Understand, use and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite. MM1G3d

Vocabulary Trapezoid: a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A B bases D C

Vocabulary Trapezoid: a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. For each of the bases of a trapezoid, there is a pair of base angles, which are the two angles that have that base as a side. A B base angles D C

Vocabulary Trapezoid: a quadrilateral with exactly one pair of parallel sides. The nonparallel sides of a trapezoid are the legs of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. A B legs D C

Vocabulary Trapezoid: a quadrilateral with exactly one pair of parallel sides. The midsegment of a trapezoid is the segment that connects the midpoints of its legs. A B midsegment D C

Theorem If a trapezoid is isosceles, then each pair of base angles is congruent. <A is congruent to <B. <C is congruent to <D. A B D C

Theorem If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A B D C

Theorem A trapezoid is isosceles if and only if its diagonals are congruent. A B AC is congruent to BD, so ABCD is an isosceles trapezoid. D C

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. A B x = AB + CD 2 x D C

Example 1 Find the missing angle measures for the isosceles trapezoid ABCD. <A and <B are base angles, so they are congruent. m<B = 92 ABCD is a quadrilateral so the angles add up to 360°. <C and <D are base angles, so x + x + 92 + 92 = 360 2x +184 = 360 – 184 – 184 2x = 176 2 2 A B 92° 92° x x C D

Example 1 Find the missing angle measures for the isosceles trapezoid ABCD. <A and <B are base angles, so they are congruent. m<B = 92 ABCD is a quadrilateral so the angles add up to 360°. <C and <D are base angles, so x + x + 92 + 92 = 360 2x +184 = 360 – 184 – 184 2x = 176 2 2 x = 88 A B 92° 92° 88° 88° C D

Example 2 EF is the midsegment of trapezoid ABCD. Find x. x = 12 + 20

Example 3 EF is the midsegment of trapezoid ABCD. Find x. 2 * – 16 – 16 x = 18 * 2 A x B 17 E F D C 16

Vocabulary kite: a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorem If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Example 4 EFGH is a kite. Find m<F. E In a kite, one pair of opposite angles is congruent. Therefore, <H is congruent to <F. So, m<F = 83°. 124° H 83° F G

Assignment Textbook: p.325-326 (1-14) & p331 (1-6)