Curriculum / Math / 10th Grade / Unit 1: Constructions, Proof, and Rigid Motion / Lesson 11
Math
Unit 1
10th Grade
Lesson 11 of 19
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Construct parallel lines. Prove the relationship between corresponding angles. Use this relationship to find missing measures directly and algebraically.
The core standards covered in this lesson
G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.A.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.C.9 — Prove theorems about lines and angles. 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.
G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
The foundational standards covered in this lesson
8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
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.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
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 essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
In this lesson, we are focusing on only corresponding angle relationships in parallel line diagrams. In the next lesson, we will work with additional angle relationships, finding missing measures, and establishing proofs.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Watch this video of the construction of parallel lines.
Construct a pair of parallel lines and describe the angle and line relationships formed.
Explain, using the properties of translations, why the lines shown are not parallel.
Find the value of $$x$$ in the following diagram. Given: $${\overleftrightarrow {HB} \parallel \overleftrightarrow {GF}}$$
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Suppose $${{\overline{AB}}}$$ is a line segment and D is a point not on $${{\overline{AB}}}$$ as pictured below.
Let C be the point so that $${CD=AB}$$, $${\overleftrightarrow {CD} \parallel \overleftrightarrow {AB}}$$ and ABCD is a quadrilateral.
Parallelograms and Translations, accessed on Aug. 7, 2017, 2:37 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.
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.
Lesson 10
Lesson 12
Topic A: Constructions of Basic Geometric Figures
Describe the precise definition and notation for foundational geometric figures.
G.CO.A.1
Construct an equilateral triangle with only a straight-edge and a compass. Copy a line segment.
G.CO.A.1 G.CO.D.12 G.CO.D.13
Construct, bisect, and copy an angle.
G.CO.A.1 G.CO.D.12
Construct a perpendicular bisector.
Construct the altitudes and perpendicular bisectors of sides of triangles.
G.CO.C.10 G.CO.D.12
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Topic B: Justification and Proof of Angle Measure
Use principles of proof to justify each step in solving an equation.
A.REI.A.1
Use angle relationships around a point to find missing measures. Prove angle relationships around a point using geometric statements and reasons.
G.CO.C.9
Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships
Describe and identify rigid motions.
G.CO.A.1 G.CO.A.2 G.CO.B.6
Describe rigid motions. Use algebraic rules to translate points and line segments and describe translations on the coordinate plane.
G.CO.A.2 G.CO.B.6
Translate points and line segments not on the coordinate plane using constructions. Describe properties of translations with respect to line segments and angles.
G.CO.A.2 G.CO.A.4 G.CO.A.5
G.CO.A.1 G.CO.A.4 G.CO.C.9 G.CO.D.12
Prove angle relationships in parallel line diagrams.
G.CO.A.1 G.CO.C.9
Construct auxiliary parallel lines and use these in the development of proofs and identification of missing measures.
G.CO.C.9 G.CO.D.12
Topic D: Reflections and Rotations of Points, Line Segments, and Angles
Perform reflections on a coordinate plane across axes and other defined lines. Identify characteristics and an algebraic rule for the reflection.
G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.6
Use construction and patty paper to reflect a figure not on the coordinate plane. Describe the properties of a reflection.
Perform rotations on a coordinate plane. Identify characteristics and algebraic rules for the rotation.
Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.
G.CO.A.2 G.CO.A.5 G.CO.B.6
Describe a sequence of rigid motions that will map a point, line segment, or angle onto another figure.
G.CO.A.5 G.CO.B.6
Describe rigid motions, or sequences of rigid motions that have the same effect on a figure.
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