Math 20-2

Chapter 2 Properties of Angles and Triangles

Lesson 2.1 Exploring Parallel Lines

GOAL

The goal of this lesson is stated at the top of page 70 in your textbook, and is also written below.
Keep this in mind as you work through the lesson material.

Identify relationships among the measures of angles formed by intersecting lines.

Getting Started

If you haven’t had a chance to look at the Getting Started part of this unit,  please do that now!
In particular,  we will be using the concept of exterior angles in this unit.
(see the bottom of Page 69)

LINK  Exterior Angles in a triangle

LINK   Exterior angles in a triangle

Video Tutorial

Geometry Facts

The link below is for a Geometry Facts document you can download and use in your math class.
You could print it, though it has been created to be viewed on computer.
Your teacher will have the final word,  but the idea is that you can use this document just like a formula/reference page for any work you do in  your class.
If you have ideas to make it more useful, let your teacher know, and the document can be edited accordingly!
LINK
https://drive.google.com/file/d/1Ub6rjs6KJo91tNInkbeuv_gugZy64_4G/view?usp=sharing

Video Tutorial

Explore the Math

A chance to explore the ideas of the lesson

There is so much information here, be sure to go through all the sections below and actually Explore these ideas.
Use the provided links for interactive ways to connect with the ideas of the lesson.
Then view the video tutorials to further your understanding.

TEXT Pages 70 - 71

Read/Work through the Explore the Math section on Pages 70-71 in your textbook.

Introduction

Video Tutorial

?  When a transversal intersects two parallel lines, how are the angle measures related?

Explore the ideas online
LINK  (transversal)
LINK  (transversal)

Video Tutorial

Demonstration

Using the demonstration below, you can explore all the ideas of this lesson.

Video Tutorial

Manipulate[

DynamicModule[{a={-1,0.25},b={1,ptb},c={-1,-0.25},d={1,ptb-0.5},e={pte,1},f={-0.25,-1},g,h},

angle[pt1_,pt2_,pt3_]:=SetAccuracy[Round[ArcCos[(pt1-pt2).(pt3-pt2)/(Norm[(pt1-pt2)]*Norm[(pt3-pt2)])]*180/Pi,0.01],3];

linetwo[pt1_,pt2_]:=(y-pt1[[2]])==(pt2[[2]]-pt1[[2]])/(pt2[[1]]-pt1[[1]])(x-pt1[[1]]);

intsec[{pt1_,pt2_},{pt3_,pt4_}]:=Flatten[{x,y}/.Solve[{linetwo[pt1,pt2],linetwo[pt3,pt4]},{x,y}]];

angletext[e_,g_,b_,color_,adj_,adj2_,num_]:=Graphics[{Rotate[Text[Style[If[num===0,Row[{angle[e,g,b],"°"}],num],14,"Label",color],g+0.2Through[{Cos,Sin}[If[adj2==Pi,-1,1]*(Sign[b[[2]]-g[[2]]]angle[b,g,{b[[1]],g[[2]]}])Pi/180+adj*Pi/10+adj2]]],If[adj2==Pi,-1,1]*(Sign[b[[2]]-g[[2]]]angle[b,g,{b[[1]],g[[2]]}])Pi/180]}];

Deploy[
Grid[{{
LocatorPane[Dynamic[{a,b,c,d,e,f},
(a={-1,#[[1,2]]};
b={1,#[[2,2]]};c={-1,Min[{#[[3,2]],a[[2]]}]};d={1,b[[2]]-(a[[2]]-c[[2]])};e={#[[5,1]],1};f={If[#[[5,1]]==#[[6,1]],#[[6,1]]+.001,#[[6,1]]],-1})&],
Dynamic[Refresh[Show[
Plot[2,{x,-1,1}],
Graphics[{Darker[Green,0.3],Thick,Line[{{a,b},{c,d},{e,f}}]}],

Graphics[{PointSize[0.025],Black,Opacity[0],Point[g=intsec[{a,b},{e,f}]],Point[h=intsec[{c,d},{e,f}]]}],

ang2[clr_,num_]:=angletext[e,g,b,clr,1,0,num];ang6[clr_,num_]:=angletext[e,h,d,clr,1,0,num];
ang3[clr_,num_]:=angletext[f,g,a,clr,1,Pi,num];
ang7[clr_,num_]:=angletext[f,h,c,clr,1,Pi,num];ang4[clr_,num_]:=angletext[f,g,b,clr,-1,0,num];ang8[clr_,num_]:=angletext[f,h,d,clr,-1,0,num];
ang1[clr_,num_]:=angletext[e,g,a,clr,-1,Pi,num];ang5[clr_,num_]:=angletext[e,h,c,clr,-1,Pi,num];

