M.Doll Name: ____________________________________ Block: _________

Geometry/TrigUnit 3 – Parallel Line and Planes

Notes Packet

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Parallel Lines and Planes Day 1 Notes

Section 3.1: Parallel Lines and Planes NotesParallel Lines: Diagram:

Notation/Symbol:

Skew Lines: Diagram:

Parallel Planes: Diagram:

Theorem 3-1:

Transversal: Diagram:

Pairs of Angles

Corresponding Angles:

Alternate Interior Angles:

Alternate Exterior Angles:

Same Side Interior Angles:

Same Side Exterior Angles:

Day 1 Practice

Complete each statement with the word always, sometimes, or never.

1. Two lines in the same plane are __________ parallel.

2. Two lines in the same plane are __________ skew.

3. Two noncoplanar lines __________ intersect.

4. Two planes __________ intersect.

5. A line and a plane __________ have exactly one point of intersection.

6. If two planes do not intersect, then they are __________ parallel.

Name a) the two lines and b) the transversal that form each pair of angles.

7. 1 and 4

a) _____ , _____ b) _____

8. 6 and 8

a) _____ , _____ b) _____

9. 3 and 2

a) _____ , _____ b) _____

10. 5 and 7

a) _____ , _____ b) _____

Classify each pair of angles as (AI) alternate interior angles, (SSI) same-side interior angles, or (Corr) corresponding angles.

_______11) 2 and 4 _______12) 7 and 12

_______13) 10 and 11 _______14) 5 and 10

_______15) 14 and 15 _______16) 3 and 11

_______17) STX and TXY _______18) WXZ and STZ

_______19) UTZ and YXZ _______20) WXT and XTR

_______21) STX and TXW _______22) URX and RXW

Properties of Parallel Lines Day 2 Notes

HW: pg 75 #1-10 all

Postulate 10: If two parallel lines are cut by a transversal, then __________________________ are ________________.

Theorem 3-2: If two parallel lines are cut by a transversal, then __________________________ are _______________.

Theorem: If two parallel lines are cut by a transversal, then __________________________ are ________________.

Theorem 3-3: If two parallel lines are cut by a transversal, then _________________________ are ________________.

Theorem: If two parallel lines are cut by a transversal, then __________________________ are ________________.

Theorem 3-4: If a transversal is ________________ to one of two parallel lines, then it is ________________ to the other parallel line.

Theorem 3-2 Proof: Theorem Proof:

Given: Given:

Prove: Prove:

Statements Reasons Statements Reasons

Theorem 3-3 Proof: Theorem 3.4 Proof:

Given: Given:

Prove: Prove:

Statements Reasons Statements Reasons

Day 2 Practice

State the postulate or theorem that justifies each statement.

_____________________________________ 1.) 3 7

_____________________________________ 2.) a k

_____________________________________ 3.) m6 + m7 = 180

_____________________________________ 4.) 6 8

_____________________________________ 5.) 4 6

_____________________________________ 6.) 1 2

_____________________________________ 7.) m5 = m7

_____________________________________ 8.) 5 is supp. to 8

Use the diagram to complete the following.

9.) If a || b, name all the angles that must be congruent to 1.

10.) If c || d, name all the angles that must be congruent to 1.

Assume that a || b and c || d.

11.) Name the seven angles that must be congruent to 2.

_____, _____, _____, _____, _____, _____, _____

12.) Name the eight angles that must be supplementary to 2.

_____, _____, _____, _____, _____, _____, _____, _____

13.) If m6 = 60, what are the measures of the other numbered s?

1=____, 2=____, 3=____, 4=____, 5=____, 7=____, 8=____, 9=____,

10=____, 11=____, 12=____, 13=____, 14=____, 15=____, 16=____

14.) If m2 = 80, then m4 = _____ and m8 = _____.

15.) If m9 = 105, then m10 = _____ and m15 = _____.

16.) If m6 = m5 + 30, find m 3. _____

17.) If m15 = m11 – 20, find m14. _____

Day 2 Homework

a b

k

j1

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3 46 57 8

a b

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41516

Use the diagram below to complete each exercise. All justifications must be written in a formal matter. Example – It would be unacceptable to write “Corresponding ∠s are ≅.” You must instead write, “If two ll lines are cut by a trans, then corresponding ∠s are ≅.”

1.) ∠3 ≅ ∠7

Justification: _________________________________

_______________________________________________________________________________________________________________________________________

2.) ∠3 and ∠5 are supplementary

Justification: ___________________________________ __________________________________________________________________________________________________________________________________________

3.) ∠4 ≅ ∠5

Justification: ___________________________________________________________ _____________________________________________________________________.

4.) ∠1 ≅ ∠8

Justification: ___________________________________________________________ _____________________________________________________________________.

5.) m∠7 + m∠8 = 180

Justification: ___________________________________________________________ _____________________________________________________________________.

6.) ∠5 ≅ ∠8

Justification: ___________________________________________________________ _____________________________________________________________________.

Part 2 – Use the diagram to complete each algebra connection problem. You must show all work.

Day 3 Notes/Practice

Find the values of x and y.

