{"id":82430,"date":"2022-06-02T18:30:13","date_gmt":"2022-06-02T13:00:13","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=82430"},"modified":"2022-05-10T14:19:20","modified_gmt":"2022-05-10T08:49:20","slug":"eureka-math-grade-8-module-2-lesson-13","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/eureka-math-grade-8-module-2-lesson-13\/","title":{"rendered":"Eureka Math Grade 8 Module 2 Lesson 13 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math 8th Grade Module 2 Lesson 13 Answer Key<\/h2>\n

Eureka Math Grade 8 Module 2 Lesson 13 Exploratory Challenge Answer Key<\/h3>\n

Classwork
\nConcept Development
\n\"Eureka
\nm\u22201+m\u22202+m\u22203=m\u22204+m\u22205+m\u22206=m\u22207+m\u22208+m\u22209=180\u00b0
\nNote that the sum of the measures of angles 7 and 9 must equal 90\u00b0 because of the known right angle in the right triangle.<\/p>\n

We want to prove that the angle sum of any triangle is 180\u00b0. To do so, we use some facts that we already know about geometry:
\n\u2192 A straight angle is 180\u00b0 in measure.
\n\u2192 Corresponding angles of parallel lines are equal in measure (corr. \u2220\u2019s, \\(\\overline{A B}\\) || \\(\\overline{C D}\\).
\n\u2192 Alternate interior angles of parallel lines are equal in measure (alt. \u2220\u2019s, \\(\\overline{A B}\\) || \\(\\overline{C D}\\).<\/p>\n

Exploratory Challenge 1.
\nLet triangle ABC be given. On the ray from B to C, take a point D so that C is between B and D. Through point C, draw a segment parallel to \\(\\overline{A B}\\), as shown. Extend the segments AB and CE. Line AC is the transversal that intersects the parallel lines.
\n\"Eureka<\/p>\n

a. Name the three interior angles of triangle ABC.
\nAnswer:
\n\u2220ABC, \u2220BAC, \u2220BCA<\/p>\n

b. Name the straight angle.
\nAnswer:
\n\u2220BCD
\nOur goal is to show that the measures of the three interior angles of triangle ABC are equal to the measures of the angles that make up the straight angle. We already know that a straight angle is 180\u00b0 in measure.
\nIf we can show that the interior angles of the triangle are the same as the angles of the straight angle, then we will have proven that the sum of the measures of the interior angles of the triangle have a sum of 180\u00b0.<\/p>\n

c. What kinds of angles are \u2220ABC and \u2220ECD? What does that mean about their measures?
\nAnswer:
\n\u2220ABC and \u2220ECD are corresponding angles. Corresponding angles of parallel lines are equal in measure (corr. \u2220\u2019s, \\(\\overline{A B}\\) || \\(\\overline{\\boldsymbol{C E}}\\)).<\/p>\n

d. What kinds of angles are \u2220BAC and \u2220ECA? What does that mean about their measures?
\nAnswer:
\n\u2220BAC and \u2220ECA are alternate interior angles. Alternate interior angles of parallel lines are equal in measure (alt. \u2220\u2019s, \\(\\overline{\\boldsymbol{A B}}\\) || \\(\\overline{\\boldsymbol{C E}}\\)).<\/p>\n

d. We know that m\u2220BCD=m\u2220BCA+m\u2220ECA+m\u2220ECD=180\u00b0. Use substitution to show that the measures of the three interior angles of the triangle have a sum of 180\u00b0.
\nAnswer:
\nm\u2220BCD=m\u2220BCA+m\u2220BAC+m\u2220ABC=180\u00b0 (\u2220 sum of \u25b3)<\/p>\n

