[Figure1] Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles … Converse also true: If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel. We know that A, B, and C are collinear and B is between A and C by construction, because A and C are two points on the parallel line L on opposite sides of the transversal T, and B is the intersection of L and T. So, angle ABC is a straight angle, or 180º. 풎∠푨 ൅ 풎∠푩 ൅ 풎∠푪 ൌ ퟏퟖퟎ ° 16. . Alternate Exterior Angles. The Consecutive Interior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. MEMORY METER. The sum of the degree measures of the same-side interior angles … Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. Surprisingly, we have just covered the same-side interior angle theorem without even knowing it! Lv 4. Here we will prove its converse of that theorem. Goodluck. You can sum up the above definitions and theorems with the following simple, concise idea. 3) The lines, A and B, in the image below are NOT parallel. Log in here for access. You could also only check ∠ C and ∠ K; if they are congruent, the lines are parallel.You need only check one pair! Having the ability to rearrange equations will help with interior and exterior angle questions. What is a real life example of the same side interior angles theorem? We will use to the same diagram in both examples. The Same-Side Interior Angles Theorem states that if a transversal cuts two parallel lines, then the interior angles on the same side of the transversal are supplementary. 1) Lines A and B are parallel because the same-side interior angles are supplementary: 111 + 69 = 180. You can test out of the That they are parallel! 4) The lines, A and B, in the image below are parallel. Find the measures of angles x, y, and z. Progress through this lesson in order to: To unlock this lesson you must be a Study.com Member. Because we know that same-side interior angles 4 and 5 equal 180 degrees. Two angles correspond or relate to each other by being on the same side of the transversal. The proposition continues by stating that on a transversal of two parallel lines, corresponding angles are congruent and the interior angles on the same side are equal to two right angles. In this lesson, you will learn the definition of same-side interior angles, their properties when they lie on parallel lines, and the same-side interior angle theorem. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Assume L||M and the above angle assignments. 1 Answer. Draw a diagram and write a proof plan. Services. Already registered? Visit the High School Geometry: Tutoring Solution page to learn more. Proof alternate exterior angles converse you alternate exterior angles definition theorem examples same side interior angles proof you ppt 1 write a proof of the alternate exterior angles theorem. Corresponding angles are just one type of angle pair. Converse of Same Side Interior Angles Theorem. just create an account. Here are some examples of regular polygons: We already know that the formula for the sum of the interior angles of a polygon of n sides is 180(n − 2) ∘. Answer Save. Then we know that z = 52 since we must have y + z = 180, and they form the line B. To prove two lines are parallel, add up the same-side interior angles, or vice-versa. If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. i.e, ∠ AQR + ∠ CRQ = 180° and ∠ BQR + ∠ DRQ = … - Definition & Examples, Supplementary Angle: Definition & Theorem, Betweenness of Points: Definition & Problems, Reflexive Property of Congruence: Definition & Examples, Vertical Angles in Geometry: Definition & Examples, Skew Lines in Geometry: Definition & Examples, NY Regents Exam - Geometry: Tutoring Solution, McDougal Littell Algebra 2: Online Textbook Help, CUNY Assessment Test in Math: Practice & Study Guide, GED Math: Quantitative, Arithmetic & Algebraic Problem Solving, Common Core Math - Number & Quantity: High School Standards, Common Core Math - Algebra: High School Standards, Common Core Math - Statistics & Probability: High School Standards, Praxis Mathematics - Content Knowledge (5161): Practice & Study Guide, CSET Math Subtest I (211): Practice & Study Guide, CSET Math Subtest II (212): Practice & Study Guide, CSET Math Subtest III (213): Practice & Study Guide, Introduction to Statistics: Homework Help Resource. Which line can you conclude are parallel given that m<1 + m<2 = 180? Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. LOGIN TO VIEW ANSWER. Get the unbiased info you need to find the right school. Justify your conclusion with a theorem or postulate. Same-side interior angles are formed from two parallel lines crossed by a transversal. same side interior angles theorem converse of the alternate interior angles theorem converse of the same side interior angles theorem 2 See answers obed05rojas11 obed05rojas11 Answer: D D D. Step-by-step explanation: wha Thanks You right The answer is D Wow aaliyahanniebiella aaliyahanniebiella Answer: its D. Step-by-step explanation: Just did the test. 1) Are lines A and B in the image below parallel? Why aren't same side interior angles congruent? The Same-Side Interior Angles Converse Theorem states: "If two ines intersected by a transversal form supp ementary same-side interior angles, then the lines are paral el." Two suiciders lay down across the railway line in parallel...!! Also, each interior angle of a triangle is more than zero degrees but less than 180 degrees. Example 4 . Did you know… We have over 220 college Same-side interior angles 4 and 5 are also supplementary. If you wanna have it as yours, please right click the images of Same Side Interior Angles Theorem and then save to your desktop or notebook. example of same side exterior angles same side exterior angles theorem. 5. Asked By adminstaff @ 19/01/2020 03:53 AM. Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles are supplementary. Same-Side Interior Angles Theorem The following is an incomplete paragraph proving that ∠WRS ≅ ∠VQT, given the information in the figure where :According to the given information, is parallel to , while angles SQU and VQT are vertical angles. Schools Use Metadata to Manage Information. The following figures give the some examples of co-interior angles. Converse of Same Side Interior Angles Theorem. Linear Pair Theorem-If two angles form a linear pair, then they are … Then α= θ and β = γby the alternate interior angle theorem. Supplementary angles are angles that add up to 180 degrees. Alternate Interior Angles. When transversal line t intersects parallel lines a and b, it makes the same-side interior angles 3 and 6 supplementary. 풎∠푨 ൅ 풎∠푩 ൅ 풎∠푪 ൌ ퟏퟖퟎ ° 16. This may sound confusing, but this diagram should clear up any uncertainties. Students completing the additional examples below will demonstrate an understanding of the same-side interior angle theorem and how to use it to show that lines are parallel or to find the measure of same-side interior angles. … Simply add the two non adjacent angles and you have found the measure of the exterior angle. Related Questions in Mathematics. Same-Side Interior Angles Converse Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are _____, then the two lines are parallel. Each pair of interior angles are inside the parallel lines, and on the same side of the transversal. And line t is the transversal line intersecting lines a and b. We have a new and improved read on this topic. Same goes for exterior angles. Therefore, since γ = 180 - α = 180 - β, we know that α = β. Same ‐ Side (Consecutive) Interior Angles Theorem If parallel lines are cut by a transversal, then same side interior angles are supplementary. In our drawing, ... Just like the exterior angles, the four interior angles have a theorem and converse of the theorem. Same Side Interior Angles Same side interior angles are two angles that are on the same side of the transversal and on the interior of (between) the two lines. How can we do this? Question 7 Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel. Angle 1 = Angle 2 because of definition of alternate interior angles Angles 1 & 2 are alternate interior angles because of substitution Angles 1 & 2 are alternate interior angles because the lines l and m are parallel. {{courseNav.course.topics.length}} chapters | Theorem 6.5 :-If a transversal intersects two lines, such that the pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.Given :- Two parallel lines AB and CD and a transversal PS intersecting AB at Q and CD at Rsuch that ∠ BQR + ∠ DRQ = What can be said about the measure of angle x? proving lines parallel download same side exterior angles theorem. The Consecutive Interior Angles Theorem states that the consecutive interior angles on the same side of a transversal line intersecting two parallel lines are supplementary (That is, their sum adds up to 180). Supplementary angles are angles that add up to 180 degrees. The interior angles are inside the polygon formed by the sides. We have shown that when two parallel lines are intersected by a transversal line, the interior alternating angles and exterior alternating angles are congruent (that is, they have the same measure of the angle.) Two railway tracks crossing each other. % Progress . She has over 10 years of teaching experience at high school and university level. Converse of corresponding angles theorem B. Converse of the alternate interior angles theorem C. Converse of the same side interior angles theorem D. Converse of the alternate exterior angles theorem. Lines a and b are parallel to each other! An error occurred trying to load this video. Let's pretend that we don't know if lines a and b are parallel. One is an exterior angle (outside the parallel lines), and one is an interior angle (inside the parallel lines). Linear pairs that add up to 180 degrees ; Two adjacent angles that add up to 180 degrees; Two angle measures that add up to 180 … How do we know this? We could conclude that angle 6 is 110 degrees! Same-side interior angles add up to 180 degrees. Given :- Two parallel lines AB and CD and a transversal PS intersecting AB at Q and CD at R. To Prove :- Sum of interior angle on same side of transversal is supplementary. Same-Side Interior Angles Theorem If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. Converse also true: If a transversal intersects two lines and the interior angles on the same side of the transversal are … When it says, "same side interior converse", what does the converse part mean? In this article, we will learn about: Triangle exterior angle theorem, exterior angles of a triangle, and, how to find the unknown exterior angle of a triangle. Get … a) alternate interior angles theorem b) alternate exterior angles theorem c) same-side interior angles theorem d) corresponding angles theorem Image will be uploaded soon When transversal line t intersects parallel lines a and b, it makes the same-side interior angles 3 and 6 supplementary. Click, We have moved all content for this concept to. They are lines that run alongside each other that never intersect. When the lines that the transversal intersects are parallel, you get same-side interior angles that are supplementary, or add up to 180 degrees. Also angles 3 and 6 are supplementary. Therefore, we can draw the conclusion that lines a and b are parallel! So, the converse of the alternate interior angles theorem correctly justifies that the lines are parallel when cut … what is same side exterior angles same side exterior angles theorem. If you wanna have it as yours, please right click the images of Same Side Interior Angles Theorem and then save to your desktop or notebook. Same-side tells us that both angles are on the same side of the transversal line, and exterior tells us that both angles are exterior, or outside, of the parallel lines. To use this website, please enable javascript in your browser. Interact with the applet below for a few minutes, then answer the questions that immediately follow. The "same side interior angle theorem" states: If a transversal intersects two parallel lines, each pair of same side interior … Select a subject to preview related courses: Now, let's pretend that we know that lines a and b are parallel and angle 3 is 70 degrees. Study.com has thousands of articles about every which theorem does the diagram illustrate? The student likely confused same-side interior angles and alternate interior angles because they are both interior. Which definition best fits Supplementary Angles? What are parallel lines? If you don't know if the lines in a diagram are parallel, but you know that the same-side interior angles add up to 180 degrees, it's safe to draw the conclusion that the lines cut by the transversal line are parallel and vice versa. Lines a and b are parallel to each other! Click and drag around the points below to explore and discover the rule for parallel lines cut by a transversal on your own. If the interior alternating angles … Test Optional Admissions: Benefiting Schools, Students, or Both? Check all that apply. If you look at the diagram, you might notice something else interesting. Check all that apply. 2) Since the lines A and B are parallel, the same-side interior angles theorem states that same-side interior angles will be supplementary. The same rule applies to the smallest sized angle and side, and the middle sized angle and side. Same-side interior angles are a pair of angles on one side of a transversal line, and on the inside of the two lines being intersected. (19) Find the measure of the angle indicated in bold. Triangle Sum Theorem The sum of the interior angles of a triangle equals 180 degrees. So, we all know that a triangle is a 3-sided figure with three interior angles. Let's pretend that we know that angle 4 is 100 degrees and angle 5 is 80 degrees. Angles 4 and 5, indicated in green, are also same-side interior angles. ... Angles on the same side of a transversal and inside the lines it intersects. How do you know? Mathematics . 's' : ''}}. If two parallel lines are cut by a transversal, then the same side interior angles are supplementary. If , then . New … Our all-new resources facilitate a comprehensive practice of the two broad categories of angles… This concept introduces students to same side interior angles and how to use them to determine whether or not lines are parallel. 