## Monday, February 25, 2008

### Geometry Proofs - unit 2 Triangles

Triangles – Proofs – Unit 2
1. Δ ABC is congruent to Δ ABC , this is by the Reflexive Postulate
2. if Δ ABC is congruent to Δ DEF then Δ DEF is congruent to Δ ABC, this is by the Symmetric Postulate
3. if Δ ABC is congruent to Δ DEF and Δ DEF is congruent to Δ GHI, Then Δ ABC is congruent to Δ GHI by the Transitivity Postulate

5 ways to show Triangle Congruence:
1. SAS = SAS Congruence Postulate - if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and angle B = angle E, then ΔABC = ΔDEF.

2. SSS = SSS Congruence Postulate - if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and AC = DF, then ΔABC = ΔDEF.

3. ASA = ASA Congruence Postulate - if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if angle A = angle D, AB = DE, and angle B = angle E, then Δ ABC = Δ DEF.

4. AAS = AAS Congruence Postulate - if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
Example: Given Δ ABC and Δ DEF,
if angle A = angle D, angle C = angle F, and BC = EF , then Δ ABC = Δ DEF.

5. HL = HL Congruence Postulate - if the leg and hypotenuse of one right triangle is congruent to the corresponding leg and hypotenuse of another right triangle, then the two triangles are congruent by hypotenuse - leg postulate.
Example: Given right Δ ABC and right Δ DEF, if angle B and angle E are both right angles and leg AB = leg DE and hypotenuse AC = hypotenuse DF, then Δ ABC = Δ DEF.

I) Vocabulary:

A) When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.
Example: Given Δ ABC is congruent to Δ PQR then we know
1) angle A = angle P, angle B = angle Q, and angle C = angle R
2) AB = PQ, BC = QR, and AC = PR By Corresponding Parts of Congruent Triangles are Congruent (CPCTC) - which means that if 2 triangles are congruent, then their corresponding parts are congruent

Make sure that you list the corresponding angles in the same order with the triangle congruence.
Example: ΔABC = ΔDEF is not the same as ΔABC = ΔEFD because
ΔABC = ΔDEF has angle A = angle D, angle B = angle E and angle C = angle F
while ΔABC = ΔEFD has angle A = angle E, angle B = angle F and angle C = angle D

B) Third Angles Theorem - if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

C) Reflexive Postulate - Every triangle is congruent to itself

D) Symmetric Postulate - If ΔABC = ΔDEF, then ΔDEF = ΔABC

E) Transitive Postulate - If ΔABC = ΔDEF and ΔDEF = ΔJKL, then ΔABC = ΔJKL

Theorems to remember:

If 2 angles of one triangle are congruent then the sides opposite are congruent.
If 2 sides of one triangle are congruent then the angles opposite are congruent.
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Practice Proofs:

## Friday, February 15, 2008

### Precalculus Unit 8 - chapter 6.3, 6.4, 10.5, 10.6, 10.7

PRECALCULUS
Unit 8
Vectors and Parametric/Polar Equations

Section 6.3A HW# 53; Pg.453; #1 – 45 by 4’s (1, 5, 9, …)

Section 6.3B; HW # 54; *Pg.453; #3, 11, 19, 23, 31, 39, 43
Pg.454; #49, 53, 57, 61, 65, 69, 73, 74, 82

Section 6.4; HW # 55; *Pg. 454; #51, 55, 59, 63, 67, 71,
Pg. 464; #1, 5, 9, 13, 17, 21, 25, 29, 33, 37
Quiz on Sections 6.3A – 6.3B Next Class

Section 10.5; HW# 56; *Pg.464; #3, 7, 11, 15, 19, 23, 27, 31
Pg.736; #1, 5, 9, 13, 21, 29, 41, 47

Section 10.6; HW# 57; *Pg.736; #3, 7, 11, 17, 27, 43
Pg.743; #5 – 41 by 4’s, 45, 50, 53 – 57 odds
Quiz on Sections 6.4 – 10.5 Next Class

Section 10.7; HW# 58; *Pg.743; #7, 11, 15, 23, 27, 39, 47, 59
Pg.752; #21 – 33 odds, 55, 59

Review HW #59
Pg.480; 39, 43, 47, 51, 59, 69, 79, 83, 85
Pg.762; 47, 49, 59, 65, 69, 73, 75, 81, 85, 89

HW #60; Unit 8 Test

## Tuesday, February 5, 2008

### Unit One for Proofs - Angles and Lines

If a = b and c = d, then a + c = b + d

The Partition Postulate:

AB + BC = AC

These are two different concepts. Let's try a proof:

As you can see, in step 2, we added equal quantities to each other. This is Addition Postulate.
But in step 3, we added a part plus a part equals a whole so this is Partition.
Let's try another example:
Here we used the Subtraction Postulate: subtracting equal quantities from equal quantities.
In step 3, we had the whole minus a part equals a part. This is still Partition Postulate.