The Quadrilateral Family Tree Friday, 1/7/11

1. TRAPEZOID 1. Four-sided polygon Q UADRILATERAL 1. Opposite sides are congruent 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supplementary PARALLELOGRAM 2. Properties of a rhombus 1. Properties of a rectangle S QUARE 2. Four right angles 3. Diagonals are congruent 1. All properties of a parallelogram R ECTANGLE 1. Properties of a parallelogram 2. All sides are congruent 3. Diagonals are perpendicular 4. Diagonals bisect opposite angles R HOMBUS K ITE I SOSCELES T RAPEZOID

Trapezoid A quadrilateral with exactly one pair of parallel sides

1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle S QUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS Opposite sides are congruent All properties of a rhombus

Example: Find m  C. C D m  C =114°

Trapezoid If the legs are congruent, the trapezoid is an isosceles trapezoid

1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Q UADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle SQUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are R HOMBUS 1. Legs are congruent ISOSCELES TRAPEZOID 1. Opposite sides are congruent All properties of a rhombus

Isosceles Trapezoid

In an isosceles trapezoid, the base angles are congruent

1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Q UADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle SQUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are R HOMBUS 1. Legs are congruent 2. Base <s are congruent 3. ISOSCELES TRAPEZOID 1. Opposite sides are congruent All properties of a rhombus

Isosceles Trapezoid What should be true about the diagonals of an isosceles trapezoid?

Isosceles Trapezoid What should be true about the diagonals of an isosceles trapezoid?

1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Q UADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle S QUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are R HOMBUS 1. Legs are congruent 2. Base <s are congruent 3. Diagonals are congruent ISOSCELES TRAPEZOID 1. Opposite sides are congruent All properties of a rhombus

Isos.  trap. s base  Same-Side Int. s Thm. Def. of  s Substitute 49 for mE. mF + mE = 180° E  H mE = mH mF = 131° mF + 49° = 180° Simplify. Check It Out! Example 3a Find mF.

Check It Out! Example 3b JN = 10.6, and NL = Find KM. Def. of  segs. Segment Add Postulate Substitute. Substitute and simplify. Isos.  trap. s base  KM = JL JL = JN + NL KM = JN + NL KM = = 25.4

Example 4A: Applying Conditions for Isosceles Trapezoids Find the value of a so that PQRS is isosceles. a = 9 or a = –9 Trap. with pair base s   isosc. trap. Def. of  s Substitute 2a 2 – 54 for mS and a for mP. Subtract a 2 from both sides and add 54 to both sides. Find the square root of both sides. S  PS  P mS = mP 2a 2 – 54 = a a 2 = 81

Trapezoid Midsegment The segment formed by the midpoints of the legs

Trapezoid Midsegment Its length is the average of the two bases.

Trapezoid Midsegment Its length is the average of the two bases.

Example 5: Finding Lengths Using Midsegments Find EF. Trap. Midsegment Thm. Substitute the given values. Solve. EF = 10.75

Check It Out! Example 5 Find EH. Trap. Midsegment Thm. Substitute the given values. Simplify. Multiply both sides by = 25 + EH Subtract 25 from both sides. 13 = EH = ( 25 + EH ) 2

Kite A quadrilateral with exactly two pairs of congruent consecutive sides

1. One pair of parallel sides T RAPEZOID 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle S QUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS K ITE 1. Legs are congruent 2. Base <s are congruent 3. Diagonals are congruent I SOSCELES T RAPEZOID 1. Opposite sides are congruent pairs of congruent consecutive sides 2. All properties of a rhombus

Kite Diagonals are perpendicular

1. One pair of parallel sides T RAPEZOID 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle S QUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS K ITE 1. Legs are congruent 2. Base <s are congruent 3. Diagonals are congruent I SOSCELES T RAPEZOID 1. Opposite sides are congruent 2. Diagonals are _ pairs of congruent consecutive sides 2. All properties of a rhombus

Kite Exactly one pair of opposite angles are congruent

1. One pair of parallel sides T RAPEZOID 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. P ARALLELOGRAM 1. All properties of a rectangle S QUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 2. All properties above 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS K ITE 1. Legs are congruent 2. Base <s are congruent 3. Diagonals are congruent I SOSCELES T RAPEZOID 1. Opposite sides are congruent 2. Diagonals are _ 3.One pair of opposite <s congruent 1. 2 pairs of congruent consecutive sides 2. All properties of a rhombus

Example: Find KL. Hint* JM bisected by diagonal KL KL = 44

Check It Out! Example 2b In kite PQRS, mPQR = 78°, and mTRS = 59°. Find mQPS. Hint* Non congruent <s bisected by diagonal (<Q & <S) Kite  one pair opp. s   Add. Post. Substitute. QPS  QRS mQPS = mQRT + mTRS mQPS = mQRT + 59° mQPS = 51° + 59° mQPS = 110°

1. Exactly one pair of parallel sides TRAPEZOID 1. Four-sided polygon Q UADRILATERAL 1. Opposite sides are congruent 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supplementary PARALLELOGRAM 2. Properties of a rhombus 1. Properties of a rectangle S QUARE 2. Four right angles 3. Diagonals are congruent 1. All properties of a parallelogram R ECTANGLE 1. Properties of a parallelogram 2. All sides are congruent 3. Diagonals are perpendicular 4. Diagonals bisect opposite angles R HOMBUS 1. Exactly two pairs of congruent sides 2. Exactly one pair of congruent opposite angles 3. Diagonals are perpendicular K ITE 1. Legs are congruent 2. Diagonals are congruent 3. Two pairs of congruent base angles I SOSCELES T RAPEZOID