By Ethan Arteaga and Alex Goldschmidt Chapter 5 By Ethan Arteaga and Alex Goldschmidt

5-1 :-: Properties of a Parallelogram :-: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. The symbol for a parallelogram is

Theorem 5-1 Opposite sides of parallelograms are congruent. Given: EFGH Prove: EF ≅ HG ; FG ≅ EH E H F G

Theorem 5-1 Proving Opposite sides of parallelograms are congruent. H 4 Given ∠1 ≅ ∠2 ∠3 ≅ ∠4 3. ΔEHG ≅ ΔGFE 4. EH ≅ GF HG ≅ EF 1. EFGH 2. AIA 3. ASA 4. CPCTC 2 1 E 3 F

Theorem 5–2 & 5-3 Opposite angles of a parallelogram are congruent. Diagonals of a parallelogram bisect each other. E H 3 ∠1 ≅ ∠2 ∠3 ≅ ∠4 2 1 4 F G

Example 1 10 x y 8 a 62 b Solve for all the variables X = 62 because of of theorem 5-2 Y = 118 by subtracting 180 by 62 because you can derive a triangle by cutting the parallelogram in half. 10 x y 8 a 62 b

Practice Solve for all the variables, assume the quadrilaterals are parallelograms. 15 80 30 y 50 8 b a a x 70 9 11 33 b

Theorem 5-4 This theorem proves that a quadrilateral is a a parallelogram. It states if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-5 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Theorem 5-6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-7 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Example Is the following quadrilateral a parallelogram? Yes because opposite angles are congruent.

Theorem 5-8 If two lines are parallel, then all points on one line are equidistant from the other line.

Theorem 5-9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Theorem 5-10 & 11 A line that contains the midpoint of one side o f a triangle and is parallel to another side passes through the midpoint of the third side. Any segment of a triangle that joins the midpoints of two sides of a triangle is not only parallel to the third side but is half as long as the third side as well.

Example Solve for a a = 6 because since the segment joins the midpoints of the sides it is half of the third side (12). a 12

Practice Find the values of x and y. The red segment is the midpoint of the triangle. 4y + 2 3x + 5 7(y-1) 12x-8

Theorem 5-12 & 5-16 The diagonals of a rectangle are congruent. If a parallelogram has a right then the parallelogram is a rectangle.

Theorem 5-13 & 5-14 The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 5-17 If two consecutive sides of a parallelogram are congruent then parallelogram is a rhombus.

Theorem 5-15 The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices

Trapezoids A quadrilateral with exactly one pair of parallel sides is called a trapezoid. The parallel sides are called the bases. The other sides are legs. Base Leg Leg Base

Theorem 5-18 In another type of trapezoid, an isosceles trapezoid, the base angles and legs are congruent.

Theorem 5-19 A median of a trapezoid is the segment that joins the midpoints of the legs. This median is parallel to the bases and has a length equal to the average of the base lengths. ½(b1+b2) will give you the median length.

Example 8 12 Find the length of the median. The median is 10 because ½(8+12) = ½(20) = 10 8 12

Practice Solve for x, assume the median. 18 8 x + 4

End - Credits Power Point Directed by Ethan Arteaga & Alex Goldschmidt.