Geo Review Chapter 8
Lesson 8.l: Angle measures in polygons
 The interior angles of a polygon ---(n-2) ∙ 180 (n = # of sides)
 The sum of the interior angles of a quadrilateral = 360°
 The sum of the exterior angles of all convex polygons = 360°
Lesson 8.2: Properties of Parallelograms
 A parallelogram is a quad with both pairs of opposite sides parallel
 Theorems:
o A quad is a parallelogram if its opposite sides are congruent
o A quad is a p-gram if its opposite angles are congruent
o Supplementary Angles Theorem: The consecutive angles of a p-gram are
supplementary
o Diagonals Theorem: The diagonals of a p-gram bisect each other
Lesson 8.3: show that a Quad is a Parallelogram
 Theorems:
o If both pairs of opposite sides of a quad are congruent, then the quad is a p-gram
o If both pairs of opposite angles of a quad are congruent, then the quad is a pgram
o If one pair of opposite sides of a quad are congruent and parallel, then the quad is
a p-gram
o If the diagonals of a quad bisect each other then the quad is a p-gram
 5 ways to show that a quadrilateral is a parallelogram
o 1. Show both pairs of opposite sides are parallel
o 2. Show both pairs of opposite sides are congruent
o 3. Show both pairs of opposite angles are congruent
o 4. Show one pair of opposite sides are congruent and parallel (in coordinate
geometry, show they are parallel by comparing slopes m =
o 5. Show the diagonals bisect each other
Lesson 8.4: Properties of Rhombuses, Rectangles and Squares
 A rhombus is a p-gram with 4 congruent sides
 A rectangle is a p-gram with 4 right angles
 A square is a p-gram with 4 congruent sides and 4 right angles
 Corollaries:
𝑦2 −𝑦1
𝑥2 −𝑥1
o A quad is a rhombus if and only if it has 4 congruent sides
o A quad is a rectangle if and only if it has 4 right angles
o A quad is a square if and only if it is has 4 congruent sides and 4 right angles (if it’s
a rhombus and a rectangle
o A p-gram is a rhombus if and only if its diagonals are perpendicular
o A p-gram is a rhombus if and only if each diagonal bisects a pair of opposite
angles
o A p-gram is a rectangle if and only if its diagonals are congruent
 Rectangles:
o Have 4 right angles
o Is a parallelogram:
 Opposite sides are congruent
 Opposite sides are parallel
 Diagonals are congruent
 Diagonals bisect each other
 Opposite angles are congruent
 Opposite angles are supplementary
 Rhombus:
o Has 4 congruent sides
o It’s a parallelogram
 Opposite sides are congruent
 Opposite sides are parallel
 Diagonals are congruent
 Diagonals bisect each other
 Opposite angles are congruent
 Opposite angles are supplementary
Lesson 8.5: Trapezoids and Kites
 Trapezoid:
o Is a quadrilateral with EXACTLY one pair of parallel sides (not both pairs)
 Theorems:
o If a trapezoid is isosceles, then each pair of base angles are congruent
o If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid
o A trapezoid is isosceles if and only if its diagonals are congruent
o Midsegment Theorem for trapezoids: Midsegment = ½(base + base)
 Kite:
o Is a quadrilateral that has 2 sides of consecutive congruent sides, but opposite
sides are not congruent
 Theorems:
o If a quad is a kite, then its diagonals are perpendicular
o If a quad is a kite, then exactly one pair of opposite angles are congruent
Quadrilaterals
4 sided polygon
Parallelgrams
--opp sides are ≅ 𝑎𝑛𝑑 ∥
--opp ∡𝑠 𝑎𝑟𝑒 ≅
--diagonals bisect each
Other
Rectangle
--4 right ∡𝑠
Rhombus
--All sides ≅
--opp ∡𝑠 𝑎𝑟𝑒 ≅
But not 90°
Square
All sides are ≅
4 right angles
Trapezoids
one pair of ∥ sides
Kites
2 consecutive sides ≅
1 pair of opp ∡𝑠 𝑎𝑟𝑒 ≅
Isosceles Trapezoid
both pairs of base ∡𝑠 𝑎𝑟𝑒 ≅
nonparallel sides are ≅
Q