Download Unit 2B Parallelograms - Clark Magnet High School

Unit 2B Parallelograms
Day 5 Tuesday, Nov 8, 2016
Agenda 11/8/2016
1) Bulletin
2) Pythagorean Problems - what did you come
up with?
3) Unit 2B Review Part 1
4) Unit 2B Review Part 2
5) Thursday Unit 2B test (45mins).
Pythagorean theorem problem
A. We know that x + y + z = 180° since they form
a straight angle (line).
We know that the four right triangles are
congruent so they can be labelled with the
angles x and y and the right angle.
So since the sum of the 3 angles in a triangle is
always 180°, we know that x + y = 90°
And so by substitution (90°) + y = 180°, or y= 90°
A. continued.
So the inside quadrilateral has 4 congruent sides
and 4 right angles, so it must be a square.
B. 1) Determine the area of the large square
using the length of its sides.
Area = length x width
A = (a + b)(a + b)
A = a2 + 2ab + b2
B. 2) Determine the area of the large square as
the sum of the areas of the four triangles and the
inside square.
Area = 4 (½ ab) + c(c)
A = 2ab + c2
C. Use results of Parts A and B to demonstrate
that the Pythagorean theorem is true.
From 1 and 2 above we have 2 equations for the
area of the same large square.
A = 2ab + c2
A = a2 + 2ab + b2
So by substitution 2ab + c2 = a2 + 2ab + b2
C. Use results of Parts A and B to demonstrate
that the Pythagorean theorem is true.
So by substitution 2ab + c2 = a2 + 2ab + b2
If we Subtract 2ab from both sides of this
equation
We get:
c 2 = a2 + b2
C.
We get:
c 2 = a2 + b2
c
b
a
Which is how we write the Pythagorean theorem
for a right triangle labelled as in the original
diagram.
Possible extension for class activity.
How did you finish labelling the diagram to help
you write an alternative proof of the
Pythagorean theorem?
White board
c2 = 4 (½ ab) + (b - a)2
c2 = 2ab + (b - a)(b - a)
c2 = 4 (½ ab) + (b - a)2
c2 = 2ab + (b - a)(b - a)
c2 = 2ab + b2 - 2ab + a2
c2 = 4 (½ ab) + (b - a)2
c2 = 2ab + (b - a)(b - a)
c2 = 2ab + b2 - 2ab + a2
c2 = b2 + a2
Fill in the blanks with sometimes, always or
never. State your reasons.
1. A square is always a rhombus.
Reasons: A rhombus is a quadrilateral with four
congruent sides and diagonals that are
perpendicular and bisect each other (and bisect
a pair of opposite angles). A square has all these
same properties, so it can always be classified
as a rhombus.
(pg 351)
Fill in the blanks with sometimes, always or
never. State your reasons.
2. The diagonals of a parallelogram are
sometimes congruent.
Reasons:
Fill in the blanks with sometimes, always or
never. State your reasons.
2. The diagonals of a parallelogram are
sometimes congruent.
Reasons:
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
If a parallelogram is a rectangle then the
diagonals are congruent.
Parallelograms are only rectangles sometimes.
Fill in the blanks with sometimes, always or
never. State your reasons.
3. A quadrilateral with (only) one pair of
congruent sides and (different) one pair of
parallel sides is never a parallelogram.
Reasons: Would be an isosceles trapezoid.
However, it might be a parallelogram if one pair
of opposites sides was parallel and that pair
were also congruent.
If one pair of opposite sides of a quadrilateral
is both parallel and congruent, then the
quadrilateral is a parallelogram. P 334
Fill in the blanks with sometimes, always or
never. State your reasons.
4. The diagonals of a parallelogram always
bisect each other. p 335
Reasons: This is one of the theorems we use
when we want to prove a quadrilateral is a
parallelogram.
Fill in the blanks with sometimes, always or
never. State your reasons.
5. A quadrilateral with 4 congruent sides is
sometimes a square.
Reasons: A rhombus is also a quadrilateral with
4 congruent sides.
6. Give 2 facts about a rectangle that
are not true for all parallelograms.
a)
b)
HINT: USE THE TABLE ON PAGE 62 OF
YOUR WORKBOOK TO WRITE SOME
CORRECT RESPONSES THAT YOU WILL
BE ABLE TO REMEMBER.
7. Give 2 facts about a rhombus that
are not true for all parallelograms.
a)
b)
Use the table on page 62 to help you find facts
that are true for rhombi but not for all
parallelograms.
Complete these theorems:
8. If one pair of opposite sides of a quadrilateral
are both ___________ and ___________, then
it is a parallelogram.
9. Each pair of base angles of a(n)
____________ are congruent.
Complete these theorems:
8. If one pair of opposite sides of a quadrilateral
are both parallel and congruent, then it is a
parallelogram.
9. Each pair of base angles of a(n) isosceles
trapezoid are congruent.
10. Quadrilaterals with diagonals that
are perpendicular.
10. Quadrilaterals with diagonals that
are perpendicular.
kite
rhombus
square
11. Given WXYZ is a parallelogram,
and angle XWZ is congruent to angle
YRZ
Prove: Line segment YR is congruent to line
segment XW.
Draw the given information on the diagram first.
Unit 2B Review Part 2
For Additional practice see the
Mid-Chapter Quiz on page 347 of your textbook
Questions
6, 7, 8, 10, 11, 14, 15.
Homework
Continue to Practice for the 2B test on
Thursday.
I will post some worked answers in the google
classroom tomorrow after school.