3-3 Proving Lines Parallel

3-3 Proving Lines Parallel

Today we learned 2 new postulates and four new theorems.

The new theorems were all converses of the theorems in lesson

3-2 (the ones we learned the day before).

The two new postulates were;

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

And the converse.

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

The first theorem was:If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

The next theorem was:
If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

In the example above, since ∠4 and ∠5 are supplementary, and ∠6 and ∠7 are also supplementary, then lines p and r are parallel.

The next theorem involves perpendicular lines:
In a plane two lines perpendicular to the same line are parallel.

The final theorem was:

Two lines parallel to a third line are parallel to each other.

Overall we learned the different ways to prove two lines are parallel

1. 1. Show that a pair of corresponding angles are congruent.
2. 2. Show that a pair of alternate interior angles are congruent.
3. 3. Show that a pair of same-side angles are supplementary.
4. 4. In a plane show that both lines are perpendicular to a third line.
5. 5. Show that both lines are parallel to a third line.

Andddd,the next scribe is….

Kristaa:33

Chapter 3.1: When Lines and Planes Are Parallel

2 lines that do not intersect are either parallel or skew.

Parallel lines: Coplanar lines that do not intersect.

Skew lines: Noncoplanar lines. Therefore, they are neither parallel nor intersecting.

For examples, look in page 73 in you textbook.

Segments and rays contained in parallel lines are also called parallel.

Parallel lines do not intersect.

A line and a plane are parallel if they do not intersect.

NEW THEOREM ALERT! NEW THEOREM ALERT! NEW THEOREM ALERT!

If 2 parallel planes are cut by a third planes, then the lines of intersection are parallel.

For proof, look on page 74 in your textbook.

A transversal is a line that intersects 2 or more coplanar lines in different points.

Alternate interior angles are 2 nonadjacent angles on the same side of the transversal.

Corresponding angles are 2 angles in corresponding positions relative to the 2 lines.

For examples, look on page 74 in your textbook.

The next scribe is Alivia Dew.

Chapter 2 Quizzes

So tommorow is our chapter test and i thought we should go over our chapter 2 quizzes to help us since they will probably be like the test we will have tommorow.

Quiz 2.1-2.3

For each conditional, underline the hypotyhesis once and the conclusion twice.

1. If Mellisa practices the flute, then she plays well.

We know Mellisa practices the flute is the hypothesis, because if is before it and we know she plays well is the conclusion, because that is the only possible answer left, plus then is before it.

2. The lake begins to freeze only if the temperature drops below 0 degrees celcius.

I know the lake begins to freeze is the conclusion because only if is after that and in the text book it says: q only if p

So the temperature drops below 0 degrees celcius is the hypothesis.

3. S is the midpoint of RT implies that RS=ST

I know that S is the midpoint of RT is the hypothesis because in the book it says that p implies q and that means that RS=ST is the conclusion.

4. The car will not start if the battery is discharged.

I know that the battery is discharged is the hypothesis, because if is before it so that leaves the car will not start as the conclusion.

Provide a counter example to show that each statement is false.

5. If the sum of 2 integers is even, then the integers are even.

This is false because 3+5=8 3 is odd 5 is odd and 8 is even.

6. An angle is an obtuse if its measure is greater than 90.

This statement is false because the angle could measure 181 degrees and an obtuse angle is one that measures between 90 and 180 degrees.

Tell whether the converse of each conditonal is true or false.

7. If an integer is greater than 10, then it is a positive integer.

It is false because the converse would be if it's a positive integer then it is greater than 10. Well, what if the integer is 5?

8. An angle is a right angle if its measure is 90.

True, because the converse is the angle's measure is 90 degrees if it's a right angle which is true.

9. Given: AC+=BD

Prove: AB=CD

1. AC=BD, given. 2. AB+BC=AC, BC+CD=BD, segment addition postulate 3. AB+BC=BC+CD, substitution property of equality. 4. AB=CD, subtraction property of equality. The proof explains itself.

10. Given measure of angle1 = the measure of angle 3

1. Angle 1 =angle 3, given. 2. Angle 1 + angle 2= angle WOY, angle 2+ angle 3=angle XOZ, angle addition postulate. 3. Angle 2 =angle WOY, Angle 2 =angle XOZ, subtraction property of equality. 4. Angle WOY=angle XOZ, substitution property of equality. The proof plan explains itself.

Prove: the measure of angle WOY = the measure of angle ZOX

In Questions 11-14 is the midpoint of QT, S is the midpoint of RT, RX biscts angle WRZ and RY bisects angle XRZ.

