Geometry Notes PDF

English system of measurement - It is a way of measuring things like length, weight, and volume - It is mostly used in the United States and a few other countries Metric system - It is an internationally agreed decimal system of measurement created in France during the year 1799 Conversion; Quadr...

English system of measurement - It is a way of measuring things like length, weight, and volume - It is mostly used in the United States and a few other countries Metric system - It is an internationally agreed decimal system of measurement created in France during the year 1799 Conversion; Quadrilateral - It is a two-dimensional figure with four sides and four angles - A quadrilateral should be closed shape with 4 sides - All the interior angles of a quadrilateral sum up to 360 Parallelogram - It is a quadrilateral that has opposite sides that are equal and parallel - The opposite sides of a parallelogram are parallel and congruent - Opposite angles are also congruent - Consecutive angles are supplementary - The diagonals bisect each other - Each diagonal divides a parallelogram into two congruent triangles Opposite sides theorem Opposite sides of a parallelogram are congruent Opposite angles theorem - Opposite angles of a parallelogram are congruent Consecutive angles - Each angle is consecutive to two other angles Consecutive angles in parallelogram theorem - Consecutive angles in a parallelogram are supplementary Diagonals - Diagonals are segments that join non-consecutive vertices Diagonal property - When the diagonals of a parallelogram intersect they meet at the midpoint of each diagonal - Diagonals are not congruent - The diagonals of a parallelogram bisect each other - The diagonals of a parallelogram form two congruent triangles Theorem of Rectangles - If a parallelogram is a rectangle, then its diagonals are congruent - If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle - If a parallelogram has at least one right angle then the parallelogram is a rectangle Theorem of Rhombus - The diagonals of a rhombus are perpendicular - Each diagonal of a rhombus bisects the opposite angles - If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus Theorems of a square - The diagonals of a square bisect the vertex angle Midline theorem - The line segment in a triangle joining the midpoint of any two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side Trapezoid - A quadrilateral with exactly one pair of parallel sides. - Parallel sides are called bases - Angles formed by a base and a leg are base angles - Non parallel sides are called legs Isosceles Trapezoid - A trapezoid with congruent legs Properties of Isosceles Trapezoid - Both pairs of base angles of an isosceles trapezoid are congruent - The diagonals of an isosceles trapezoid are congruent Isosceles Trapezoid Theorem 1; Base angles of an isosceles trapezoid are congruent Theorem 2; Diagonals of an isosceles trapezoid are congruent Theorem 3; If the base angles of a trapezoid are congruent, then the trapezoid is isosceles Theorem 4; If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid Median of a Trapezoid - The median of a trapezoid is the segment joining the midpoints of the legs Theorem 1; The median of a trapezoid is parallel to the bases Theorem 2; The length of the median is one-half the sum of the lengths of the bases DAY 6: KITE - A quadrilateral with two distinct pairs of congruent adjacent sides. Theorem: - If a quadrilateral is a kite, then its diagonals are perpendicular. - If exactly one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. - The area of a kite is half the product of the lengths of its diagonals. Problems Involving Trapezoids and Kites DAY 8: PARTS OF A CIRCLE - the set of all points in a plane that are the same distance from a given point, called the center of a circle. This distance is called the radius of the circle. - A circle is named by its center. Arcs: - consists of two points on the circle and all the points on the circle between those two points. Minor arc – measure is less than 180 degrees (named with 2 points) Major arc – measure is greater than 180 degrees (named with 3 points) Semicircle– measure equals 180 degrees (named with 3 points) Arc AdditionPostulate: - The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC= mAB+ mBC Central angle & intercepted arc: - The measure of a central angle is EQUAL to the measure of the INTERCEPTED ARC. - A central angle is an angle whose vertex is the CENTER of the circle. Inscribed angles: - an angle whose vertex is on a circle and whose sides contain chords. - The measure of an inscribed angle is half the measure of the intercepted arc. DAY 9: Theorems about Chords, Arcs, Central Angles, and Inscribed Angles - An angle formed by a chord and a tangent can be considered an inscribed angle. - An angle inscribed in a semicircle is a right angle. Congruent arcs – are two arcs of the same circle or of congruent circles that have the same measure. THEOREM: - If two chords of a circle or of congruent circles are congruent, then its corresponding minor arcs are congruent. - If two minor arcs of a circle or of congruent circles are congruent, then its corresponding chords are congruent. - If two central angles of a circle or of congruent circles are congruent, then its corresponding arcs are congruent. - If two minor arcs of a circle or of congruent circles are congruent, then its corresponding central angles are congruent. - If two central angles of a circle or of congruent circles are congruent, then its corresponding chords are congruent. - If two chords of a circle or of congruent circles are congruent, then its corresponding central angles are congruent. Congruent Chords Theorem: - In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. DAY 10: Angles formed by Tangents, Secants, and Chords Interior Intersection Theorem: - If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Interior angle = 1⁄2 (arc 1 + arc 2) Exterior intersection theorem: - If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. Note: The sum of the exterior angle and the arc near is equal to 180 degrees. This only applies if the exterior angle is made by two tangent lines. Exterior angle = 1⁄2 (far arc – near arc)