If[angletypes===none,{},{}],
If[angletypes===numbers,{ang1[Black,1],ang4[Black,4],ang2[Black,2],ang3[Black,3],ang5[Black,5],ang8[Black,8],ang6[Black,6],ang7[Black,7]},{}],
If[angletypes===altext,{ang1[Red,0],ang8[Red,0],ang2[Blue,0],ang7[Blue,0]},{}],
If[angletypes===altint,{ang4[Red,0],ang5[Red,0],ang3[Blue,0],ang6[Blue,0]},{}],
If[angletypes===consecext,{ang1[Red,0],ang7[Red,0],ang2[Blue,0],ang8[Blue,0]},{}],
If[angletypes===consecint,{ang3[Red,0],ang5[Red,0],ang4[Blue,0],ang6[Blue,0]},{}],
If[angletypes===corresp,{ang1[Red,0],ang5[Red,0],ang2[Purple,0],ang6[Purple,0],ang3[Blue,0],ang7[Blue,0],ang4[Orange,0],ang8[Orange,0]},{}],
If[angletypes===vert,{ang1[Red,0],ang4[Red,0],ang2[Blue,0],ang3[Blue,0],ang5[Orange,0],ang8[Orange,0],ang6[Purple,0],ang7[Purple,0]},{}],
If[angletypes===allequal,{ang1[Red,0],ang4[Red,0],ang2[Blue,0],ang3[Blue,0],ang5[Red,0],ang8[Red,0],ang6[Blue,0],ang7[Blue,0]},{}],

Graphics[{Style[#,Italic,14,Bold,"Label"]&/@{Text["A",{-1.1,a[[2]]}],Text["B",{1.1,b[[2]]}],Text["C",{-1.1,c[[2]]}],Text["D",{1.1,b[[2]]-(a[[2]]-c[[2]])}],Text["E",{e[[1]],1.1}],Text["F",{f[[1]],-1.1}]},{{PointSize[0.05],Opacity[0.75],White,Point[g]},Style[#,Italic,14,Bold,"Label"]&@Text["G",g]},{{PointSize[0.05],Opacity[0.75],White,Point[h]},Style[#,Italic,14,Bold,"Label"]&@Text["H",h]}}],

PlotRange->1,AspectRatio->Automatic,ImagePadding->20,ImageSize->350,PlotRangeClipping->False,Axes->False],
TrackedSymbols:>{angletypes,a,b,c,d,e,f}]],Appearance->{Automatic,Automatic,Automatic,None,Automatic}
],
Column[{"",""
}]

}},ItemSize->{"Columns"->{27,0}},Alignment->{"Rows"->{Top}}]
]
],
{{pte,0.25},-1,1,ControlType->None},{{ptb,0.25},-1,1,ControlType->None},
"","",
{{angletypes,none,""},{none->"no angle labels",numbers->"angle numbers",altext->"alternate exterior",altint->"alternate interior",consecext->"consecutive exterior",consecint->"consecutive interior",corresp->"corresponding",vert->"vertical angles",allequal->"congruent angles"},ControlType->RadioButtonBar,Appearance->"Vertical"},ControlPlacement->Right,TrackedSymbols:>{pte,ptb,angletypes}]

Reflecting

Video Tutorial

In Summary

Key Ideas and Need to Know 
Presents and summarizes  the main new concepts of the lesson

•	Carefully read through each of the bullet points in this section.

•	If you are unclear about any of these ideas,  ASK YOUR TEACHER FOR HELP!

Video Tutorial   (Using the Geometry Facts Sheet)

Further Your Understanding

Further Your Understanding problems reinforce the math presented in this lesson.

These are just practice problems with a different heading....

•	Work through these practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

Notes

Question 4,    I didn’t do a video for this one.
An adjustable T-Bevel is a real thing.   
Watch THIS video for a solution with the real thing, or do an internet search for T Bevel and you will get hundreds of hits.

Question 6,    a link to the online page I used in my video solution.
LINK

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

({{"Question 1", "Question 2", "Question 3",                               , "Question 5"}, {"Question 6", , , , }})