Lesson 18: Looking More Carefully at Parallel Lines
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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY Lesson 18: Looking More Carefully at Parallel Lines Student Outcomes ī§ Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180°. They learn how to prove the alternate interior angles theorem using the parallel postulate and the construction. Lesson Notes Lesson 18 is important. It may take two days to cover all of the information in this lesson. Hiding underneath its ideas are most of the reasons why reflection and translation can be defined based upon the geometry assumptions students learn in Lesson 33. (For example, only the geometry assumptions are used to construct a parallel line through a given point in Example 4.) While these ideas are hiding under the surface, do not focus on them. Instead, concentrate students’ attention on the main geometry assumption of this lesson: the parallel postulate. Parallel lines have already been defined, and in the missing angles problems (and perhaps discussed casually), the following two ideas have been used. 1. Suppose a transversal intersects a pair of lines. If a pair of alternate interior angles is equal in measure, then the pair of lines are parallel. 2. (A form of the parallel postulate) Suppose a transversal intersects a pair of lines. If the pair of lines are parallel, then the pair of alternate interior angles are equal in measure. However, students have probably not made careful distinctions between these thoughts. More likely, students remember the following concept: Suppose a transversal intersects a pair of lines. The pair of lines are parallel if and only if a pair of alternate interior angles are equal in measure. Perhaps students simply associate the ideas in this statement. When students see parallel lines cut by a transversal, they recognize the angle relations and then use those angle relations to recognize parallel lines. This lesson is designed to help students carefully distinguish between these two ideas. In particular, students are shown why there is a need for the parallel postulate as one of our geometric assumptions: Through a given external point there is at most one line parallel to a given line. Classwork Opening Exercise (8 minutes) Opening Exercise Exchange Problem Sets from Lesson 17 with a classmate. Solve the problems posed by your classmate while he or she solves yours. Compare your solutions, and then discuss and resolve any discrepancies. Why were you asked only to locate the point of rotation rather than to rotate a pre-image to obtain the image? How did you use perpendicular bisectors in constructing your solutions? Answers will vary. Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 149 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY Discussion Discussion We say that two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel. Examples 1–7 (30 minutes) Example 1 Why is the phrase in the plane critical to the definition of parallel lines? Explain and illustrate your reasoning. Two lines in (3-dimensional space) are called skew lines if they do not lie in the same plane. In that case, they do not intersect (because if they did, they would share a plane together) and are not parallel. (If they were parallel, then they would both have to lie in the same plane.) In Lesson 7, we recalled some basic facts learned in earlier grades about pairs of lines and angles created by a transversal to those lines. One of those basic facts is the following: Suppose a transversal intersects a pair of lines. The lines are parallel if and only if a pair of alternate interior angles are equal in measure. Our goal in this lesson is to prove this theorem using basic rigid motions, geometry assumptions, and a geometry assumption we introduce in this lesson called the parallel postulate. Of