Lesson 6-1

Warm-up

Solve the following triangles using the Pythagorean Theorem a 2 + b 2 = c √3

Warm-up Find the missing point given the following information. 1. Point 1 (3, 8), Point 2 (5, 12), Midpoint (x, y) 2. Point 1 (-2, 5), Point 2 (3, -3), Midpoint (x, y) 3. Point 1 (2, 4), Point 2 (x, y), Midpoint (5, -1) 4. Point 1 (-1, 2), Point 2 (2, y), distance = 5

Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel

Properties of Parallelograms Its opposite sides are congruent Its opposite angles are congruent Its consecutive angles are supplementary (add to 180°) Its diagonals bisect each other. (Cut each other into 2 equal sections)

Let’s Practice Find the value of each variable in the parallelogram.

Let’s Practice Find the value of each variable in the parallelogram.

Types of Parallelograms Rhombus – a parallelogram with four congruent sides. Rectangle – a parallelogram with four right angles. Square – a parallelogram four congruent sides and four right angles. Rhombus Corollary – a quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary – a quadrilateral is a rectangle if and only if it has four right angles. Square Corollary – a quadrilateral is a square if and only if it is a rhombus and a rectangle.

Special Parallelogram Properties If a parallelogram is a rhombus, its diagonals are perpendicular. If a parallelogram is a rhombus, each diagonal bisects a pair of opposite angles. If a parallelogram is a rectangle, its diagonals are congruent.

Let’s Practice Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y.

Let’s Practice Classify the special quadrilateral. Explain your reasoning. Then find the values of x and y.

Other Quadrilaterals Trapezoid – a quadrilateral with exactly one pair of parallel sides. Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Trapezoid Vocabulary Base - the parallel sides are the bases. Base Angles - in a trapezoid, the two angles that have that base as a side. Legs – the non-parallel sides of a trapezoid. Isosceles Trapezoid – a trapezoid where both legs are congruent. Midsegment of a Trapezoid – the segment that connects the midpoints of the legs of a trapezoid.

Trapezoid Properties For an isosceles trapezoid, each pair of base angles is congruent. For an isosceles trapezoid, the diagonals are congruent. Midsegment Theorem for Trapezoids – the midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

Kite Properties Its diagonals are perpendicular Exactly one pair of opposite angles are congruent. The diagonal between the non-congruent angles bisects the diagonal between the congruent angles.

Let’s Practice Find “x”.

Let’s Practice