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Lesson 6-1

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Warm-up

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Solve the following triangles using the Pythagorean Theorem a 2 + b 2 = c 2 9 5 12 9 8 8√3

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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

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

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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)

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

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

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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.

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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.

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

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

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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.

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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.

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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.

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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.

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Let’s Practice Find “x”.

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Let’s Practice

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