An Introduction to Plane Geometry -- Part 3 by lyxng

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steemiteducation · @lyxng · Mar 26 '18 (edited)

$4.52

An Introduction to Plane Geometry -- Part 3

<html>
<h1>Lesson III. Kinds of Angles</h1>
<center>
<img src=https://steemitimages.com/DQmbzdvNxuu1c3KrypUYRCk8NUjzRnhnh1Ld666s8fEU6ru/abstract-wallpapers-phombo-wallpaper-photo-hqpictures-photocombo-audi.jpg><br>
<sup><i><a href="https://www.walldevil.com/geometry-wallpaper-421066">Image Source</a></i> 
</sup>
</center>
<p><br></p>
<h3><em><strong>Kinds of angles according to their measures:</strong></em></h3>
<p><br></p>
<p>1.) <em><strong>Acute angle</strong></em> is an angle whose measure is less than 90°.</p>
<p><i>e.g.</i></p>
<ul>
  <li>5°, 27°, 89°, 15.5°</li>
</ul>
<p><br></p>
<p>&nbsp;2. ) <em><strong>Right angle</strong></em> is an angle whose measure is 90°.</p>
<p><br></p>
<p>&nbsp;3. ) <em><strong>Obtuse angle</strong></em> is an angle whose measure is greater than 90°.</p>
<p><i>e.g.</i></p>
<ul>
  <li>91°, 120°, 100°, 121°.</li>
</ul>
<p><br></p>
<h3><em><strong>Kinds of angle pairs:</strong></em></h3>
<p><br></p>
<p>&nbsp;1. ) <em><strong>Complementary pair</strong></em></p>
<blockquote><i>Complementary pair of angles are two angles whose sum of their measures is 90°.</i></blockquote>
<p><i>e.g.</i></p>
<ul>
  <li>20° and 70°, 50° and 40°, 10° and 80°</li>
  <li>The complement of 30° is 60°.</li>
  <li>15° and 75° are complementary angles.</li>
</ul>
<p><br></p>
<p>&nbsp;2. ) <em><strong>Supplementary pair</strong></em></p>
<blockquote><i>Supplementary pair of angles are two angles whose sum of their measures is 180°.</i></blockquote>
<p><i>e.g.</i></p>
<ul>
  <li>60° and 120°, 100° and 80°, 110° and 70°</li>
  <li>The supplement of 30° is 150°.</li>
  <li>Two right angles are supplementary.</li>
</ul>
<p><br></p>
<p>&nbsp;3. ) <em><strong>Vertical angles</strong></em></p>
<blockquote><i>Vertical angles are two opposite angles formed by intersecting lines or ray.</i></blockquote>
<center>
<img src=https://s19.postimg.org/9kddn1p83/Pics_Art_03-26-06.45.53.jpg>
</center>
<p><br></p>
 <p><i>Line m and line r intersect, forming two pairs of vertical angles.</i></p>
<ul>
  <li>∠1 and ∠3 are vertical pair of angles.</li>
  <li>∠2 and ∠4 are also pair of vertical angles.</li>
</ul>
<p><br></p>
<p><i>Vertical angles are congruent, therefore</i>:</p>
<ul>
  <li>∠1 ≅ ∠3 ; angle 1 is congruent to angle 3.</li>
  <li>∠2 ≅ ∠4 ; angle 2 is congruent to angle 4.</li>
<p><br></p>
<p>&nbsp;4. ) <em><strong>Linear pair</strong></em></p>
<blockquote><i>Linear pair of angles are two angles about a point on a straight line.</i></blockquote>
<p><i>Note:</i></p>
<p>Linear pairs are always supplementary.</p>
<p><br></p>
<center>
<img src=https://s19.postimg.org/h0cn8vuxv/Pics_Art_03-26-07.04.28.jpg>
</center>
<p><br></p>
<p>∠2 and ∠5 are two angles about point B and a common side along line AC.</p>
<p>Therefore ∠2 and ∠5 are vertical angles and m∠2 + m∠5 = 180°</p>
<p><br></p>
<h3><em><strong>Angles formed by Parallel Lines and its Transversal</strong></em></h3>
<p><br></p>
<blockquote><i>A <strong>transversal line</strong> is a line that cuts two parallel lines at different points.</i></blockquote>
<p><br></p>
<p>Given: <em>m</em><strong> ||</strong> <em>p</em> with <em>u</em> as transversal.</p>
<center>
<img src=https://s19.postimg.org/wylcz0wvn/Pics_Art_03-26-07.27.56.jpg>
</center>
<ul>
  <li><em>Vertical angle</em> pairs in the figure:<br>
&nbsp;∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8</li>
  <li><em>Linear pairs:</em><br>
&nbsp;∠1 and ∠2, ∠2 and ∠3, ∠5 and ∠6, ∠6 and ∠7<br>
&nbsp;∠1 and ∠4, ∠3 and ∠4, ∠5 and ∠8, ∠7 and ∠8</li>
  <li><em>Alternate-interior angles</em>:<br>
&nbsp;∠3 and ∠5, ∠4 and ∠6</li>
  <li><em>Alternate-exterior angles</em>:<br>
&nbsp;∠1 and ∠7, ∠2 and ∠8</li>
</ul>
<p><br></p>
<center>
<img src=https://steemitimages.com/DQmQPRBLmhWag36FBxYVNJNKZRUsBbpPeZvm7FVixnWVXhk/13.png>
</center>
<p><br></p>
<h3>Answer the following:</h3>
<p>&nbsp;<code>1. ) Are two right angles supplementary? Why?</code></p>
<p><code>&nbsp;2. ) How many right angles do we have? Why?</code></p>
<p><code>&nbsp;3. ) If ∠A and ∠B are congruent and complementary, what is the measure of ∠A? Why?</code></p>
<p><code>&nbsp;4. ) What is the measure of ∠C if it is a linear pair of 80°?</code></p>
<p><code>&nbsp;5. ) If two angles are complementary but one angle is twice as large as its pair, what are the measures of these angles?</code></p>
<p><br></p>
</hr>
<p><br></p>
<h2>References:</2>
<ul>
  <li><i>Grolier International Encyclopedia Vol. 9, Copyright 1991</i></li>
  <li><i>The New Book of Popular Science Vol. 1, Copyright 1992</i></li>
</ul>
<p><br></p>
<center>
<img src=https://steemitimages.com/DQmQPRBLmhWag36FBxYVNJNKZRUsBbpPeZvm7FVixnWVXhk/13.png>
</center>
<p><br></p>
<center>
<h1>Lessons 4, 5 and 6 will be posted next week!</h1><br>
<h1>❤❤❤</h1>
</center>
<p><br></p>
</html>