1) x = _____ y = _____ 2) x = _____ y = _____

3) x = _____ y = _____ 4) x = _____ y = _____

5) x = _____ y = _____ 6) x = _____ y = _____

x 40 y

70

50

40

x

y

Find the values of x, y, and z.

7) x = _____ y = _____ z = _____ 8) x = _____ y = _____ z = _____

9) x = _____ y = _____ z = _____ 10) x = _____ y = _____ z = _____

Day

Day 4/5 Notes Section 3.3: Proving Lines Parallel Notes

1. Postulate 11:

2. Theorem 3-5:

3. Theorem 3-6:

4. Theorem 3-7:

5. Theorem 3-8:

6. Theorem 3-9:

7. Theorem 3-10:

Theorem 3-5 Proof : Theorem 3-6 Proof:

Given: Given:

Prove: Prove:

Statements Reasons Statements Reasons

Theorem 3-7 Proof: Given:Prove:

Statements Reasons

Practice Problems

What two lines are parallel (if any) according to the given information?

1.) m1 = m4 ___________

2.) m5 + m6 = 180 __________

3.) m8 = m1 ___________

4.) m5 + m4 = 180 __________

Five Ways to Prove Lines are Parallel

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2.

3.

4.

5.

5. ) m1 = m7 _____________

6.) m8 + m2 + m3 = 180 ________

7.) m7 = m4 _____________

8.) m2 + m3 = 180 _____________

9. ) m6 + m4 = 180. _____________

Practice Problems (Continued)

For the following, state which segments must be parallel and state the postulate or theorem that justifies your answer.

10) _______ 11) _______

______________________________ ______________________________

12) _______ 13) _______

_____________________________ ______________________________

Find the values of x that makes j || k.

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A B C D53 44 53 45

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14) x = _______ 15) x = _______

Find the values of x and y that make AC || DF and AE || BF.

16) x = _____ y = _____ 17) x = _____ y = _____

Complete the following proofs by supplying the missing statements and reasons.

18) Given: 3 is supplementary to 5

Prove: BD || FE

Statements Reasons

1) ________________________ 1) ________________________2) ________________________ 2) ________________________3) ________________________ 3) ________________________4) m4 + m5 = 180 4) ________________________5) 4 is supplementary to 5 5) ________________________6) ________________________ 6) ________________________

19) Given: j || k;

1 3

6x (5x+15)

j k

(3x+10)

(5x-10)

j

k

Prove: || n

Statements Reasons1) ________________________ 1) ________________________2) ________________________ 2) ________________________3) 1 3 3) ________________________4) ________________________ 4) Substitution5) ________________________ 5) ________________________

Section 3.4 Angles of a Triangle Day 5/6 Notes

A triangle is the figure formed by _______ segments joining _______ noncollinear points. Each of the points is a ___________ of the triangle.

Vertices : ___________________ Angles : ___________________

The segments are the ___________ of the triangle. Sides : ___________________

Side ____ is opposite A. Side AB is included between ___ and ___Side ____ is opposite B. Side AC is included between ___ and ___Side ____ is opposite C. Side BC is included between ___ and ___

Triangles can be classified by… the number of congruent sides it has.

___________ ___________ ___________ No sides congruent. Two sides congruent. Three sides

congruent.

their angles._________ _________ _________ __________ 3 acute s. One obtuse . One right . All congruent s.

Theorem 3-11:The sum of the angles of a triangle is ______.

An __________ angle is formed when one side of a triangle is extended.

The __________ __________ angles are two angles of a

triangle not adjacent to the exterior angle.

Theorem 3-12: The measure of an ___________ angle of a triangle equals the sum of the measures of the two __________ __________ angles.

Day 5/6 Practice

Complete the following.

1) If A B C, then 2) If mA = 90, then 3) If A X and B Y,

mA = mB = mC = mB + mC = _____. Then _____ _____.

______.

Find the value of x.

4) x = ______ 5) x = ______ 6) x = ______ 7) x = ______

8) x = ______

Find the values of x and y.

8) x = ______ 9) x = ______ 10) x = ______

y = ______ y = ______ y = ______

Day 7 Notes

Section 3.5: Angles of a Polygon Notes

VocabularyPostulate:

Theorem:

Corollary:

Corollaries for this Section:

Corollary 1: If the ____ angles of one triangle are _____________ to _____ angles of another triangle, then the ________ angles are congruent

Corollary 2: Each angle of an _________________ triangle has a measure of ________ degrees

Corollary 3: In a triangle, there can be at most one ________ angle or one ___________ angle

Corollary 4: The __________ angles of a right triangle are _________________

Definitions:

Polygon: a closed figure meaning ______________________________.

Convex Polygon:________________________________________________________________________.

Examples: Convex Not Convex

Regular Polygon: ________________________________________________________________________.

Diagonal: ________________________________________. Diagram:

Theorem 3-13: The sum of the measures of the angles of a convex polygon with ___ sides is ____________.

Theorem 3-14: The sum of the measures of the exterior angles of any convex polygon, one angle at each

vertex, is _________.