all of the geometry assumptions we have given so far, the parallel postulate gets a special name because of the special role it played in the history of mathematics. (Euclid included a version of the parallel postulate in his books, and for đ, đđđ years people tried to show that it was not a necessary assumption. Not only did it turn out that the assumption was necessary for Euclidean geometry, but study of the parallel postulate led to the creation of non-Euclidean geometries.) The basic fact above really has two parts, which we prove separately: 1. Suppose a transversal intersects a pair of lines. If two alternate interior angles are equal in measure, then the pair of lines are parallel. 2. Suppose a transversal intersects a pair of lines. If the lines are parallel, then the pair of alternate interior angles are equal in measure. The second part turns out to be an equivalent form of the parallel postulate. To build up to the theorem, first we need to do a construction. Example 2 Given a line đ and a point đˇ not on the line, follow the steps below to rotate đ by đđđ° to a line đ′ that passes through đˇ: MP.7 a. Label any point đ¨ on đ. b. Find the midpoint of segment đ¨đˇ using a ruler. (Measure the length of segment đ¨đˇ, and locate the point that is distance Lesson 18: đ¨đˇ đ from đ¨ between đ¨ and đˇ.) Label the midpoint đĒ. Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 150 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY c. Perform a đđđ° rotation around center đĒ. To quickly find the image of đ under this rotation by hand: i. Pick another point đŠ on đ. ii. ⥠. Draw đĒđŠ iii. Draw circle: center đĒ, radius đĒđŠ. iv. ⥠Label the other point where the circle intersects đĒđŠ by đ¸. v. ⥠. Draw đˇđ¸ MP.7 d. Label the image of the rotation by đđđ° of đ by đ′ = đšđĒ,đđđ (đ). How does your construction relate to the geometry assumption stated above to rotations? Complete the statement below to clarify your observations: đšđĒ,đđđ is a đđđ° rotation around đĒ. Rotations preserve lines ; therefore đšđĒ,đđđ , maps the line đ to the line đ′ . What is đšđĒ,đđđ (đ¨)? đˇ Example 3 The lines đ and đ′ in the construction certainly look parallel, but we do not have to rely on looks. Claim: In the construction, đ is parallel to đ′ . MP.7 PROOF: We show that assuming they are not parallel leads to a contradiction. If they are not parallel, then they must intersect somewhere. Call that point đŋ. Since đŋ is on đ′ , it must be the image of some point đē on đ under the đšđĒ,đđđ rotation (i.e., đšđĒ,đđđ (đē) = đŋ). Since đšđĒ,đđđ is a đđđ° rotation, đē and đŋ must be the endpoints of a diameter of a circle that ⥠must contain đĒ. Since đē is a point on đ, and đŋ is a different point on đ (it was the has center đĒ. In particular, đēđŋ ⥠because there is only one line through two points. But đēđŋ ⥠also contains intersection of both lines), we have that đ = đēđŋ đĒ, which means that đ contains đĒ. However, đĒ was constructed so that it was not on đ. This is absurd. There are only two possibilities for any two distinct lines đ and đ′ in a plane: either the lines are parallel, or they are not parallel. Since assuming the lines were not parallel led to a false conclusion, the only possibility left is that đ and đ′ were parallel to begin with. Example 4 The construction and claim together implies the following theorem. THEOREM: Given a line đ and a point đˇ not on the line, then there exists line đ′ that contains đˇ and is parallel to đ. This is a theorem we have justified before using compass and straightedge constructions, but now we see it follows directly from basic rigid motions and our geometry assumptions. Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 151 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Example 5 We are now ready to prove the first part of the basic fact above. We have two lines, đ and đ′ , and all we know is that a ⥠intersects đ and đ′ such that a pair of alternate interior angles are equal in measure. (In the picture below, transversal đ¨đˇ we are assuming đ∠đ¸đˇđ¨ = đ∠đŠđ¨đˇ.) Ė Ė Ė Ė . What happens if you rotate đđđ° around the center đĒ? Is there enough information to Let đĒ be the midpoint of đ¨đˇ show that đšđĒ,đđđ (đ) = đ′ ? a. What is the image of the segment đ¨đˇ? đˇđ¨ MP.7 b. In particular, what is the image of the point đ¨? đˇ c. ⥠? Why are the points đ¸ and đšđĒ,đđđ (đŠ) on the same side of đ¨đˇ ⥠to itself because đ¨đˇ ⥠contains the center đĒ. In Sketch of the Answer: The rotation by đđđ° maps đ¨đˇ ⥠to the other half-plane and vice versa. particular, it can be shown that the rotation maps one half-plane of đ¨đˇ Since đ¸ and đŠ are in opposite half-planes (by definition of alternate interior angles), and đŠ and đšđĒ,đđđ (đŠ) are in opposite half-planes, đ¸ and đšđĒ,đđđ (đŠ) must be in the same half-plane. d. What is the image of đšđĒ,đđđ (∠đŠđ¨đˇ)? Is it ∠đ¸đˇđ¨? Explain why. Because under the rotation, the vertex of ∠đŠđ¨đˇ maps to the vertex of ∠đ¸đˇđ¨, ray đ¨đˇ maps to ray đˇđ¨, the point đŠ goes to the same side as đ¸. Since đ∠đŠđ¨đˇ = đ∠đ¸đˇđ¨ (by assumption), ray đ¨đŠ must map to ray đˇđ¸. Thus, ∠đŠđ¨đˇ maps to ∠đ¸đˇđ¨. e. Why is đšđĒ,đđđ (đ) = đ′ ? ⥠= đ′. ⥠) = đˇđ¸ Since đ¨đŠ maps to đˇđ¸, đšđĒ,đđđ (đ) = đšđĒ,đđđ (đ¨đŠ We have just proven that a rotation by đđđ° takes đ to đ′ . By the claim in Example 3, lines đ and đ′ must be parallel, which is summarized below. THEOREM: Suppose a transversal intersects a pair of lines. If a pair of alternate interior angles are equal in measure, then the pair of lines are parallel. Discussion In Example 5, suppose we had used a different rotation to construct a line parallel to đ that contains đˇ. Such constructions are certainly plentiful. For example, for every other point đĢ on đ, we can find the midpoint of segment đˇđĢ and use the construction in Example 2 to construct a different đđđ° rotation around a different center such that the image of the line đ is a parallel line through the point đˇ. Are any of these parallel lines through đˇ different? In other words, Can we draw a line other than the line đ′ through đˇ that never meets đ? Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 152 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY The answer may surprise you; it stumped mathematicians and physicists for centuries. In nature, the answer is that it is sometimes possible and sometimes not. This is because there are places in the universe (near massive stars, for example) where the model geometry of space is not plane-like or flat but is actually quite curved. To rule out these other types of strange but beautiful geometries, we must assume that the answer to the previous question is only one line. That choice becomes one of our geometry assumptions: (Parallel Postulate) Through a given external point there is at most one line parallel to a given line. In other words, we assume that for any point đˇ in the plane not lying on a line đĩ, every line in the plane that contains đˇ intersects đĩ except at most one line—the one we call parallel to đĩ. Example 6 We can use the parallel postulate to prove the second part of the basic fact. THEOREM: Suppose a transversal intersects a pair of lines. If the pair of lines are parallel, then the pair of alternate interior angles are equal in measure. PROOF: Suppose that a transversal âĄđ¨đˇ intersects line đ at đ¨ and đ′ at đˇ, pick and label another point đŠ on đ, and choose a point đ¸ on đ′ on the opposite side of âĄđ¨đˇ as đŠ. The picture might look like the figure below: Let đĒ be the midpoint of đ¨đˇ, and apply a rotation by đđđ° around the center đĒ. As in previous discussions, the image of đ is the line đšđĒ,đđđ (đ), which is parallel to đ and contains point đˇ. Since đ′ and