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@acont · Mar 26 '18

1. ) Are two right angles supplementary? Why?
Yes, because theyr sum is 90°+90°=180°

 2. ) How many right angles do we have? Why?
We have 2: 90° and 270°. Because in both cases the lines are perpendicular between each other, creating a right angle.

 3. ) If ∠A and ∠B are congruent and complementary, what is the measure of ∠A? Why?
45° because A+B=90° (complementary condition) and A=B (congruent condition)

 4. ) What is the measure of ∠C if it is a linear pair of 80°?
10° because linear pairs are supplementary.

 5. ) If two angles are complementary but one angle is twice as large as its pair, what are the measures of these angles?
30° and 60°. Complementary condition: A+B=90°; pair condition: B+2B=180°.

Keep spreading education! @lyxng

👍 lyxng

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@lyxng · Mar 27 '18

I hope this one stays on the top of the comment section for reference. Great job, Engineer!

👍 acont

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@chrisdsteemsport · Mar 26 '18

@lyxng i just followed you now. The project you are introducing looks superb and i look forward to learn more of plane geometery. Thanks for sharing

👍 lyxng

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@lyxng · Mar 26 '18

Wow! Thank you @chrisdsteemsport! Please check out @steemiteducation for a lot more topics to read. Followed you back. Good day!

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@kamchore · Mar 26 '18

Thanks for refreshing high school geometry. Keep sharing @lyxng

👍 lyxng

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@lyxng · Mar 26 '18

Thanks for dropping by @kamchore!

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@muhammadazzikra · Mar 26 '18

Thanks for sharing valuable post.
I continue follow your post.
I appreciate your contest @lyxng

👍 lyxng

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@pranithreddy · Mar 26 '18

Lol!! I started becoming nostalgic now. Those days we used to follow heavy textbooks, but kids of this generation are getting everything in capsule form. Thanks for making all info in catalogue form.

👍 lyxng

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@lyxng · Mar 26 '18

Thank you @pranithreddy! Exactly. When I was in gradeschool I used to carry a huge trolly around, filled with books and notebooks for 8 subjects. Now, students can just carry a tablet and everything will be there. 😀

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