![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2021\/12\/linear-pair-of-angles.png\")
<\/figure>\n\n\n\nLet\u2019s take a look at the linear pair of angles shown above. We can see that the angles, [katex]\\angle AOC[\/katex] and [katex]\\angle COB[\/katex], are adjacent angles sharing a common side of [katex]\\overline{OC}[\/katex] and a vertex of [katex]O[\/katex]. The linear pair of angles are also supplementary and form a straight angle, so [katex]\\angle AOC + \\angle COB = 180\\degree = \\angle AOB[\/katex].<\/p>\n\n\n\n This is the fundamental definition of linear pair of angles- know this concept by heart to help you understand more complex properties and applications of linear pair of angles. <\/p>\n\n\n\n Determine whether each pair of angles can form a linear pair of angles when combined.<\/p>\n\n\n\n a.<\/p>\n\n\n\n If we merge the two angles, they will form a larger angle, [katex]\\angle AOC + \\angle COB = \\angle AOB[\/katex]. Since the two angles share a common side and a common point ([katex]\\overline{OC}[\/katex] and [katex]O[\/katex]), these two are adjacent angles when combined.<\/p>\n\n\n\n Adding the two angles, we can see that their sum is 180o<\/sup> and consequently, they are also supplementary. Since we\u2019ve shown that when we combine the two angles, they become adjacent and supplementary, we can confirm that they are linear pairs of angles.<\/p>\n\n\n\n b.<\/p>\n\n\n\n Visually, we can see that the angles will not form a straight line but will form an angle larger than 180o<\/sup>. <\/p>\n\n\n\n <\/p>\n\n\n\n By adding the two angles, we\u2019ll also see that the angle measure of [katex]\\angle AOB[\/katex] is greater than 180o<\/sup>.<\/p>\n\n\n\n [katex]\\begin{aligned}\\angle AOB &= \\angle AOC + \\angle COB \\\\&= 110\\degree + 90\\degree\\\\&= 200\\degree\\end{aligned}[\/katex]<\/p>\n\n\n\n Since the pair of angles form an angle not measuring 180o<\/sup>, they do not form a straight angle.<\/p>\n\n\n\n c.<\/p>\n\n\n\n Let\u2019s now rearrange the angles and try to visualize how they would form a linear angle. From inspection, we can see that the common vertex shared is [katex]O[\/katex] and the common side shared by these angles is [katex]\\overline{OC}[\/katex].<\/p>\n\n\n\n We\u2019ve shown that the angles form a linear angle and we can also see that [katex]\\angle AOB[\/katex] has a measure of 180o<\/sup>. This means that the pair of angles is linear pair of angles.<\/p>\n\n\n\n Two lines, [katex]\\overline{AB}[\/katex] and [katex]\\overline{CD}[\/katex], intersect at point, [katex]O[\/katex]. Write down four linear pairs of angles you can find.<\/p>\n\n\n\n Sketch the figure described by the problem to help you visualize the four pairs that you\u2019re looking for. <\/p>\n\n\n\n Let\u2019s take a look at these two pairs \u2013 each pair of angles are adjacent angles forming a straight angle each. This means that both pairs are linear pair of angles. Hence, [katex]\\angle AOD[\/katex] and [katex]\\angle AOC[\/katex] as well as [katex]\\angle AOC[\/katex] and [katex]\\angle COB[\/katex] are both linear pairs of angles.<\/p>\n\n\n\n Let\u2019s take a look at these two pairs \u2013 each pair of angles are adjacent angles forming a straight angle each. This means that both pairs are linear pairs of angles. Hence, [katex]\\angle AOD[\/katex] and [katex]\\angle AOC[\/katex] as well as [katex]\\angle AOC[\/katex] and [katex]\\angle COB[\/katex] are both linear pairs of angles.