In order to observe consecutive interior angles, we need to have two lines (they could be parallel or they could eventually intersect) that are both intersected by a third, transversal<\/em> line.<\/p>\n We start with the two lines, \\(\\overleftrightarrow{x}\\) and \\(\\overleftrightarrow{y}\\).<\/p>\n And now a transversal line, \\(\\overleftrightarrow{T}\\), is crossing both original lines.<\/p>\n Notice that there are 8 angles total in this scenario. Any angle that sits in between line \\(\\overleftrightarrow{x}\\) and line \\(\\overleftrightarrow{y}\\) is an interior angle. Hence, in the above example, \\(\u2220c,\u2220d,\u2220e,\\) and \\(\u2220f\\) are all interior angles.<\/p>\n Now, let\u2019s look into consecutiveness:<\/p>\n When dealing with consecutive interior angles, it\u2019s important to understand that the word, \u201cconsecutive,\u201d is in respect to the transversal line. So, if two angles sit next to each other on the same side of the transversal, they are considered consecutive.<\/p>\n This all leads us to the following conclusions: the pair \\(\u2220c\\) and \\(\u2220e\\) are consecutive interior angles, and- similarly- \\(\u2220d\\) and \\(\u2220f\\) are consecutive interior angles as well.<\/p>\n Note that \\(\u2220c\\) and \\(\u2220f\\) ARE NOT consecutive interior angles; neither are \\(\u2220d\\) and \\(\u2220e\\), because neither pair sits on the same side of \\(\\overleftrightarrow{T}\\) together.<\/p>\n When discussing angles on any<\/em> number of intersecting lines, we have to understand what supplementary angles<\/strong> are. By definition, when the measures of two angles sum up to 180\u00b0, we call the pair \u201csupplementary.\u201d In fact, if any two angles form a linear pair (see below), then they are supplementary.<\/p>\n This concept is called the linear pair postulate<\/em>, and it is a key piece in understanding the Consecutive Interior Angles Theorem<\/strong>:<\/p>\n Let’s prove that \\(\u2220c\\) and \\(\u2220e\\) are supplementary.<\/p>\n We\u2019re given the parallel lines, \\(\\overleftrightarrow{x}\\) and \\(\\overleftrightarrow{y}\\), with the intersecting transversal, \\(\\overleftrightarrow{T}\\).<\/p>\n By the corresponding angle postulate, we know that \\(\u2220a\\) and \\(\u2220e\\) are congruent- meaning: \\(m\u2220a=m\u2220e\\) (which reads as \u201cthe measure of angle \\(a\\) equals the measure of angle \\(e\\)\u201d).<\/p>\n We also know by the linear pair postulate<\/em> that \\(m\u2220c+m\u2220a=180\u00b0\\). We can substitute \\(m\u2220e\\) for \\(m\u2220a\\): \\(m\u2220c+m\u2220e=180\u00b0\\).<\/p>\n Hence, by definition, \\(\u2220c\\) and \\(\u2220e\\) are supplementary.<\/p>\n Here\u2019s a simpler form of that proof:<\/p>\n![\"Line](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-1.png\")
<\/p>\n![\"Transversal](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-2.png\")
<\/p>\n![\"Diagram](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-3.png\")
<\/p>\n![\"2](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-4.png\")
<\/p>\n![\"Parallel](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-5.png\")
<\/p>\n
![\"Click](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/imcr20-New.png\")
<\/a>\n![\"Lines](\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Consecutive-Interior-Angles-O-PQ-6.png\")
<\/p>\n<\/div>\n\t\t\t\t\t