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Quadrilateral Family Topic Index | Geometry Index | Regents Exam Prep Center Each member of the quadrilateral family will describe its specific properties. *Quadrilateral I have exactly four sides. The sum of the interior angles of all quadrilaterals is 360º. *Trapezoid I have only one set of parallel sides. [The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.] A quadrilateral is any four sided figure. Do not assume any additional properties for a quadrilateral unless you are given additional information. A trapezoid has ONLY ONE set of parallel sides. When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are not parallel. *Isosceles Trapezoid I have: - only one set of parallel sides - base angles congruent - legs congruent - diagonals congruent - opposite angles supplementary *Parallelogram Never assume that a trapezoid is isosceles unless you are given (or can prove) that information. Notice how the properties of a parallelogram come in sets of twos: two properties about the sides; two properties about the angles; two properties about the diagonals. Use this fact to help you remember the properties. I have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles *Rectangle I have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent If you know the properties of a parallelogram, you only need to add 2 additional properties to describe a rectangle. *Rhombus A rhombus is a slanted square. It has all of the properties of a parallelogram plus three additional properties. I have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular *Square Hey, look at me! I have all of the properties of the parallelogram AND the rectangle AND the rhombus. I have it all! The square is the most specific member of the quadrilateral family. It has the largest number of properties. Theorems Dealing with Parallelograms Topic Index | Geometry Index | Regents Exam Prep Center *Parallelogram Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. I have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles Using this definition, the remaining properties regarding a parallelogram can be "proven" true and become theorems. When GIVEN a parallelogram, the definition and theorems are stated as ... A parallelogram is a quadrilateral with both pairs of opposite sides parallel. If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent. (Proof appears further down the page.) If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are congruent. If a quadrilateral is a parallelogram, the consecutive angles are supplementary. If a quadrilateral is a parallelogram, the diagonals bisect each other. If a quadrilateral is a parallelogram, the diagonals form two congruent triangles. When trying to PROVE a parallelogram, the definition and theorems are stated as ... (many of these theorems are converses of the previous theorems) A parallelogram is a quadrilateral with both pairs of opposite sides parallel. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram. If one angle is supplementary to both consecutive angles in a quadrilateral, the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram. (Proof appears further down the page.) ** Be sure to remember this last method, as it may save you time when working a proof. Proof of Theorem: If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent. (Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.) STATEMENTS 1 2 Draw segment from A to C 3 4 REASONS 1 Given 2 Two points determine exactly one line. 3 A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 4 If two parallel lines are cut by a transversal, the alternate interior angles are congruent. 5 Reflexive property: A quantity is congruent to itself. 6 ASA: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 7 CPCTC: Corresponding parts of congruent triangles are congruent. 5 6 7 Proof of Theorem: If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram. (Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.) STATEMENTS REASONS 1 1 Given 2 Draw segment from A to C 3 2 Two points determine exactly one line. 3 If two parallel lines are cut by a transversal, the alternate interior angles are congruent. 4 Reflexive property: A quantity is congruent to itself. 4 5 6 7 8 5 SAS: If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 6 CPCTC: Corresponding parts of congruent triangles are congruent. 7 If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. 8 A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Theorems Dealing with Rectangles, Rhombuses, Squares Topic Index | Geometry Index | Regents Exam Prep Center Definition: A rectangle is a parallelogram with four right angles. Using the definition, the properties of the rectangle can be "proven" true and become theorems. When dealing with a rectangle, the definition and theorems are stated as ... 1. *Rectangle I have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent A rectangle is a parallelogram with four right angles. While the definition states "parallelogram", it is sufficient to say: "A quadrilateral is a rectangle if and only if it has four right angles.", since any quadrilateral with four right angles is a parallelogram. 2. 3. If a parallelogram has one right angle it is a rectangle. A parallelogram is a rectangle if and only if its diagonals are congruent. Construction workers use the fact that the diagonals of a rectangle are congruent (equal) when attempting to build a "square" footing for a building, a patio, a fenced area, a table top, etc. Workers measure the diagonals. When the diagonals of the project are equal the building line is said to be square. Definition: A rhombus is a parallelogram with four congruent sides. Using the definition, the properties of the rhombus can be "proven" true and become theorems. When dealing with a rhombus, the definition and theorems are stated as ... *Rhombus 1. A rhombus is a parallelogram with four congruent sides. I have all of the properties of While the definition states "parallelogram", it is sufficient the parallelogram PLUS to say: "A quadrilateral is a rhombus if and only if it has four congruent sides.", since any quadrilateral with - 4 congruent sides four congruent sides is a parallelogram. - diagonals bisect angles - diagonals perpendicular 2. If a parallelogram has two consecutive sides congruent, it is a rhombus. 3. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. 4. A parallelogram is a rhombus if and only if the diagonals are perpendicular. (Proof of theorem appears further down page.) Definition: A square is a parallelogram with four congruent sides and four right angles. Using the definition, the properties of the rhombus can be "proven" true and become theorems. When dealing with a square, the definition is stated as ... A square is a parallelogram with four congruent sides and four right angles. This definition may also be stated as *Square Hey, look at me! I have all of the properties of the parallelogram AND the A quadrilateral is a square if and only if it is a rhombus and a rectangle. rectangle AND the rhombus. I have it all! Proof of Theorem: If a parallelogram is a rhombus, then the diagonals are perpendicular. (Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.) STATEMENTS REASONS 1 1 Given 2 Draw segment from A to C 3 2 Two points determine exactly one line. 3 A rhombus is a parallelogram with four congruent sides. 4 A rhombus is a parallelogram with four congruent sides. 4 5 6 7 8 9 5 If a quadrilateral is a parallelogram, the diagonals bisect each other. 6 A bisector of a segment intersects the segment at its midpoint. 7 Midpoint of a line segment is the point on that line segment that divides the segment two congruent segments. 8 Reflexive Property - A quantity is congruent to itself. 9 SSS - If three sides of one triangle are congruent to three sides of a second triangle, 10 11 12 the triangles are congruent. 10 CPCTC - Corresponding parts of congruent triangles are congruent. 11 If 2 congruent angles form a linear pair, they are right angles. 12 Perpendicular lines meet to form right angles. Theorems Dealing with Trapezoids Topic Index | Geometry Index | Regents Exam Prep Center Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides. *Trapezoid I have only one set of parallel sides. [The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.] A trapezoid has ONLY ONE set of parallel sides. When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are NOT parallel. The median (also called the mid-segment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg. ; Theorem: The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (True for ALL trapezoids.) Definition: An isosceles trapezoid is a trapezoid with congruent legs. Theorems: 1. A trapezoid is isosceles if and only if the base angles are congruent. 2. A trapezoid is isosceles if and only if the *Isosceles Trapezoid I have: - only one set of parallel sides - base angles congruent - legs congruent - diagonals congruent - opposite angles supplementary diagonals are congruent. 3. If a trapezoid is isosceles, the opposite angles are supplementary. Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.