Curriculum / Math / 8th Grade / Unit 3: Transformations and Angle Relationships / Lesson 18
Math
Unit 3
8th Grade
Lesson 18 of 22
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Lesson Notes
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Define and identify alternate interior and alternate exterior angles in parallel line diagrams. Find missing angles in parallel line diagrams.
The core standards covered in this lesson
8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
The foundational standards covered in this lesson
7.G.B.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Lesson 18 builds on Lesson 17 by introducing two new angle relationships in parallel line diagrams: alternate interior and alternate exterior angles. Students use their knowledge of transformations to prove why these angles are congruent and to deepen their understanding of the relationships of angles in parallel line diagrams.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Suppose $$l$$ and $$m$$ are parallel lines, with point $$Q$$ on line $$l$$ and point $$P$$ on line $$m$$, as pictured below. Also labeled in the picture is point $$M$$, the midpoint of $$\overline{PQ}$$, and two angles $$a$$ and $$b$$.
How can you use a rotation to prove that angles $$a$$ and $$b$$ are congruent?
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Congruence of Alternate Interior Angles via Rotations, accessed on Oct. 13, 2017, 4:15 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
In the picture below, $$m$$ and $$n$$ are parallel lines and angles $$a$$ and $$b$$ are shown. Show that angle $$a$$ is congruent to angle $$b$$ using rigid motions.
Rigid Motions and Congruent Angles, accessed on Oct. 13, 2017, 4:16 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
In the diagrams below, lines $$b$$ and $$c$$ are parallel; lines $$f$$ and $$d$$ are not parallel.
a. Identify all of the angles congruent to $${{\angle 1}}$$ in the first diagram. Explain why each angle is congruent to $${{\angle 1}}$$.
b. Identify all of the angles congruent to $${{\angle 1}}0$$ in the second diagram. Explain why each angle is congruent to $${{\angle 1}}0$$.
A set of suggested resources or problem types that teachers can turn into a problem set
15-20 minutes
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
In the figure below, $$L1$$ and $$L2$$ are parallel and $$m$$ is a transversal that intersects them both.
a. What is the relationship between the measures of ∠4 and ∠6? Explain how you know.
b. What is the relationship between the measures of ∠1 and ∠7? Explain how you know.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
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Solve for missing angle measures in parallel line diagrams using equations.
Topic A: Congruence and Rigid Transformations
Understand the rigid transformations that move figures in the plane (translation, reflection, rotation).
Standards
8.G.A.1.A8.G.A.1.B8.G.A.1.C8.G.A.2
Describe and perform translations between congruent figures. Use translations to determine if figures are congruent.
Describe and apply properties of translations. Use coordinate points to represent relationships between translated figures.
8.G.A.1.A8.G.A.1.B8.G.A.1.C8.G.A.28.G.A.3
Describe and perform reflections between congruent figures. Use reflections to determine if figures are congruent.
Describe sequences of transformations between figures using reflections and translations. Use coordinate points to represent relationships between reflected figures.
Describe and perform rotations between congruent figures.
Describe sequences of transformations between figures using rotations and other transformations.
Describe a sequence of rigid transformations that will map one figure onto another.
Describe multiple rigid transformations using coordinate points.
8.G.A.28.G.A.3
Review rigid transformations and congruence between two figures.
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Topic B: Similarity and Dilations
Define a dilation as a non-rigid transformation, and understand the impact of scale factor.
8.G.A.4
Describe and perform dilations.
Describe a sequence of dilations and rigid motions between two figures. Use coordinate points to represent relationships between similar figures.
8.G.A.38.G.A.4
Determine and informally prove or disprove if two figures are similar or congruent using transformations.
8.G.A.28.G.A.4
Find missing side lengths in similar figures. Find scale factor between similar figures.
Use properties of similar triangles to model and solve real-world problems.
Topic C: Angle Relationships
Define and identify corresponding angles in parallel line diagrams. Review vertical, supplementary, and complementary angle relationships.
8.G.A.28.G.A.5
8.G.A.5
Define and use the interior angle sum theorem for triangles.
Define and use the exterior angle theorem for triangles.
Define and use the angle-angle criterion for similar triangles.
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