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The Distance Formula • Used to find the distance between two points: A( x1, y1) and B(x2, y2) d ( x2 x1 ) y2 y1 2 2 You also could just plot the points and use the Pythagorean Theorem!! Find the distance between the two points. Round your answer to the nearest tenth. 1. T(5, 2) and R(-4, -1) 2. A( -2, -3) and B(1, 3) Take a look at example 2, p. 44 Midpoint Formula • Find the midpoint coordinates between 2 points • Find by averaging the x-coordinates and the ycoordinates of the endpoints (x2, y2) (x1, y1) x1 x2 y1 y2 M , 2 2 Find the coordinates of the midpoint QSof 1. Q(3, 5) and S(7, -9) 2. Q( -4, 4) and S(5, -1) Warm-up Draw each figure on graph paper if possible. If not possible explain why. 1. A parallelogram that is neither a rhombus nor a rectangle 2. An isosceles trapezoid with vertical and horizontal congruent sides 3. A trapezoid with only one right angle 4. A trapezoid with two right angles 5. A rhombus that is not a square 6. A kite with two right angles Special Quadrilaterals Parallelogram – A quadrilateral with both pairs of opposite sides parallel. Rhombus – A parallelogram with four congruent sides. Rectangle – A parallelogram with four right angles. Square – A parallelogram with four congruent sides and four right angles. Kite – A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. Trapezoid – A quadrilateral with exactly one pair of parallel sides. Isosceles Trapezoid – A trapezoid whose nonparallel opposite sides are congruent. Theorem 6-1 Opposite sides of a parallelogram are congruent. Theorem 6-2 Opposite angles of a parallelogram are congruent. (Consecutive angles of a parallelogram are supplementary, they are same-side interior angles!) Theorem 6-3 The diagonals of a parallelogram bisect each other. Theorem 6-4 If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. TR=12 find VR QS=10 find VS Find x and the length of the side Find all angle measures 110̇ 120 30 Proving that a quadrilateral is a parallelogram. (both pairs of opposite sides are parallel) Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Proving that a quadrilateral is a parallelogram. (Both pairs of opposite sides are parallel) Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-9 Each diagonal of a rhombus bisects two angles of the rhombus. Theorem 6-10 The diagonals of a rhombus are perpendicular. Theorem 6-11 The diagonals of a rectangle are congruent. Theorem 6-12 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6-13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6-14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Use the properties of rectangles to find all missing angle measures, list the properties you used. 32 6.5: Trapezoids and Kites Objective: To verify and use properties of trapezoids and kites Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to the bases. The length of the midsegment of a trapezoid is half the sum of the lengths of the bases. If you are given the midsegment length and one base: -double the length of the midsegment -subtract the other base. Midsegment (Median) of a Trapezoid Joins the midpoints of the nonparallel sides Is parallel to the bases Its length is ½ the sum of the bases MN || BC MN || AD MN = ½(BC+AD) 86° 3 108° 4 Find the following: EF: mBAD : mCFE : mDCB : DF : AB : Find x: x 8 12 Theorem 6-15 The base angles of an isosceles trapezoid are congruent. Theorem 6-16 The diagonals of an isosceles trapezoid are congruent. Theorem 6-17 The diagonals of a kite are perpendicular. Theorem: The diagonals of a kite are perpendicular. A kite has exactly one pair of opposite, congruent angles. Find the measure of the missing angles. 1 44° 112° 2 What is the sum of the angles in a quadrilateral? TRAPEZOID •The 2 parallel sides are the bases •The 2 non-parallel sides are the legs BASE ANGLES B D LEG LEG A C BASE ANGLES Name the following: Bases: Legs: 2 Pairs of Base Angles: Theorem: The base angles of an isosceles trapezoid are congruent B D A C B D C A m A 3 x mC x 21 Find x. Theorem: The diagonals of an isosceles trapezoid are congruent. EXAMPLE: If BD= 2x+10 and AC=x+15, find x and the length of the diagonals. 2 angles that share a leg are supplementary because they are same-side interior angles. mA mC 180 The measure of angle A= 110. find the measures of the other 3 angles. One side of a kite is 4 cm less than two times the length of another side. The perimeter of the kite is 58 cm. Find the lengths of the sides of the kite. Midsegments of trapezoids • The midsegment of an isosceles trapezoid measures 14 cm. One of the bases measures 24 cm. Find the length of the other base.