1. Ariella Lindenfeld & Nikki Colona
Math Project
Ariella Lindenfeld & Nikki Colona

2. 5.1 - Parallelograms
EFGH
EF = HG;FG=EH 1. H G 3 1 2.
EFGH
<E = <G 4 2 3.
EFGH
<3 = <4 E F
A parallelogram is a quadrilateral with both pairs of opposite parallel sides.
Theorem 5-1: Opposite sides of the parallelogram are congruent.
Theorem 5-2: Opposite angles of a parallelogram are congruent.
Theorem 5-3: Diagonals of a parallelogram bisect each other.

3. Example: 38 X y 7
120
X equals 120 degrees because of OAC which says that Opposite Angles are congruent
Y equals 22 because it supplementary to 120
So 60 minus 38 is 22

4. 5.2 – Proving 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 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 bisects each other, then the quadrilateral is a parallelogram. T
< S 3 2
<< M 1
<< 4
< Q R

5. 5.3 – Theorems Involving Parallel Lines
Theorem 5- 8: If two lines are parallel, then all points on the line are equidistant from the other line. A B
> L
> M C D
AB=CD

6. 5.3 continued…
Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A X B 1 2 3 Y C 4 5 Z
AX II BY II CZ and AB = BC
XY = YZ

7. Section 3 – theorems Involving Parallel Lines
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
Is parallel to the third side
Is half as long as the third side

8. Example: 2(4x+3) = 13x+1 8x+6=13x+1 5=5x X=1 5y+7 = 6y +3 4=y X=1, Y=4

9. 5-4 Special parallelograms
Rectangle  is a quadrilateral with four right angles.
Therefore, every rectangle is a parallelogram.
Rhombus  A quadrilateral with four congruent sides.
Therefore, every rhombus is a parallelogram.

10. 5-4 Special parallelograms
Square  a quadrilateral with four right angles and four congruent sides.
Therefore, every square is a rectangle, a rhombus, and a parallelogram
*** since rectangles, rhombuses, and squares are parallelograms, they have all the properties of parallelograms.

11. 5-4 Special parallelograms
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 are perpendicular

12. 5-4 Special parallelograms
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

13. 5-5 Trapezoids
Trapezoid a quadrilateral with exactly one pair of parallel sides
Bases  the parallel sides
Legs  the other sides
Isosceles Trapezoid  a trapezoid with congruent legs
Base angles are congruent

14. 5-5 Trapezoids Theorem 5-18 Theorem 5-19
Base angles of an isosceles trapezoid are congruent
*** Median (of a trapezoid )the segment that joins the midpoints of the legs
Theorem 5-19
The Median of a Trapezoid
Is parallel to the bases
Has a length equal to the average of the base lengths

15. Practice Questions: 1.
4x-y
7x-4y 5 9 2. z X y 75

16. Practice Question 42 3. 26
3x-2y
4x+y

17. Proof: Statement Reason Rectangle QRST RKST JQST 1.Given KS =RT
JT = QS
OSC
3. RT =QS DB
4. JT = KS
4. substitution T S J Q R K