Exploratory Challenge 2.
\nThe figure below shows parallel lines L1<\/sub> and L2<\/sub> . Let m and n be transversals that intersect L1<\/sub> at points B and C, respectively, and L2<\/sub> at point F, as shown. Let A be a point on L1<\/sub> to the left of B, D be a point on L1<\/sub> to the right of C, G be a point on L2<\/sub> to the left of F, and E be a point on L2<\/sub> to the right of F.
\n\"Eureka<\/p>\n

a. Name the triangle in the figure.
\nAnswer:
\n\u25b3BCF<\/p>\n

b. Name a straight angle that will be useful in proving that the sum of the measures of the interior angles of the triangle is 180\u00b0.
\nAnswer:
\n\u2220GFE
\nAs before, our goal is to show that the sum of the measures of the interior angles of the triangle are equal to the measure of the straight angle. Use what you learned from Exploratory Challenge 1 to show that the measures of the interior angles of a triangle have a sum of 180\u00b0.<\/p>\n

c. Write your proof below.
\nAnswer:
\nThe straight angle \u2220GFE is comprised of \u2220GFB, \u2220BFC, and \u2220EFC. Alternate interior angles of parallel lines are equal in measure (alt. \u2220\u2019s, \\(\\overline{A D}\\) || \\(\\overline{C E}\\)). For that reason, \u2220BCF=\u2220EFC and \u2220CBF=\u2220GFB. Since \u2220GFE is a straight angle, it is equal to 180\u00b0. Then, \u2220GFE=\u2220GFB+\u2220BFC+\u2220EFC=180\u00b0.
\nBy substitution, \u2220GFE=\u2220CBF+\u2220BFC+\u2220BCF=180\u00b0. Therefore, the sum of the measures of the interior angles of a triangle is 180\u00b0 (\u2220 sum of \u25b3).<\/p>\n

Eureka Math Grade 8 Module 2 Lesson 13 Exit Ticket Answer Key<\/h3>\n

Question 1.
\nIf L1<\/sub> || L2<\/sub> , and L3<\/sub> || L4<\/sub> , what is the measure of \u22201? Explain how you arrived at your answer.
\n\"Eureka
\nAnswer:
\nThe measure of angle 1 is 29\u00b0. I know that the angle sum of triangles is 180\u00b0. I already know that two of the angles of the triangle are 90\u00b0 and 61\u00b0.<\/p>\n

Question 2.
\nGiven that line AB is parallel to line CE, present an informal argument to prove that the measures of the interior angles of triangle ABC have a sum of 180\u00b0.
\n\"Eureka
\nAnswer:
\nSince \\(\\overleftrightarrow{A B}\\) is parallel to \\(\\overleftrightarrow{C E}\\), the corresponding angles \u2220BAC and \u2220ECD are equal in measure. Similarly, \u2220ABC and \u2220ECB are equal in measure because they are alternate interior angles. Since \u2220ACD is a straight angle (i.e., equal to 180\u00b0 in measure), substitution shows that triangle ABC has a sum of 180\u00b0. Specifically, the straight angle is made up of \u2220ACB, \u2220ECB, and \u2220ECD. \u2220ACB is one of the interior angles of the triangle and one of the angles of the straight angle. We know that \u2220ABC has the same measure as \u2220ECB and that \u2220BAC has the same measure as \u2220ECD. Therefore, the sum of the measures of the interior angles will be the same as the sum of the measures of the angles of the straight angle, which is 180\u00b0.<\/p>\n

Eureka Math Grade 8 Module 2 Lesson 13 Problem Set Answer Key<\/h3>\n

Students practice presenting informal arguments about the sum of the angles of a triangle using the theorem to find the measures of missing angles.<\/p>\n

Question 1.
\nIn the diagram below, line AB is parallel to line CD, that is, LAB<\/sub> \u2225LCD<\/sub>. The measure of \u2220ABC is 28\u00b0 and the measure of \u2220EDC is 42\u00b0. Find the measure of \u2220CED. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle.
\n\"Eureka
\nAnswer:
\nThe measure of \u2220CED is 110\u00b0. This is the correct measure for the angle because \u2220ABC and \u2220DCE are alternate interior angles of parallel lines. That means that the angles are congruent and have the same measure. Since the angle sum of a triangle is 180\u00b0, then \u2220CED=180\u00b0-(28\u00b0+42\u00b0) and \u2220CED is 110\u00b0.<\/p>\n

