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**Quadrilaterals**

**SECTION **

**5-1**

**Properties of **

**Parallelograms**

**• Quadrilateral a closed plane figure that has four sides**

**• Opposite sides two sides that do not share a common endpoint**

**• Opposite angles two angles that do not share a common side**

**• Parallelogram a quadrilateral with both pairs of opposite sides parallel.**

**THEOREM 5 -1**

**• Opposite sides of a parallelogram are congruent**

**THEOREM 5 - 2**

**• Opposite angles of a parallelogram are congruent**

**THEOREM 5 - 3**

**• Diagonals of a parallelogram bisect each other**

**SECTION **

**5-2**

**Ways to Prove that **

**Quadrilaterals Are **

**Parallelograms**

**THEOREM 5 - 4**

**• If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.**

**THEOREM 5 - 5**

**• If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.**

**THEOREM 5 - 6 **

**• If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.**

**THEOREM 5 - 7**

**• If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.**

**Ways to Prove that Quadrilaterals **

**Are Parallelograms**

**1.**

**Show that both pairs of opposite sides are parallel.**

**2.**

**Show that both pairs of opposite sides are congruent**

**3.**

Show that one pair of opposite sides are both congruent and parallel .

**Ways to Prove that Quadrilaterals **

**4. **

**Are Parallelograms**

**Show that both pairs of opposite angles are congruent.**

**5. Show that the diagonals bisect each other**

**SECTION **

**5-3**

**Theorems Involving **

**Parallel Lines**

**THEOREM 5 - 8**

**• If two lines are parallel, then all points on one line are equidistant from the other line.**

**THEOREM 5 - 9**

**• If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.**

**THEOREM 5 - 10**

**• A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.**

**THEOREM 5 - 11**

**• The segment that joins the midpoints of two sides of a triangle**

**(1) is parallel to the third side;**

**(2) is half as long as the third side**

**SECTION **

**5-4**

**Special Parallelograms**

**• Rectangle is a quadrilateral with four right angles.**

**• Square is a quadrilateral with four right angles and four sides of equal length.**

**• Rhombus is a quadrilateral with four sides of equal length.**

**THEOREM 5 - 12**

**• The diagonals of a rectangle are **

**congruent**.

**THEOREM 5 - 13**

**• The diagonals of a rhombus are perpendicular.**

**THEOREM 5 - 14**

**• Each diagonal of a rhombus bisects two angles of the rhombus**

**THEOREM 5 - 15**

**• The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices**

**THEOREM 5 - 16**

**• If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.**

**THEOREM 5 - 17**

**• If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.**

**SECTION **

**5-5**

**Trapezoids**

**• Trapezoid a quadrilateral with exactly one pair of parallel sides.**

**• Bases the sides that are parallel in a trapezoid. **

**• Legs the nonparallel sides of a trapezoid.**

**• Base angles angles that share a base. **

**Trapezoids have two pairs of base angles.**

**• Isosceles Trapezoid a trapezoid with legs of equal length.**

**THEOREM 5 - 18**

**• Base angles of an isosceles trapezoid are congruent .**

**• Median the segment that joins the midpoints of the legs.**

**THEOREM 5 - 19**

**The median of a trapezoid**

**1.**

**is parallel to the bases;**

**2.**

**length is equal to the **

**½(sum of the two bases**)

• END Chapter 5