When two lines intersect, vertical angles are formed. If the angles are opposite to each other,they are sometimes referred to as vertically opposite angles. Vertical angles are used in real-lifesituations such as railroad crossing signs, letter "X", open scissors pliers, and so on. To ensurethat two intersecting lines were equal, the Egyptians would draw two intersecting lines and
measure the vertical angles.
Vertical angles are always equidistant from each other. In general, when two lines intersect,then two pairs of vertical angles are formed.Let’s Know More About Vertical Angles First and Then Jump toCongruent Angles● Vertical Angles TheoremThe vertical angles, defined as angles formed by two intersecting straight lines, are congruent.Vertical angles are always congruent with one another.● Is it True that all Vertical Angles are CongruentVertical (opposite) angles are always congruent, or equal to each other, no matter how we crossour pencils or how any two intersecting lines cross. In mathematics, it is known as the verticalangles theorem.● Congruent AnglesIf the corresponding sides and angles of two triangles are of equal size, they are said to becongruent. When two angles are superimposed, they are also congruent. It means if theycoincide with each other by turning and/or moving. A parallelogram's diagonals also createcongruent vertex angles.Applications of Vertical AnglesVertical angles are used in a variety of ways that we see or experience in our daily lives.● The roller coasters are set at a specific angle to ensure proper operation. These anglesare so critical that if they are shifted by a degree above or below, an accident couldoccur. A roller coaster's maximum vertical angle (Mumbo Jumbo, Flamingo Land) is 112degrees.● We see two vapour trails that cross each other and make vertical angles at an airshow.● Railroad crossing signs (X) are placed on roads to ensure the safety of vehicles.
● A kite is held aloft by two wooden sticks that form a cross.As we have already discussed what vertical angles are, let’s learn about the properties of
congruent angles.
Facts About Vertical Angles● Angles that are congruent - Vertical angles of equal measure are always congruent.● Vertical angle sum - Both pairs of vertical angles (for a total of four angles) always addup to 360 degrees.● Angles that are adjacent - Angles formed by each pair of vertical angles are referred toas adjacent angles, and they are supplementary (the angles sum up to 180 degrees).Four Congruence ConditionsThe triangle congruence criteria are as follows:● SSS (Side-Side-Side)Two triangles are said to be congruent if three sides of one triangle are congruent with threesides of another triangle.● SAS (Side-Angle-Side)Two triangles are said to be congruent if their two sides and included angle are congruent to thecorresponding parts of another triangle.● ASA (Angle-Side-Angle)The condition is as follows : Two triangles are known to be congruent if two angles and theincluded side of one triangle are congruent to the corresponding parts of another triangle.● AAS (Angle-Angle-Side)The condition is as follows: Two triangles are said to be congruent if their two angles and non-included side are congruent to the corresponding parts of another triangle.Congruent and Vertical angles is a vast topic, which can be easily learnt through cuemath - thebest Maths online tutoring platform.● HL (Hypotenuse-Leg, Right Triangle Only)So, the condition is if the hypotenuse and leg of one right triangle are congruent to thecorresponding parts of another right triangle, the right triangles are then said to be congruent.
Tips and Tricks for Congruent Angles1. Equal angles are also known as congruent angles.2. Congruent angles are those that are vertically opposite one another.3. Congruent angles are all alternate angles and corresponding angles formed by the
intersection of two parallel lines and a transversal.
Recall that if $\angle BAC$ and $\angle BAD$ are supplementary angles, and if $\angle B'A'C'$ and $\angle B'A'D'$ are supplementary angles, and if $\angle BAC\cong\angle B'A'C'$, then also $\angle BAD\cong\angle B'A'D'$. (This is Proposition 9.2 on page 92 of Robin Hartshorne's Geometry: Euclid and Beyond.) A proof may be found here.
Now vertical angles are defined by the opposite rays on the same two lines. Suppose $\alpha$ and $\alpha'$ are vertical angles, hence each supplementary to an angle $\beta$. Since $\beta$ is congruent to itself, the above proposition shows that $\alpha\cong\alpha'$.
The Vertical Angle Theorem says the opposing angles of two intersecting lines must be congruent, or identical in value. That means no matter how or where two straight lines intersect each other, the angles opposite to each other will always be congruent, or equal in value:
Explaining the Vertical Angle Theorem
According to the Vertical Angle Theorem, when two straight lines intersect, they form two linear pairs. This means that the adjacent angles formed when two lines intersect are supplementary angles, meaning that their angles add up to 180 degrees:
You can see in the perpendicular line figure above that the two lines intersect to form two pairs of vertical angles. Vertical angles are also referred to as vertically opposite angles because they are each on the opposite side of the other. In this figure, angle D and angle B and angles A and C are each a pair of vertically opposite angles. As the Vertical Angle Theorem says, these vertical angles are congruent.