Question 2.
\nIn the diagram below, line AB is parallel to line CD, that is, LAB<\/sub> || LCD<\/sub> . The measure of \u2220ABE is 38\u00b0, and the measure of \u2220EDC is 16\u00b0. Find the measure of \u2220BED. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: Find the measure of \u2220CED first, and then use that measure to find the measure of \u2220BED.)
\n\"Eureka
\nAnswer:
\nThe measure of \u2220BED is 54\u00b0. This is the correct measure for the angle because \u2220ABC and \u2220DCE are alternate interior angles of parallel lines. That means that the angles are congruent and have the same measure. Since the angle sum of a triangle is 180\u00b0, then \u2220CED=180\u00b0-(38\u00b0+16\u00b0)=126\u00b0. The straight angle \u2220BEC is made up of \u2220CED and \u2220BED. Since we know straight angles measure 180\u00b0 and \u2220CED=126\u00b0, then \u2220BED is 54\u00b0.<\/p>\n

Question 3.
\nIn the diagram below, line AB is parallel to line CD, that is, LAB<\/sub> || LCD<\/sub> . The measure of \u2220ABE is 56\u00b0, and the measure of \u2220EDC is 22\u00b0. Find the measure of \u2220BED. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: Extend the segment BE so that it intersects line CD.)
\n\"Eureka
\nAnswer:
\nThe measure of \u2220BED is 78\u00b0. This is the correct measure for the angle because \u2220ABE and \u2220DFE are alternate interior angles of parallel lines. That means that the angles are congruent and have the same measure. Since the angle sum of a triangle is 180\u00b0, then \u2220FED=180\u00b0-(56\u00b0+22\u00b0)=102\u00b0. The straight angle \u2220BEF is made up of \u2220FED and \u2220BED. Since straight angles measure 180\u00b0 and \u2220FED is 102\u00b0, then \u2220BED is 78\u00b0.<\/p>\n

Question 4.
\nWhat is the measure of \u2220ACB?
\n\"Eureka
\nAnswer:
\nThe measure of \u2220ACB is 180\u00b0-(83\u00b0+64\u00b0), which is equal to 33\u00b0.<\/p>\n

Question 5.
\nWhat is the measure of \u2220EFD?
\n\"Eureka
\nAnswer:
\nThe measure of \u2220EFD is 180\u00b0-(101\u00b0+40\u00b0), which is equal to 39\u00b0.<\/p>\n

Question 6.
\nWhat is the measure of \u2220HIG?
\n\"Eureka
\nAnswer:
\nThe measure of \u2220HIG is 180\u00b0-(154\u00b0+14\u00b0), which is equal to 12\u00b0.<\/p>\n

Question 7.
\nWhat is the measure of \u2220ABC?
\n\"Eureka
\nAnswer:
\nThe measure of \u2220ABC is 60\u00b0 because 60\u00b0+60\u00b0+60\u00b0, which is equal to 180\u00b0.<\/p>\n

Question 8.
\nTriangle DEF is a right triangle. What is the measure of \u2220EFD?
\n\"Eureka
\nAnswer:
\nThe measure of \u2220EFD is 90\u00b0-57\u00b0, which is equal to 33\u00b0.<\/p>\n

Question 9.
\nIn the diagram below, Lines L1<\/sub> and L2<\/sub> are parallel. Transversals r and s intersect both lines at the points shown below. Determine the measure of \u2220JMK. Explain how you know you are correct.
\n\"Eureka
\nAnswer:
\nThe Lines L1<\/sub> and L2<\/sub> are parallel, which means that the alternate interior angles formed by the transversals are equal. Specifically, \u2220LMK=\u2220JKM=72\u00b0. Since \u25b3JKM has a sum of interior angles equal to 180\u00b0,
\n\u2220KJM+\u2220JMK+\u2220JKM=180\u00b0. By substitution, we have 39\u00b0+\u2220JMK+72\u00b0=180\u00b0; therefore, \u2220JMK is 69\u00b0.<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math 8th Grade Module 2 Lesson 13 Answer Key Eureka Math Grade 8 Module 2 Lesson 13 Exploratory Challenge Answer Key Classwork Concept Development m\u22201+m\u22202+m\u22203=m\u22204+m\u22205+m\u22206=m\u22207+m\u22208+m\u22209=180\u00b0 Note that the sum of the measures of angles 7 and 9 must equal 90\u00b0 because of the known right angle in the right triangle. We want … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/82430"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=82430"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/82430\/revisions"}],"predecessor-version":[{"id":137308,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/82430\/revisions\/137308"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=82430"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=82430"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=82430"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}