between two points with coordinates (x1, y1) and (x2, y2) is
d=
(x2 – x1)2 + (y2 – y1)2 .
Equation of a circle The equation of a circle with center at the origin and radius r is x2 + y2 = r2.
Law of Sines In any triangle ABC with sides of lengths a, b, and c opposite ∠A, ∠B, and ∠C, respectively:
sin A = sin B = sin C .
a
b
c
Law of Cosines In any triangle ABC with sides of lengths a, b, and c opposite ∠A, ∠B, and ∠C, respectively:
c2 = a2 + b2 – 2ab cos C.