Why We Must Know the Vertical Angle Theorem
As we know, lots of theorems and postulates used in geometry problems explain how angles work such as the Supplementary Angles Theorem, Right Angles Theorem, Angle Addition Postulate, and Triangle Congruence Postulate. Like the rest of these, the Vertical Angles Theorem serves a foundational role in the rules of geometry and trigonometry.
This theorem says that when two straight lines intersect, they form two sets of linear pairs with congruent angles. It also means that the adjacent angles formed by the intersection of these two lines are supplementary, or equal to 180 degrees.
More Math Homework Help
{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T21:05:29+00:00","modifiedTime":"2016-03-26T21:05:29+00:00","timestamp":"2022-09-14T18:09:40+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Geometry","_links":{"self":"//dummies-api.dummies.com/v2/categories/33725"},"slug":"geometry","categoryId":33725}],"title":"Proving Vertical Angles Are Congruent","strippedTitle":"proving vertical angles are congruent","slug":"proving-vertical-angles-are-congruent","canonicalUrl":"","seo":{"metaDescription":"When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem tha","noIndex":0,"noFollow":0},"content":"<p>When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem that tells you so.</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQyNy5pbWFnZTAuanBn.webp\" width=\"535\" height=\"135\" alt=\"image0.jpg\"/>\n<p><b>Vertical angles are congruent:</b><b> </b>If two angles are vertical angles, then they’re congruent (see the above figure).</p>\n<p>Vertical angles are one of the most frequently used things in proofs and other types of geometry problems, and they’re one of the easiest things to spot in a diagram. Don’t neglect to check for them!</p>\n<p>Here’s an algebraic geometry problem that illustrates this simple concept: Determine the measure of the six angles in the following figure.</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQyOC5pbWFnZTEuanBn.webp\" width=\"417\" height=\"400\" alt=\"image1.jpg\"/>\n<p>Vertical angles are congruent, so</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQyOS5pbWFnZTIucG5n.webp\" width=\"149\" height=\"19\" alt=\"image2.png\"/>\n<p>and thus you can set their measures equal to each other:</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQzMC5pbWFnZTMucG5n.webp\" width=\"276\" height=\"43\" alt=\"image3.png\"/>\n<p>Now you have a system of two equations and two unknowns. To solve the system, first solve each equation for <i>y</i>:</p>\n<p><i>y</i> = –3<i>x</i></p>\n<p><i>y</i> = –6<i>x</i> – 15</p>\n<p>Next, because both equations are solved for <i>y</i>, you can set the two <i>x</i>-expressions equal to each other and solve for <i>x</i>:</p>\n<p>–3<i>x</i> = –6<i>x</i> – 15</p>\n<p> 3<i>x</i> = –15</p>\n<p> <i>x</i> = –5</p>\n<p>To get <i>y</i>, plug in –5 for <i>x</i> in the first simplified equation:</p>\n<p><i>y</i> = –3<i>x</i></p>\n<p><i>y</i> = –3(–5)</p>\n<p><i>y</i> = 15</p>\n<p>Now plug –5 and 15 into the angle expressions to get four of the six angles:</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQzMS5pbWFnZTQucG5n.webp\" width=\"241\" height=\"43\" alt=\"image4.png\"/>\n<p>To get angle 3, note that angles 1, 2, and 3 make a straight line, so they must sum to 180°:</p>\n<img src=\"//sg.cdnki.com/any-two-vertical-angles-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzIyMDQzMi5pbWFnZTUucG5n.webp\" width=\"131\" height=\"61\" alt=\"image5.png\"/>\n<p>Finally, angle 3 and angle 6 are congruent vertical angles, so angle 6 must be 145° as well. Did you notice that the angles in the figure are absurdly out of scale? Don’t forget that you can’t assume anything about the relative sizes of angles or segments in a diagram.</p>","description":"<p>When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem that tells you so.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220427.image0.jpg\" width=\"535\" height=\"135\" alt=\"image0.jpg\"/>\n<p><b>Vertical angles are congruent:</b><b> </b>If two angles are vertical angles, then they’re congruent (see the above figure).</p>\n<p>Vertical angles are one of the most frequently used things in proofs and other types of geometry problems, and they’re one of the easiest things to spot in a diagram. Don’t neglect to check for them!</p>\n<p>Here’s an algebraic geometry problem that illustrates this simple concept: Determine the measure of the six angles in the following figure.