11. By the Midpoint Theorem RS= 1/2_

RS= 1/2 RT. This is correct because it states in the problem that S is the midpoint of RT.

12. If QT=26 then QR =_ and ST=_

QR=13 and ST =6.5. This is correct because QR is 1/2 of QT which = 26, so QR is 13. ST is 1/2 of QR, so ST is 6.5.

13. By the Angle Bisector Theorem measure of angle WRX=1/2 the measure of WRZ.

Angle WRX =1/2 angle WRZ. This is correct because it says that ray RX bisects angle WRZ, so that means that angle WRX is 1/2 of angle WRZ.

14. If the measure of angle YRZ =26 then the measure of angle XRZ =- and the measure of angle WRZ =-

Angle XRZ =52 and angle WRZ=104. Angle YRZ is 1/2 of angle XRZ, and angle YRZ =26 so angle XRZ is 52. Angle XRZ is 1/2 of angle WRZ, so angle WRZ = 104.

1. Vertical angles are _.
Vertical angles are congruent because. This is true because definition of vertical angles.

2. A complement of an acute angle is a _ angle.
A complement of an acute angle is an acutge angle. This is true because if an angle is acute and the compliment of that angle has to be less than 90 degrees making it an acute angle.

3. A supplement of a right angle is a _ angle.
A supplement of a right angle is a right angle. This is true because if one supplement is 90 degrees and 180 degrees minus 90 degrees is 90 degrees, thus it is a right angle.

4. A supplement of an acute angle is a _ angle.
A supplement of an acute angle is an obtuse angle. This is true because if two angles are supplementary then the add up to 180. If one angle is acute (less than 90) than the other angle must be obtuse (more than 90).

5. Find the measure of a supplement of an angle with measure 75.
The supplement is 105, because if one angle is 75 than the supplement has to add up to 180. Thus, 180 minus 75 is 105.

6. If the meaure of angle A=3y, and angle A and angle B are conmplementary angles, find the measure of angle B in terms of y.
90 minus 3y because angle A and angle B are both complementary and angle A is 3y so 90 minus 3y is angle B. Thus, in terms of y you say 90 minus 3y.

7. Name a right angle.
Angle 4, because ray GD is perpendicular to line EB and perpendicular lines form right angles.

8. Name two complementary angles.
Angle 3 and angle 2, because ray GD and line EB are perpendicular so angle GDB is a right angle and angle 3 and angle 2 make up angle GDB.

9. Name two congruent adjacent angles.
angle 4 and angle GDB because ray GD and line EB are perpendicular. So, they form congruent, adjacent angles.

10. Name a supplement of angle EGA.
Angle 1 because angle EGB is a straight angle and angle 1 and angle EGA make up angle EGB.

11. Is the measure of angle 5= 40, then: a, the measure of angle 2 = _ and b, the measure of angle 3=_.
a. 40 because angle 5 and angle 2 are vertical angles so they are congruent and angle 5 =40.
b. 50 because angle 2 and 3 are complements and 90 minus 40=50, so angle 3 =50.

12. If angle 7 is supplementary tosupplementary to angle 8, what can you conclude about angle 7 and angle 9?
Angle 7 is congruent to angle 9 because if two angle share the same compliment, they are congruent.

13. If angle 5 is complementary to angle 3 and angle 3 is congruent to angle 4, what can you conclude about angle 5 and angle 4?
Angle 5 is complementary to angle 4 because angle 3 is the same as angle 4, and angle 5 is complementary to angle 3 and it is also complementary to angle 4.

14. Complete the proof.
Given: Ray AC bisects angle BAD;
angle 1 and angle 2 are comps.;
angle 3 and angle 4 are comps.
Prove measure of angle 2 = measure of angle 4.
1. Ray AC bisects angle BAD,
angle 1 and angle 2 are comps
angle 3 and angle 4 are comps, given
2. angle 1 =angle 3, angle bisector theorem
3. angle 1 + angle 2 =90, angle 3 + angle 4=90, definition of complementary angles.
4.angle 1 +angle 2=angle 3 +angle4, substitution property of equality.
5. angle 2+anlge 4, subtraction property of equality.
Unable to do 15 and 16 because they were graphs.

17. The sum of the measures of a complement and a supplement of an angle is 160. Find the measure of the angle.
(180-x)+(90-x)=160, combine like terms
270-2x=160,subtract 270 from both sides
-2x=-110, divide by -2
x=55

and the scribe is emma!!!!!!!




just kidding its david

Worksheet Review (By Kerry~Zahra's just being nice:3)

Yesterday, we did three worksheets as a review for our quiz today.