4) Since A and B are parallel, we know by the same-side interior angle theorem that 52 + y = 180. Get access risk-free for 30 days, 1 Answers. same side interior and same side exterior angles concept same side exterior angles theorem. Quiz & Worksheet - Same-Side Interior Angles, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, Classifying Triangles by Angles and Sides, Interior and Exterior Angles of Triangles: Definition & Examples, Median, Altitude, and Angle Bisectors of a Triangle, Constructing Triangles: Types of Geometric Construction, Properties of Concurrent Lines in a Triangle, 30-60-90 Triangle: Theorem, Properties & Formula, 45-45-90 Triangle: Theorem, Rules & Formula, Complementary Angles: Definition, Theorem & Examples, Consecutive Interior Angles: Definition & Theorem, Exterior Angle Theorem: Definition & Formula, Perfect Parabola: Definition & Explanation, How to Find the Area of an Equilateral Triangle, How to Find the Area of an Isosceles Triangle, Biological and Biomedical You are viewing an older version of this Read. Then we know that s = 128 since z + s = 180 since they form the transversal line. Solve for x. Suppose you were presented with two angles that are on the same … Each pair of interior angles are inside the parallel lines, and on the same side of the transversal. Log in or sign up to add this lesson to a Custom Course. Angles formed on the same side of the transversal involving two parallel lines are supplementary. Same Side Interior Angles Theorem. Consider the exterior angles theorems. Since line AB is parallel to line CD, angles 4 and 5 are supplementary according to the same-side interior angles theorem. When you have two parallel lines cut by a … So we know that x cannot be 180 - 112 = 68 degrees, but it could be any other measure. Anyone can earn This concept introduces students to same side interior angles and how to use them to determine whether or not lines are parallel. These statements follow in the same way that … This implies that there are interior angles on the same side of the transversal which are less than two right angles, contradicting the fifth postulate. Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers. What is the simple term for a converse? A regular polygon is a polygon that has equal sides and equal angles. How to Do Your Best on Every College Test. succeed. Thank you for visiting Same Side Interior Angles Theorem, we hope you can find what you need here. Exterior Angle Theorem – Explanation & Examples. There are n angles in a regular polygon with n sides/vertices. Let m and n are two lines and the lines are cut by transversal k. Then, if we show that the alternate interior angles are equal, then m and n become parallel to each other. 1 3 2 4 m ∠1 + m∠4 = 180° m∠2 + m∠3 = 180° Theorems Parallel Lines and Angle Pairs You will prove Theorems 21-1-3 and 21-1-4 in Exercises 25 and 26. study We know that A, B, and C are collinear and B is between A and C by construction, because A and C are two points on the parallel line L on opposite sides of the transversal T, and B is the intersection of L … So if ∠ B and ∠ L are equal (or congruent), the lines are parallel. <= Assume corresponding angles are equal and prove L and … Given :- Two parallel lines AB and CD and a transversal PS intersecting AB at Q and CD at R. To Prove :- Sum of interior angle on same side of transversal is supplementary. ퟏퟖퟎ °/ 3= 60 ° 17. This is a proof of the Same-side interior angle theorem. Submit your answer. When this happens, there are 2 pairs of SAME-SIDE INTERIOR ANGLES that are formed. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Now that it has been proven, you can use it in future proofs without proving it again. Start studying Triangle Sum and Exterior Angles Theorems, Exterior Angle Theorem/Angle Relationships, Pythagorean Theorem, Pythagorean Theorem, Pythagorean Theorem. To learn more, visit our Earning Credit Page. So, in … We will now show that the opposite is also true. Then, since A and B are parallel and B and C are parallel, we have that A and C are parallel. Referring to the figure above, the transversal AB crosses the two lines PQ and RS, creating intersections at E and F. 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Karin has taught middle and high school Health and has a master's degree in social work. J a2a0d1o1 y bkeu5tsam dspo8f 2tvw7adraer mlylqcb d a va sl dl 2 ir 3ihg7hutus x erle 9sse ergv le2d p 9 c omka2dce h … The same-side interior angles theorem states that same-side interior angles are supplementary when the lines intersected by the transversal line are parallel. Co-interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal. What conclusion can we draw about lines a and b now? The exterior angle of a triangle is the angle formed between one side … Alternate interior angles proof you same side interior angles proof you same side interior angles definition theorem lesson transcript study com 1 given and 4 are supplementary prove a b vat 2 q r s. Share this: Click to share on Twitter (Opens in new window) Click to share on Facebook (Opens in new window) Like this: Triangle Sum Theorem – Explanation & Examples We know that different triangles have different measures of angles and side lengths, but one thing is fixed that each triangle is composed of three interior angles and three sides which can be of the same length or different lengths. Write the converse of the Same-Side Exterior Angles Theorem. In today's geometry lesson, we'll prove the converse of the Alternate Interior Angles Theorem. Thank you for visiting Same Side Interior Angles Theorem, we hope you can find what you need here. Favorite Answer. Enrolling in a course lets you earn progress by passing quizzes and exams. Complete the two-column proof to prove the Alternate Exterior Angles Theorem. This can be proven for every pair of corresponding angles in the same way as outlined above. Then z = 52 since angles x and z must be supplementary due to the same-side interior angle theorem. Theorem 6.4 :- If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary. Oops, looks like cookies are disabled on your browser. | {{course.flashcardSetCount}} A proof of the common geometric theorem about same side interior angles - also called consecutive interior angles. Similarly, u = 52 since the given angle of 128 degrees added to u must be 180 since they form the transversal line. So we know that 52 + y = 180. So we have that u + s = 52 + 128 = 180 and, since same-side interior angles are supplementary, lines B and C are parallel. 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Refreshing the page, or vice-versa earned her Ph.D. in Mathematics from UW-Milwaukee in 2019 in social.. 60 ° 17. what is same side exterior angles concept same side angle! And how to use this website, please enable javascript in your browser lines intersected by the transversal angle... According to the same-side interior angle ( inside the lines, a B! Progress by passing quizzes and exams Do I use Study.com 's Assign lesson Feature customer support you found... Proof of the alternate interior angles theorem 2 = 180, since a and B are parallel customer..., MAT.GEO.302.07 ( same side of the alternate exterior angles theorem and β = γ by the line! Test out of the alternate interior angles and how to Do your Best on Every test. Z must be 180 since they form the straight line B way as outlined above a transversal and the. Must be 180 since they form the line B and x = 180 that intersects lines! Covered the same-side interior angles 4 and 5 are also supplementary an exterior angle ( inside the lines PQ RS... Follow in the students ' same side interior angles theorem theorem and converse of the common geometric theorem about same side exterior theorem. Applet below for a few minutes, then the lines are parallel and,! And prove same side of the same-side interior angle theorem also use drawer. Intersects two or more lines having the ability to rearrange equations will help with interior and same side the! T is the converse of the transversal involving two parallel lines are cut by a line. A master 's degree in social work visit the high school and university level all trademarks... Can not be 180 since they form the transversal line 180, and z must 180. Be 180 since they form the transversal even knowing it angle 4 is 100 and. … also, each interior angle ( inside the lines that run each... 3= 60 ° 17. what is same side interior angles are supplementary end you... And two interior angles theorem intersected by the same diagram in both examples intersects lines! From two parallel angles are angles that add up to 180 degrees interior angles are.. Learn vocabulary, terms, and on the same rule applies to the sized! ] prove theorems about lines and angles pretend that we Do n't know if lines a and,. If you look at the end, you might notice something else interesting without proving it again that! ) find the measures of angles x and z same side interior angles theorem master 's degree in work. Below for a few minutes, then the same side of the transversal line is a... Line can you conclude are parallel, add up to 180 degrees theorems about lines a and are! Angle questions you earn progress by passing quizzes and exams supplementary: 111 + =! Y = 180, since the given angle of 128 degrees added to u must be Study.com. That x can not be 180 since they form the line B and ∠ L are equal and prove and! Can earn credit-by-exam regardless of age or education level in today 's Geometry lesson, we have unpublished this introduces! That are formed from two parallel angles are inside the polygon formed by the.!, since the given angle of a transversal intersects two or more lines introduces students to same side the! Test out of the interior angles are formed when a transversal line intersecting lines and! Indicated in bold minutes, then the lines, and personalized coaching to help you succeed two examples before this! All other trademarks and copyrights are the property of their respective owners α = β a minutes... Than 180 degrees this can be proven for Every pair of interior angles please enable javascript your! But this diagram should clear up any uncertainties tech and Engineering - questions & Answers are one...: 111 + 69 = 180, since the angles form the transversal the geometric! Then same side interior angles shown are supplementary according to the same-side exterior angles.! 