<\/p>\n\n\n\n Let\u2019s now take a look at the two remaining pairs of linear angles as shown above. Similar to the previous pairs, since each pair is adjacent angles that form straight angles, we can conclude that these two pairs are also linear pairs. This means that we have the following four pairs of linear angles:<\/p>\n\n\n\n [katex] \\begin{aligned}\\angle AOC \\,\\&\\, \\angle AOD\\\\\\angle AOC \\,\\&\\, \\angle AOB\\\\\\angle BOC \\,\\&\\, \\angle BOD\\\\\\angle AOD \\,\\&\\, \\angle DOB\\end{aligned} [\/katex]<\/p>\n\n\n\n These two types of angles are sometimes get interchanged, but it\u2019s important that we know the difference between these two pairs of angles.<\/p>\n\n\n\n Take a look at these two examples: for a pair of angles to be a linear pair, they must always be adjacent angles and are supplementary angles. Meanwhile, supplementary angles are simply pairs of angles that add up to 180o<\/sup>. This means that the angles do not have to be merged and form a straight line.<\/p>\n\n\n\n Hence, all linear pairs of angles are supplementary but not all supplementary angles are linear pairs of angles. This is why it\u2019s important to keep this difference in mind when working with different problems involving angles adding up to 180o<\/sup>.<\/p>\n\n\n\n There is a wide range of problems involving linear pairs of angles. More often than not, we\u2019re asked to find missing angles\u2019 measures or solve for the unknown values given linear pair of angles. But before trying out different problems, let us first introduce you to the linear pair postulate. It\u2019s an important postulate to use when solving problems involving linear pairs of angles.<\/p>\n\n\n\n [katex]\\begin{aligned}\\angle AOC + \\angle COB = 180\\degree\\end{aligned}[\/katex]<\/p>\n\n\n\n According to the linear pair postulate, if a ray stands on and divides a line, the two adjacent angles that were formed will always have a sum of 180o<\/sup>. The converse statement remains true as well \u2013when to adjacent angles add up to 180o<\/sup>, they form a line. In the next section, we\u2019ll show you different problems involving linear pairs of angles.<\/p>\n\n\n\n Suppose that the line [katex]PQ[\/katex] is divided by the ray, [katex]OC[\/katex]. The ray divides the linear angle into two equal adjacent angles. Show that the angles formed are right angles.<\/p>\n\n\n\n After sketching the illustration similar to the one we\u2019ve shown above, you\u2019ll see that the ray, [katex]OC[\/katex], divides the linear angle in half and reflects into L-shaped angles similar to right angles. But of course, we want to make sure that the linear pair of angles, [katex]\\angle POC[\/katex] and [katex]\\angle COQ[\/katex] do form angles each measuring 90o<\/sup>. <\/p>\n\n\n\n [katex] \\begin{aligned}\\angle POC &= \\angle COQ \\end{aligned} [\/katex]<\/p>\n\n\n\n We know that the two angles are equal and that they are supplementary, so we can set up the equation shown below.<\/p>\n\n\n\n [katex] \\begin{aligned}\\angle POC +\\angle COQ &= \\angle POQ\\\\ 2\\angle POC &= 180\\degree\\\\\\angle POC &= 90 \\degree\\\\\\angle COQ &= 90\\degree\\end{aligned} [\/katex]<\/p>\n\n\n\n Since both angles measure 90o<\/sup>, we can conclude that when a ray divides a line equally, the angles will always be right angles or measuring 90o<\/sup>.