The first, (2.4) Special Pair of Angles, which talked about complementary, supplementary and vertical angles. Complementary angles are 90 degrees in measure while supplementary are 180 degrees in measure. Vertical angles are angles right across from each other.

1&3 are vertical angles. 2 &4 are also vertical angles. One thing about vertical angles you have to remember is that they are ALWAYS congruent.

Since b & d are vertical angles, then b equals 40 degrees.

The second worksheet, (2.5) Perpendicular Lines, focused on perpendicular lines mostly. Perpendicular lines are when two lines intersect and create 90 degree angles, or right angles. The definition of perpendicular lines are:

If two lines are perpendicular, then they form right angles.

If two lines form right angles, then the lines are perpendicular.

Important properties to remember are:

If two are perpendicular, then they form congruent adjacent angles.

If two lines form congruent adjacent angles, then the lines are perpendicular.

If the exterior sides of two adjacent acute angles are perpendicular, then the angles complementary.

On our last worksheet, (2.6) Planning a Proof, we reviewed on planning proofs. Which is where you explain what you will do in your proof before you do it.

All of these were on the quiz today. Hopefully you all did amazing.

Oh and go but Mindless Behavior’s album #1 Girl. I’m listening to it now and it’s amazing.

NEXT SCRIBE IS J.D

It's posted under Zahra,but Kerry really did all the writing.She just had trouble posting it~~

--Zahrrrraa.

Review Time For Quiz

Review For Quiz!!

Today we reviewed our homework, pg. 63 1-16 all,18-22 evens, and pg. 65 1-9.

In this homework we mainly reviewed the postulates, definitions, properties or theorem. We also covered supplementary angles and complementary angles and how they can be used to find out if two angles are congruent or not.

Angle ABD and angle DBC are supplementary
Angle ABD and angle XYZ are supplementary

Or

/ABD + /DBC=180
/ABD+/XYZ=180

which after some substituting and such you can conclude that /DBC = /XYZ


We also talked about proofs and how to logically use definitions, theorems, properties and postulates to prove a statement.

We also reviewed verticle angles and how they are of equal measure.

like angle r is equal to angle q. and angle x is equal to angle y.






In the review sheets we were given in class today, they went over 2-4, 2-5 and 2-6.



2-4 Special pairs of angles


In this worksheet we reviewed mainly supplementary and complementary angles and how they are used to find the solution of x or y.

Ex. /C = 3y-5 /D = y+15 /D and /C are complementary.

/C + /D = 90
(3y – 5) + (y + 15) = 90
4y + 10 = 90
4y = 80
y = 20

You can do the same with supplementary angles.

Something minor that was in this worksheet is how a line segment or ray or line can bisect a angle or line and from that you can figure out what the angles are or what the distances are.



2-5 Perpendicular lines


In this worksheet we reviewed…well perpendicular lines.


We reviewed that perpendicular lines make 90 degree angles or right angles. And if the line goes through the other line then it might look something like this.
As shown all the angles are 90 degree angles.


2-6 Planning Proofs

In this review we went over proofs, proofs and… more proofs. We reviewed how to logically go through the problem by using postulates, definitions, theorems and properties to prove a statement.

When doing a Proof always start with the given or givens. Do this so you can have a foundation of what to go off of. Then after that look t what your supposed to prove and think through different ways of getting to that statement by using the theorems, postulates, definitions and properties and that’s really all to say.

Here is an example proof.

Notice he stated the givens first.

And that is all for today, until next time and GOOD LUCK ON THE TEST!!






The next scribe is………



KERRY!!!!!!!!…..jk
Its Katie Bwahahhahahaaaa

2.5 Perpendicular Lines

Today we learned about perpendicular lines and theorems concerning them. I will tell you the definition of perpendicular, state the theorems, give an example proof, and of course, pick the next scribe.

Perpendicular- a term used to describe two intersecting lines forming right angles
These lines are perpendicular.
Here are the theorems

If two lines are perpendicular, then they form congruent adjacent angles (or angles of the same size sharing a ray).
If two lines for congruent adjacent angles, then the lines are perpendicular (this is the converse of the conditional listed as the first theorem).
If the exterior sides of two adjacent angles are perpendicular, the the angles are complementary.
The two dark lines are the exterior sides and they are perpendicular.

On a side note, the symbol for showing perpendicularity (if that's a word) is _l_

So, here's an example proof relating to perpendicular lines:


Given: ray CA _l_ ray CB
Prove: