1. Unit 2 - Linear Programming, Sequences, and Theorems and Proofs
By Anna Thomas and Mary Kolbas

2. L.P. - Graphing Systems of Inequalities
To graph a system of inequalities, graph as if it is an equation, then shade either less than or more than.
These may also be listed as constraints, but they serve the same purpose - to create the edges of the feasible region.
y ≤ 3x-5
y < (-½)x +4
Both shade less than, so the feasible region is where the inequalities overlap.
Remember: When it is < or >, the line should be dotted!

3. L. P. - Systems Can Create Figures
Take the example…
It creates a parallelogram!
Make sure you shade the right direction:
Shade down (for less than)
Shade up (for more than)
The only spot they all shade (all colors overlap) is the spot colored black.
x ≤ 2
x ≥ -3
y ≤ 2x + 2
y ≥ 2x-1

4. L. P. - Max and Mins of Systems of Inequalities
The maximum and minimum of a system of inequalities is always on the edge of the feasible region.
Maximize P = 5x + 2y
Constraints:
y > 0
x ≥ 0
y ≤ -x + 10
y ≤ 2x + 1

5. L.P. - Parallel and Perpendicular Lines
If equations have the same slope, they will be parallel
If equations have negative reciprocals of each other, they will be perpendicular
These are lines are parallel.
They will never intersect.
These are lines are perpendicular.
They intersect at 90°

6. L. P. & Proofs - Parallel lines cut by a transversal
∠1 ⩭ ∠4 because they are vertical angles.
∠1 ⩭ ∠5 because they are corresponding angles.
∠1 ⩭ ∠8 because they are alternate exterior angles.
∠4 ⩭ ∠5 because they are alternate interior angles.
Remember: Vertical, corresponding, and alternate angles will be ⩭

7. L. P. & Proofs - Parallel lines cut by a transversal
∠1 and ∠ 2 are supplementary because they add up to 180° (1 straight angle)
∠1 and ∠ 3 are supplementary because they add up to 180° (1 straight angle)
∠3 and ∠5 are supplementary because they are same-side interior angles. (∠3 and ∠6 are ⩭ because Opp. Int. ∠s, then solve)
∠1 and ∠7 supplementary because they are same-side exterior angles.

8. Sequences - Arithmetic
Arithmetic Sequences follow a pattern of adding or subtracting.
Arithmetic Sequences are linear.
d is the common difference, or how much is being added or subtracted
Recursive:
a1 = starting value
an = an-1+ d
Explicit:
an = a1+ (n-1)d

9. r is the common ratio, or the factor of growth (multiply or divide)
Sequences - Geometric
Geometric Sequences follow a pattern of multiplying or dividing.
Geometric Sequences are exponential.
r is the common ratio, or the factor of growth (multiply or divide)
Recursive:
a1 = starting value
an = an-1 * r
Explicit:
an = a1 (r)n-1

10. Sequences - Arithmetic Examples
3, 11, 19, 27, 35,...
d = 8
Explicit:
an = 3 + (n-1)8 If n = 6
a6 = 3 + (6-1)8
a6 =43
Recursive:
a1 = 3 If n = 6
an = an a6 = a6-1+ 8
a6 = 43

11. Sequences - Geometric Examples
1/2, 1, 2, 4, 8,...
r = 2
x2 x2 x x2
Recursive:
a1 = ½ If n = 6
an = an-1 * 2 a6 = a6-1 * 2
a6 = 16
Explicit:
an = ½ (2)n-1 If n = 6
a6 = ½ (2)6-1
a6 =16

12. Proofs - Triangle Congruence
SSS (Side-Side-Side) - If all three sides of both triangle are the same, then they are ⩭.
SAS (Side-Angle-Side) - If two sides and an angle in between are ⩭ in both triangles, the triangles are ⩭.
ASA (Angle-Side-Angle) - If two angles and a side in between are ⩭ in both triangles, the triangles are ⩭.
AAS (Angle-Angle-Side) - If two angles and one side are ⩭ in both triangles, the triangles are ⩭.
HL (Hypotenuse Leg or SSA) -If the hypotenuse and leg (two sides) of a right triangle are congruent to another right triangle, they are ⩭. (because of the right angle in all right triangles, there is a congruent angle)

13. Proofs - Triangle Similarity
AA Similarity - If two angles of one triangle are ⩭ to two angles of the other, the triangles are similar.
SSS for Similarity - If all 3 sides are proportional to another triangle’s sides, the triangles are similar.
SAS for Similarity - If an angle is congruent to a corresponding angle and if the lengths of two sides are proportional, the triangles are similar. X y 2x 2y

14. Proofs - Theorems and Rules
Mid-segment (mid-line) Theorem - The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Line DE is bisecting side AB. (Lines AD and DB are equal lengths)
Line DE is bisecting side AC (Lines AE and EC are equal lengths)

15. Proofs - Sum of Two Sides
The sum of the lengths of any two sides of a triangle must be greater than the third side. (Otherwise, the triangle won’t “close”!)
> 50
> 29
> 44
3 + 4 > 5
4+ 5 > 3
3 + 5 > 4
> 10
< 70
> 55

16. Proofs - Longest Side
In a triangle the longest side is across from the largest angle and vise versa.
You may remember using this when proving that tangent lines for right angles! 45
Longest side 90 45

17. Proofs - Base Angle
Base Angle Theorem - If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse - If two angles of a triangle are congruent, the side opposite these angles are congruent.

18. Proofs - Parallelogram Proofs
Statement
Reason
ABCD is a parallelogram
Given
AB ∥ DC
Def. of parallelogram
AD ∥ BC
∠BAC ⩭ ∠ACD
Alt. Int. ∠s are ⩭
∠BCA ⩭ ∠CAD
AC = AC
Reflexive property
△ADC ⩭ △CBA
By ASA
AB ⩭ DC
By CPCTC
AD ⩭ BC
Property 1 - Given a parallelogram, prove opposite sides are congruent.

19. Proofs - Parallelogram Proofs
Property 2 - Given a parallelogram, prove opposite angles are congruent.
A B
D C
You can also prove this by proving both triangles are congruent through ASA and then using CPCTC to prove the angles are ⩭
Statement
Reason
ABCD is a parallelogram
Given
∠A + ∠D = 180°
Same side interior angles are supplementary
∠D + ∠C = 180°
∠A + ∠D = ∠D + ∠C
Substitution
∠A ⩭ ∠C
Elimination
∥ly ∠B ⩭ ∠D “

20. Proofs - Parallelogram Proofs
Statement
Reason
ABCD is a parallelogram
Given
AB ∥ DC ; AD ∥ BC
Defn. of parallelogram
AB ⩭ DC ; AD ⩭ BC
Property 1
∠DAC ⩭ ∠BCA
Alt. Int. ∠s
∠BDC ⩭ ∠DBA
∠DCA ⩭ ∠BAC
∠ADB ⩭ ∠DBC
△ADO ⩭ △COB
ASA
△AOB ⩭ △COD
AO ⩭ OC
CPCTC
DO ⩭ DB
Proofs - Parallelogram Proofs
Property 3 - Given a parallelogram, prove that the transversal lines bisect each other to create two even line segments.

21. Proofs - Parallelogram Proofs
Property 4 - Given a parallelogram, prove angles on the same side (that share a side) are supplementary.
Statement
Reason
ABCD is a parallelogram
Given
AB ∥ DC ; AD ∥ BC
Defn. of parallelogram
m∠BAD + m∠ADC = 180°
Same Side Int. ∠s
∠A and ∠D are supplementary
Defn. of supplementary

22. Proofs - Converse
A converse is when you instead are given the “answer” in order to prove the “given”. For example, for Property 2, it is…
Same with Base Angle Theorem...
PROPERTY 2
Given the parallelogram,
Prove opposite angles are ⩭
CONVERSE OF PROPERTY 2
Given opposite angles are ⩭,
Prove the figure is a parallelogram
BASE ANGLE THEOREM
Two sides are ⩭,
So you know the angles opposite these sides are ⩭
CONVERSE BASE ANGLE THEOREM
Two angles are ⩭,
So you know the sides opposite these angles are ⩭

23. Thanks For Listening!
We hope this review was helpful on refreshing your memory for the end of year tests! Good luck with the homework! ^-^
HINT: Some of the proofs on the HW can be proven using converse of alternate interior angles and Property 2!
BONUS: Prove the square is a circle using pythagorean’s theorem
(Answer on the next slide)

24. Answer: The proof is trivial, and is left as an exercise.