<\/p>\n\n\n\n Two angles form a linear pair of angles and the ratio of their angle measures is 5:7. What are the measures of the two angles?<\/p>\n\n\n\n Constructing a sketch illustrating the problem helps us figure out what needs to be done. Since two angles form a linear pair of angles, we construct a ray dividing the line into a pair of angles. The ratio of these angles is 5:7, so if we let [katex]x[\/katex] be the common factor shared by the angles, the two angles can be represented as [katex]5x[\/katex] and [katex]7x[\/katex].<\/p>\n\n\n\n Recall that linear pairs of angles are always supplementary, so equate the sum of the angles to 180o<\/sup> then solve for [katex]x[\/katex].<\/p>\n\n\n\n [katex] \\begin{aligned}(5x)\\degree + (7x)\\degree &= 180\\degree\\\\ 12x &= 180\\\\x &= \\dfrac{180}{12}\\\\&= 15\\end{aligned} [\/katex]<\/p>\n\n\n\n To find the measures of the angles, simply multiply 15 to each of the values from the given ratio. <\/p>\n\n\n\n [katex] \\begin{aligned}5:7&= 15 \\times 5 : 15 \\times 7\\\\&= 75\\degree: 105\\degree\\end{aligned} [\/katex]<\/p>\n\n\n\n This means that the angle measures are [katex]75\\degree[\/katex] and [katex]105\\degree[\/katex].<\/p>\n\n\n\n Suppose that [katex]\\angle POC[\/katex] and [katex]\\angle COQ[\/katex] are linear pair of angles sharing a common point [katex]O [\/katex] and a line segment of [katex]\\overline{OC}[\/katex]. If the difference between the two angles is 40o<\/sup>, what are the two angles\u2019 measures if [katex]\\angle POC[\/katex] is the larger angle?<\/p>\n\n\n\n Sketch an illustration to help you when solving the problem. We can let [katex]x[\/katex] represent the smaller angle, [katex]\\angle COQ[\/katex]. Since the difference between the two angles is 40o<\/sup>, the [katex]\\angle POC[\/katex] is 40o<\/sup> larger. This means that [katex]\\angle POC = (x + 40)\\degree[\/katex].<\/p>\n\n\n\n Always remember that linear pair of angles will always add up to 180o<\/sup>. Add the two angles measures\u2019 expression in terms of [katex]x[\/katex] then equate the sum to 180o<\/sup>. <\/p>\n\n\n\n [katex] \\begin{aligned}\\angle POC + \\angle COQ &= 180\\degree\\\\(x +40)\\degree + x\\degree &= 180\\degree\\\\2x+ 40&= 180\\\\2x&= 140\\\\x&= 70\\end{aligned}[\/katex]<\/p>\n\n\n\n Use the value of [katex]x[\/katex] to find measures of [katex]\\angle POC[\/katex] and [katex]\\angle COQ[\/katex].<\/p>\n\n\n\n [katex] \\begin{aligned}\\angle COQ &= x\\degree\\\\&= 70\\degree\\\\\\\\\\angle POC &= (x + 40)\\degree\\\\&=110\\degree \\end{aligned} [\/katex]<\/p>\n\n\n\n Hence, the linear pairs of angles\u2019 measures are: [katex]\\angle POC = 110\\degree[\/katex] and [katex]\\angle COQ = 70\\degree[\/katex].<\/p>\n\n\n\n These examples showcase just three of the many problems that are opened to us once we learn about linear pairs of angles. The problems may be simpler or may even be more complex than the ones we\u2019ve shown, but what\u2019s important is that you feel confident in applying the definition of linear pair of angles to solve these problems!<\/p>\n","protected":false},"excerpt":{"rendered":" Linear pair of angles are two angles that form a straight angle (angle measuring 180 degrees). In this article, we\u2019ll review what makes linear pair of angles unique, show you the important conditions before we can confirm a pair of angles are considered linear, and also let you try out different problems involving linear pair of angles.