</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220428.image1.jpg\" width=\"417\" height=\"400\" alt=\"image1.jpg\"/>\n<p>Vertical angles are congruent, so</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220429.image2.png\" width=\"149\" height=\"19\" alt=\"image2.png\"/>\n<p>and thus you can set their measures equal to each other:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220430.image3.png\" width=\"276\" height=\"43\" alt=\"image3.png\"/>\n<p>Now you have a system of two equations and two unknowns. To solve the system, first solve each equation for <i>y</i>:</p>\n<p><i>y</i> = –3<i>x</i></p>\n<p><i>y</i> = –6<i>x</i> – 15</p>\n<p>Next, because both equations are solved for <i>y</i>, you can set the two <i>x</i>-expressions equal to each other and solve for <i>x</i>:</p>\n<p>–3<i>x</i> = –6<i>x</i> – 15</p>\n<p> 3<i>x</i> = –15</p>\n<p> <i>x</i> = –5</p>\n<p>To get <i>y</i>, plug in –5 for <i>x</i> in the first simplified equation:</p>\n<p><i>y</i> = –3<i>x</i></p>\n<p><i>y</i> = –3(–5)</p>\n<p><i>y</i> = 15</p>\n<p>Now plug –5 and 15 into the angle expressions to get four of the six angles:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220431.image4.png\" width=\"241\" height=\"43\" alt=\"image4.png\"/>\n<p>To get angle 3, note that angles 1, 2, and 3 make a straight line, so they must sum to 180°:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/220432.image5.png\" width=\"131\" height=\"61\" alt=\"image5.png\"/>\n<p>Finally, angle 3 and angle 6 are congruent vertical angles, so angle 6 must be 145° as well. Did you notice that the angles in the figure are absurdly out of scale? Don’t forget that you can’t assume anything about the relative sizes of angles or segments in a diagram.</p>","blurb":"","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses as well as standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. ","hasArticle":false,"_links":{"self":"//dummies-api.dummies.com/v2/authors/8957"}}],"primaryCategoryTaxonomy":{"categoryId":33725,"title":"Geometry","slug":"geometry","_links":{"self":"//dummies-api.dummies.com/v2/categories/33725"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":230077,"title":"How to Copy an Angle Using a Compass","slug":"copy-angle-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230077"}},{"articleId":230072,"title":"How to Copy a Line Segment Using a Compass","slug":"copy-line-segment-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230072"}},{"articleId":230069,"title":"How to Find the Right Angle to Two Points","slug":"find-right-angle-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230069"}},{"articleId":230066,"title":"Find the Locus of Points Equidistant from Two Points","slug":"find-locus-points-equidistant-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230066"}},{"articleId":230063,"title":"How to Solve a Two-Dimensional Locus Problem","slug":"solve-two-dimensional-locus-problem","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230063"}}]},"hasRelatedBookFromSearch":true,"relatedBook":{"bookId":282230,"slug":"geometry-for-dummies-3rd-edition","isbn":"9781119181552","categoryList":["academics-the-arts","math","geometry"],"amazon":{"default":"//www.amazon.com/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119181550-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//catalogimages.wiley.com/images/db/jimages/9781119181552.jpg","width":250,"height":350},"title":"Geometry For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><p><b><b data-author-id=\"8957\">Mark Ryan</b></b> is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre-algebra to calculus. 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When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal, and here’s the official theorem that tells you so.
Vertical angles are congruent: If two angles are vertical angles, then they’re congruent (see the above figure).
Vertical angles are one of the most frequently used things in proofs and other types of geometry problems, and they’re one of the easiest things to spot in a diagram. Don’t neglect to check for them!
Here’s an algebraic geometry problem that illustrates this simple concept: Determine the measure of the six angles in the following figure.
Vertical angles are congruent, so
and thus you can set their measures equal to each other:
Now you have a system of two equations and two unknowns. To solve the system, first solve each equation for y:
y = –3x
y = –6x – 15
Next, because both equations are solved for y, you can set the two x-expressions equal to each other and solve for x:
–3x = –6x – 15
3x = –15
x = –5
To get y, plug in –5 for x in the first simplified equation:
y = –3x
y = –3(–5)
y = 15
Now plug –5 and 15 into the angle expressions to get four of the six angles:
To get angle 3, note that angles 1, 2, and 3 make a straight line, so they must sum to 180°:
Finally, angle 3 and angle 6 are congruent vertical angles, so angle 6 must be 145° as well. Did you notice that the angles in the figure are absurdly out of scale? Don’t forget that you can’t assume anything about the relative sizes of angles or segments in a diagram.