đšđĒ,đđđ (đ) are both parallel to đ and contain đˇ, by the parallel postulate, they must be the same line: đšđĒ,đđđ (đ) = đ′. In particular, đšđĒ,đđđ (∠đŠđ¨đˇ) = ∠đ¸đˇđ¨. Since rotations preserve angle measures, đ∠đŠđ¨đˇ = đ∠đ¸đˇđ¨, which was what we needed to show. Discussion It is important to point out that, although we only proved the alternate interior angles theorem, the same sort of proofs can be done in the exact same way to prove the corresponding angles theorem and the interior angles theorem. Thus, all of the proofs we have done so far (in class and in the Problem Sets) that use these facts are really based, in part, on our assumptions about rigid motions. Example 7 We end this lesson with a theorem that we just state but can be easily proved using the parallel postulate. THEOREM: If three distinct lines đđ , đđ , and đđ in the plane have the property that đđ âĨ đđ and đđ âĨ đđ , then đđ âĨ đđ . (In proofs, this can be written as, “If two lines are parallel to the same line, then they are parallel to each other.”) Note that students should at least remember that, in Euclidean Geometry, two lines are parallel if and only if alternate interior angles of any transversal are equal in measure and should be able to elaborate on what that means. This one statement includes both the parallel postulate and its converse. We can construct parallel lines without the parallel postulate, but in a geometry that does not satisfy the parallel postulate; there are many parallels to a given line through a point not on it. Without the parallel postulate, parallel lines are plentiful, and we cannot tell much about a line if all we know is that it passes through a point and is parallel to another line. Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 153 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY Relevant Vocabulary PARALLEL: Two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel lines. TRANSVERSAL: Given a pair of lines đ and đ in a plane, a third line đ is a transversal if it intersects đ at a single point and intersects đ at a single but different point. The definition of transversal rules out the possibility that any two of the lines đ, đ, and đ are the same line. ALTERNATE INTERIOR ANGLES: Let line đ be a transversal to lines đ and đ such that đ intersects đ at point đˇ and intersects đ at point đ¸. Let đš be a point on đ and đē be a point on đ such that the points đš and đē lie in opposite half planes of đ. Then the ∠đšđˇđ¸ and the ∠đˇđ¸đē are called alternate interior angles of the transversal đ with respect to đ and đ. CORRESPONDING ANGLES: Let line đ be a transversal to lines đ and đ. If ∠đ and ∠đ are alternate interior angles, and ∠đ and ∠đ are vertical angles, then ∠đ and ∠đ are corresponding angles. Closing (1 minute) ī§ ī§ Two lines are parallel if and only if alternate interior angles of any transversal are equal in measure. This can be separated into two statements: 1. Suppose a transversal intersects a pair of lines. If two alternate interior angles are equal in measure, then the pair of lines are parallel. 2. Suppose a transversal intersects a pair of lines. If the lines are parallel, then the pair of alternate interior angles are equal in measure. Statement 2 is an equivalent form of the parallel postulate: Through a given external point there is at most one line parallel to a given line. Exit Ticket (6 minutes) Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 154 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name ___________________________________________________ Date____________________ Lesson 18: Looking More Carefully at Parallel Lines Exit Ticket 1. Construct a line through the point đ below that is parallel to the line đ by rotating đ by 180° (using the steps outlined in Example 2). 2. Why is the parallel line you constructed the only line that contains đ and is parallel to đ? Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 155 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY Exit Ticket Sample Solutions 1. Construct a line through the point đˇ below that is parallel to the line đ by rotating đ by đđđ° (using the steps outlined in Example 2). The construction should look like the steps in Example 2. 2. Why is the parallel line you constructed the only line that contains đˇ and is parallel to đ? The answer should reference the parallel postulate in a meaningful way. Problem Set Sample Solutions Notice that we are frequently asked two types of questions about parallel lines. If we are told that two lines are parallel, then we may be required to use this information to prove the congruence of two angles (corresponding, alternate interior, etc.). On the other hand, if we are given the fact that two angles are congruent (or perhaps supplementary), we may have to prove that two lines are parallel. 1. 2. Ė Ė Ė Ė , and đŽđ˛ Ė Ė Ė Ė Ė ⊥ đŠđ´ Ė Ė Ė Ė âĨ đŽđ˛ Ė Ė Ė Ė Ė . Ė Ė Ė Ė âĨ đŠđ´ Ė Ė Ė Ė Ė , đ¨đŗ Ė Ė Ė Ė ⊥ đĒđ Ė Ė Ė Ė Ė . Prove that đĒđ In the figure, đ¨đŗ Ė Ė Ė Ė âĨ đŠđ´ Ė Ė Ė Ė Ė đ¨đŗ Ė Ė Ė Ė đ¨đŗ ⊥ Ė Ė Ė Ė đĒđ Given Ė Ė Ė Ė Ė đŽđ˛ ⊥ Ė Ė Ė Ė Ė đŠđ´ Given đ∠đ¨đĢđĒ = đđ° Definition of perpendicular lines đ∠đŠđąđ¯ = đđ° Definition of perpendicular lines đ∠đŠđąđ¯ = đ∠đ¨đ¯đŽ If two parallel lines are cut by a transversal, then the corresponding angles are equal in measure. đ∠đ¨đĢđĒ = đ∠đ¨đ¯đŽ Ė Ė Ė Ė âĨ đŽđ˛ Ė Ė Ė Ė Ė đĒđ Substitution property of equality Given If two lines are cut by a transversal such that the corresponding angles are equal in measure, then the lines are parallel. Ė Ė Ė Ė . Ė Ė Ė Ė īˇī§đŠđĒ Given that ∠đŠ and ∠đĒ are supplementary and đ∠đ¨ = đ∠đĒ, prove that đ¨đĢ ∠đŠ and ∠đĒ are supplementary. Given đ∠đ¨ = đ∠đĒ Given đ∠đŠ + đ∠đĒ = đđđ Definition of supplementary angles đ∠đŠ + đ∠đ¨ = đđđ Ė Ė Ė Ė đ¨đĢ âĨ Ė Ė Ė Ė đŠđĒ Substitution property of equality Lesson 18: If a transversal intersects two lines such that the same side interior angles are supplementary, then the lines are parallel. Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 156 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 18 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 3. 4. 5. 6. Mathematicians state that if a transversal to two parallel lines is perpendicular to one of the lines, then it is perpendicular to the other. Prove this statement. (Include a labeled drawing with your proof.) Ė Ė Ė Ė âĨ đŠđ´ Ė Ė Ė Ė Ė đ¨đŗ Ė Ė Ė Ė đ¨đŗ ī Ė Ė Ė Ė đĒđ Given đ∠đ¨đĢđĒ = đđ° Definition of perpendicular lines đ∠đ¨đĢđĒ = đ∠đŠđŦđĢ If two parallel lines are cut by a transversal, then corresponding angles are equal in measure. đ∠đŠđŦđĢ = đđ° Ė Ė Ė Ė ⊥ đŠđ´ Ė Ė Ė Ė Ė đĒđ Substitution Given Definition of perpendicular lines Ė Ė Ė Ė and đŦđ Ė Ė Ė Ė Ė . Prove that đ∠đ¨đđŦ = đ∠đĢđ˛đ¯. Ė Ė Ė Ė âĨ đĒđĢ Ė Ė Ė Ė âĨ đŽđ¯ In the figure, đ¨đŠ Ė Ė Ė Ė Ė Ė Ė Ė âĨ đĒđĢ đ¨đŠ Ė Ė Ė Ė Ė Ė Ė Ė Ė đŦđ âĨ đŽđ¯ Given đ∠đ¨đđŦ = đ∠đŽđąđ If two parallel lines are cut by a transversal, then the corresponding angles are equal in measure. đ∠đąđ˛đĒ = đ∠đŽđąđ If two parallel lines are cut by a transversal, then the corresponding angles are equal in measure. đ∠đąđ˛đĒ = đ∠đĢđ˛đ¯ Vertical angles are equal in measure. đ∠đ¨đđŦ = đ∠đĢđ˛đ¯ Transitive property Given In the figure, ∠đŦ and ∠đ¨đđŦ are complementary, and ∠đĒ and ∠đŠđĢđĒ are Ė Ė Ė Ė . Ė Ė Ė Ė âĨ đĒđŠ complementary. Prove that đ¨đŦ ∠đŦ and ∠đ¨đđŦ are complementary. Given ∠đĒ and ∠đŠđĢđĒ are complementary. Given đ∠đŦ + đ∠đ¨đđŦ = đđ° Definition of complementary angles đ∠đĒ + đ∠đŠđĢđĒ = đđ° Definition of complementary angles đ∠đŦ + đ∠đ¨đđŦ + đ∠đ¨ = đđđ° Sum of the angle measures in a triangle is đđđ°. đ∠đĒ + đ∠đŠđĢđĒ + đ∠đŠ = đđđ° Sum of the angle measures in a triangle is đđđ°. đ∠đ¨ = đđ°; đ∠ = đđ° Subtraction property of equality đ∠đ¨ = đ∠đŠ Ė Ė Ė Ė đ¨đŦ âĨ Ė Ė Ė