Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms


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1 Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Getting Ready: How will you know whether or not a figure is a parallelogram? By definition, a quadrilateral is a parallelogram if it has two pairs of parallel sides. The following theorems which are converses of the theorems previously taken give the conditions that guarantee a quadrilateral to be a parallelogram. Theorem A quadrilateral is a parallelogram if both pairs of opposite sides are congruent. Theorem A quadrilateral is a parallelogram if both pairs of opposite angles are congruent. Theorem A quadrilateral is a parallelogram if two sides are both parallel and congruent.
2 Theorem If an angle of a quadrilateral is supplementary to both consecutive angles, then the quadrilateral is a parallelogram. Example If a diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Let's Practice: Direction: Find the value of x and y that would ensure that each quadrilateral is a parallelogram Lesson 4.4 Special Parallelograms Definition Convex Quadrilateral It is a convex polygon with four sides. Definition Diagonal It is a segment joining two nonconsecutive vertices of a polygon. Definition Parallelogram It is a quadrilateral with both pairs of opposite sides parallel to each other.
3 Definition Rectangle It is a parallelogram with a right angle. Definition Rhombus It is a parallelogram with two consecutive sides congruent. Definition Square It is a rectangle with two consecutive sides congruent. It is a rhombus with a right angle. Theorem If a parallelogram has one right angle, then it has four right angles and the parallelogram is a rectangle. Theorem The diagonals of a rectangle are congruent.
4 Theorem In a rhombus, the diagonals are perpendicular and they bisect each other. Theorem Each diagonal of a rhombus bisects a pair of opposite angles. Theorem The diagonals of a square bisect each other, are congruent, and perpendicular. Properties of a Rectangle 1. Opposite sides are congruent. 2. Opposite sides are parallel. 3. Each diagonal separated the rectangle into two congruent triangles. 4. Opposite angles are congruent. 5. Consecutive angles are supplementary. 6. All four angles are right angles. 7. Diagonals bisect each other and are congruent.
5 Properties of a Rhombus 1. All four sides are congruent. 2. Opposite sides are parallel. 3. Each diagonal separates the rhombus into two congruent triangles. 4. Opposite angles are congruent. 5. Consecutive angles are supplementary. 6. Diagonals bisect each other and are perpendicular. 7. Each diagonal bisects a pair of opposite angles. Properties of a Square 1. All four sides are congruent. 2. All angles are right angles, 3. Each diagonal separates the square into two congruent triangles. 4. Opposite angles are congruent and supplementary. 5. Consecutive angles are supplementary and congruent. 6. Diagonals bisect each other, are perpendicular and congruent. Let s Practice: Direction: Find the value of the variable (s) for each parallelogram. Kindly consider the theorems we studied in solving each of the following
6 Lesson 4.5 Trapezoids Figure Trapezoid Definition Trapezoid It is a convex quadrilateral with exactly one pair of parallel sides. Definition Bases of a Trapezoid The bases of a trapezoid are the two parallel sides. Definition Legs of a Trapezoid The legs of a trapezoid are the two nonparallel sides. Definition Isosceles Trapezoid It is a trapezoid with congruent legs. Definition Median of a Trapezoid The median of a trapezoid is the segment that joins the midpoint of the legs. It is also called the midline of a trapezoid. Definition Altitude of a Trapezoid An altitude of a trapezoid is any segment from a point on one base perpendicular to the line containing the other base.
7 Theorem The midline of a trapezoid is parallel to its bases, and its length is half the sum of the lengths of the bases. Let's Practice: Direction: Using the Theorem 4.5.1, find the following: Given: TONE is a trapezoid with median RS. 1. If TO = 14 and EN = 18, find RS. 2. If TO = 3x 8, RS = 15, and EN = 4x + 10, find x. Theorem The base of an isosceles trapezoid are congruent. Theorem If the base angles of a trapezoid are congruent, then the trapezoid is isosceles. Theorem 4.5.4
8 The diagonals of an isosceles trapezoid are congruent. Theorem If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. Let's Practice: Direction: Write a twocolumn proof for each of the following
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