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-381","post","type-post","status-publish","format-standard","hentry","category-geometry"],"_links":{"self":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/381","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/comments?post=381"}],"version-history":[{"count":3,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/381\/revisions"}],"predecessor-version":[{"id":541,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/381\/revisions\/541"}],"wp:attachment":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/media?parent=381"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/categories?post=381"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/tags?post=381"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2021\/12\/linear-pair-of-angles-as-adjacent-and-supplementary-angles.png\")
<\/figure>\n\n\n\nProblem 1<\/h3>\n\n\n\n
![\"\"\/](\"https:\/\/lh4.googleusercontent.com\/0BcdYh7aNM6MjrtiECdEmnIV4PSV1qHvJwMGdHQWfOkIZxoBO9C8H70tC3Wka2mqUPLS6SiJNDsbcQOabxhnrfZyIRhztaEAs1c6Q4a0CLeNlj_KQn61Gn8JHh7SriDzxZ4I8zs\")
![\"\"\/](\"https:\/\/lh3.googleusercontent.com\/ygRmF8OFoutDUvTWiYwNA16lnYq2qZ513HGsjXUf2Nnf8BflSJYHV90eukJJ8SpTdrlv4yI4-I_jehLeXV8F03fPFI-ZEBTHfXF3aiIZRvNqzIZaSwSEDZGCBf91iGXQbHRrTx0\")
![\"\"\/](\"https:\/\/lh3.googleusercontent.com\/0eJga3_m_-kuQiF9oHowXUAHp5B__pzSrG7SgsVZLgDfdk3azTk7PuogUcZlpqBrC-L4ruGw-EjB9ngF3xDGeSMg5v8H41YDPNscMGmGr3XU6imO8MPeHbklngvw44cB8qEsxcw\")
![\"\"\/](\"https:\/\/lh5.googleusercontent.com\/cq74sFgETXqU9CPJDy5hZR7oVrzQnvu54NMlvwDq-vv4FaL1YpSUzTP-jnzeITZVzMbDNPWlA4NDpXZ2Q7GptkPw2UZWjaiv0DUATmuHf7Kl-LJEuLs3KyjGLSgavJV4jVAzDaY\")
![\"\"\/](\"https:\/\/lh5.googleusercontent.com\/t9H5udF_8IsbxWlt2DhUXHPoJw_HDL6idOZBacRAjK8cxy5CGN8VpE01n995dke_00Ttcbnrf-qVvsx2vuCrsD8xi14NAnGonVxLghyHRDy2tYIrSaSFpZ0RRmkrdVy56vrnkdM\")
![\"\"\/](\"https:\/\/lh4.googleusercontent.com\/ne62o3-c1KZCTDpwUOZS0aktjX6laqa4JYGNCjgUYNiwk9TmEL42ohsXNC8zGLPDvTgGz1zUjmfmSn4I5hpch5k0Um9y46uNsnyW5b1KlLDK75Z-WiERi7Crq_nY-LF-OIVBhmU\")
Problem 2<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2021\/12\/identifying-linear-pair-of-angles.png\")
<\/figure>\n\n\n\n![\"\"\/](\"https:\/\/lh6.googleusercontent.com\/FgqdT3A8_cxuFIQQN09hsJI0ytDARcY608kL5h8qo4yXJ5vyg5sBiogoPeV1ARdMLrJLs35qQ2FeJK3BHKEu3F7WnI9XFGBWOGBuVZby1-orFeEDLM7LOTouA-19cdYfHEd37vE\")
![\"\"\/](\"https:\/\/lh3.googleusercontent.com\/OmPa2BDT6qGnxea1SVTJeIFzSKpzugog_oZ4HXMdLj4Qi09DS7I1SD0ow8LhRnNeVzOJIEjkQNmc-1PTS6lTN0-Y6ZDjnsHuv3a-fOz24MNUzbs6xtT-3I-_PSA447Oi7cdy-_c\")
Understanding the Difference Between Linear Pair of Angles and Supplementary Angles<\/h2>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2021\/12\/linear-pair-of-angles-vs-supplementary-angles.png\")
<\/figure>\n\n\n\nWhat Are Problems Involving Linear Pair of Angles?<\/h2>\n\n\n\n
<\/figure>\n\n\n\nProblem 3<\/h3>\n\n\n\n
![\"\"\/](\"https:\/\/lh4.googleusercontent.com\/1ZZB1lYkUsdzmaMH6vEymT_jrVEV6c2kIn_QoPfmAmPNRSv6GfKnQtsGEb6P3dF_p-DuU10iBeAOcHTTXfQK5sPCB7ER5PHVM0MQTqQRfy6nTAkLGsIB5oMCoYZB_JG7ijsEeR8\")
Problem 4<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2021\/12\/solving-problems-involving-linear-pair-of-angles.png\")
<\/figure>\n\n\n\nProblem 5<\/h3>\n\n\n\n
![\"\"\/](\"https:\/\/lh3.googleusercontent.com\/4oq5JucxN8Jlrn1PV2bEdcyRsXYJIx1f1cxkdNPJ5FWtTih0mGLBipg3BID4jE9JNwasWmpw0mQIq9gjgMfGYnuo7qW9wen-2FDAX4VhalVdMS8Mf5WGAiZyFQ288ehkoYB8vx8\")