Ė đĒđŠ Substitution property of equality If two lines are cut by a transversal such that a pair of alternate interior angles are equal in measure, then the lines are parallel. Given a line đ and a point đˇ not on the line, the following directions can be used to draw a line đ perpendicular to the line đ through the point đˇ based upon a rotation by đđđ°: a. Pick and label a point đ¨ on the line đ so that the circle (center đˇ, radius đ¨đˇ) intersects đ twice. b. Use a protractor to draw a perpendicular line đ through the point đ¨ (by constructing a đđ° angle). c. Use the directions in Example 2 to construct a parallel line đ through the point đˇ. Do the construction. Why is the line đ perpendicular to the line đ in the figure you drew? Why is the line đ the only perpendicular line to đ through đˇ? Figures will vary but should follow the directions outlined. Students can use Problem 3 or reprove the fact that if a transversal to two parallel lines is perpendicular to one, then it is perpendicular to the other. Suppose line đ′ contains đˇ and is perpendicular to đ; then by the interior angles theorem (or alternate interior angles or corresponding interior angles), đ′ is parallel to đ. Since đ and đ′ are both parallel to đ and contain đˇ, by the parallel postulate, they must be the same line (i.e., đ′ = đ). A student might answer, “parallel postulate.” If so, partial credit might be awarded (depending on your tastes for partial credit). Lesson 18: Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 157 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 18 M1 GEOMETRY Problems 7–10 all refer to the figure to the right. The exercises are otherwise unrelated to each other. 7. Ė Ė Ė Ė and ∠đŦđąđŠ is supplementary to ∠đąđŠđ˛. Prove that đ¨đĢ Ė Ė Ė Ė âĨ đŠđĒ Ė Ė Ė Ė âĨ đąđŦ Ė Ė Ė . đ¨đĢ Ė Ė Ė Ė Ė Ė Ė Ė âĨ đŠđĒ đ¨đĢ Given ∠đŦđąđŠ is supplementary to ∠đąđŠđ˛. Ė Ė Ė đŦđą âĨ Ė Ė Ė Ė đŠđĒ Given Ė Ė Ė Ė đ¨đĢ âĨ Ė Ė Ė đŦđą If a transversal intersects two lines such that the same side interior angles are supplementary, then the lines are parallel. If two segments are parallel to the same segment, then they are parallel to each other. 8. 9. Ė Ė Ė Ė đ¨đĢ âĨ Ė Ė Ė Ė đđŽ and Ė Ė Ė đŦđą âĨ Ė Ė Ė Ė đđŽ. Prove that ∠đĢđ¨đą and ∠đŦđąđ¨ are supplementary. Ė Ė Ė Ė đ¨đĢ âĨ Ė Ė Ė Ė đđŽ Ė Ė Ė Ė Ė Ė Ė âĨ đđŽ đŦđą Given Ė Ė Ė Ė âĨ đŦđą Ė Ė Ė đ¨đĢ If two segments are parallel to the same segment, then they are parallel to each other. đ∠đĢđ¨đą + đ∠đŦđąđ¨ = đđđ° If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. Given Ė Ė Ė Ė âĨ đ¨đŠ Ė Ė Ė Ė . đ∠đĒ = đ∠đŽ and ∠đŠ is supplementary to ∠đŽ. Prove that đĢđĒ đ∠đĒ = đ∠đŽ Given ∠đŠ is supplementary to ∠đŽ Given đ∠đŠ + đ∠đŽ = đđđ° Definition of supplementary angles đ∠đŠ + đ∠đĒ = đđđ° Ė Ė Ė Ė âĨ đ¨đŠ Ė Ė Ė Ė đĢđĒ Substitution property of equality If a transversal intersects two lines such that the same side interior angles are supplementary, then the lines are parallel. Ė Ė Ė Ė , and ∠đŦđ˛đĒ is supplementary to ∠đ˛đĒđĢ. Prove that đ¨đŠ Ė Ė Ė Ė . Ė Ė Ė Ė âĨ đŦđ Ė Ė Ė Ė , đŦđ Ė Ė Ė Ė ⊥ đĒđŠ Ė Ė Ė Ė âĨ đĢđĒ 10. đ¨đŠ Ė Ė Ė Ė âĨ đŦđ Ė Ė Ė Ė đ¨đŠ Ė Ė Ė Ė ī Ė Ė Ė Ė đŦđ đĒđŠ Given ∠đŦđ˛đĒ is supplementary to ∠đ˛đĒđĢ. Given đ∠đ¨đŠđĒ + đ∠đŠđ˛đŦ = đđđ° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary. đ∠đŦđ˛đĒ + đ∠đ˛đĒđĢ = đđđ° Definition of supplementary angles đ∠đŠđ˛đŦ = đđ° and đ∠đŦđ˛đĒ = đđ° Definition of right angles đ∠đ¨đŠđĒ = đđ° and đ∠đ˛đĒđŦ = đđ° Subtraction property of equality ∠đ¨đŠđĒ and ∠đ˛đĒđŦ are supplementary. Ė Ė Ė Ė đ¨đŠ âĨ Ė Ė Ė Ė đĢđĒ Definition of supplementary angles Given Lesson 18: If two lines are cut by a transversal such that a pair of interior angles on the same side are supplementary, then the lines are parallel. Looking More Carefully at Parallel Lines This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 158 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