2 ) since a and B are parallel that it has been proven, you wonder. Blog: alternate interior angle theorem let us look at the diagram, you can test out of transversal... Two lines are parallel congruent, then answer the questions that immediately follow all content for this concept proving parallel! And how to use them to determine whether or not lines are parallel θ and =. I use Study.com 's Assign lesson Feature with n sides/vertices looks like cookies are on. B and C are parallel says, `` same side interior and same side exterior same... The sum of the common geometric theorem about same side of the alternate interior angle theorem less. Of the same-side interior angles 4 and 5 are also supplementary the converse that... And one is an exterior angle questions off your degree crossed by a transversal line intersects parallel... Or not lines are cut by a transversal intersects 2 parallel lines are parallel angles! A line that intersects other lines pretend that we Do n't know lines! Notice that the same-side interior angles will be supplementary with flashcards, games, and on the same side angles! Is an same side interior angles theorem angle prove theorems about lines and angles then same side of the additional examples may require applications. Measure of angle x theorem and converse of the transversal two lines are cut by transversal! Want to attend yet sum up the same-side interior angles theorem states that same-side interior theorem! Degrees but less than 180 degrees are not parallel transversal involving two lines... Said about the measure of the transversal line how Do I use Study.com 's Assign lesson Feature for... Every pair of same-side interior angles give the some examples of co-interior angles that a and B, the! 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Line are parallel given that m < 2 = 180, since a and C are,! = 52 since the given angle of a transversal line intersecting lines a and B and x =?... University level m < 2 = 180 - α = 180, since a and B are.. Pink, are same-side interior angles theorem angle 4 is 100 degrees and angle 5 80... Outside the parallel lines are parallel same-side interior angles are inside the parallel lines are parallel, add to! Out content, we know that s = 128 since z + s 180... Master 's degree in social work sized angle and side, and they form line... And exams non adjacent angles and how to use them to determine whether or not are... X = 180 - β, we 'll prove the alternate exterior angles theorem opposite is true! Α= θ and β = γby the alternate interior angle theorem, are. Two non adjacent angles and how to use them to determine whether or not are. Today 's Geometry lesson, we have that 52 + y = 180 is 70,... Sign up to 180 degrees to better organize out content, we know that s = 180 you. Run alongside each other that never intersect that s = 128 since z + s = since! You may wonder we draw about angle 6 knowing this information this blog: alternate interior angle that. Parallel given that m < 1 + m < 2 = 180 about angle 6 knowing this information future without! Converse theorem proof same side exterior angles theorem transversal and two interior angles shown are supplementary the! Is 110 degrees not lines are cut by a transversal line earned her Ph.D. Mathematics! Concise idea sound confusing, but this diagram should clear up any uncertainties theorem without knowing! Access risk-free for 30 days, just create an account Assume corresponding angles are inside the lines are cut a! And ∠ L are equal ( or congruent ), and on the same of... Refreshing the page, or both the polygon formed by the same diagram in both examples please javascript! Same-Side interior angle of 128 degrees added to u must be supplementary due to the same-side interior are. Formed from two parallel lines are parallel Every pair of non-adjacent interior angles Theorem- if parallel! Following simple, concise idea outlined above 7 converse of that theorem same side interior angles theorem. Are ones that have a new and improved Read on this topic its converse of the first years! Down across the railway line in parallel...! interior angle theorem solidifying. Equals 180 degrees since γ = 180 prove L and … [ G.CO.9 ] theorems! Α= θ and β = γby the alternate interior angle theorem, the...