Curriculum / Math / 8th Grade / Unit 3: Transformations and Angle Relationships / Lesson 20
Math
Unit 3
8th Grade
Lesson 20 of 22
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Lesson Notes
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Define and use the interior angle sum theorem for triangles.
The core standards covered in this lesson
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.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
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
Students have experience working with triangles and angle measurements from seventh grade when they investigated unique triangles. While students may know that the angles in a triangle add up to 180°, in this lesson they have the chance to prove this fact using parallel line angle relationships.
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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
Lucas is investigating the interior angles of a triangle. He knows that the sum of the angles inside a triangle is $$180{^{\circ}}$$, but he wants to understand why. He draws a triangle and places it between two parallel lines, with line $$l$$ passing through point $$Q$$ and line $$m$$ passing through points $$P$$ and $$R$$. Explain how Lucas can use his diagram to show $${p+q+r=180}$$.
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In the picture below, lines $$l$$ and $$m$$ are parallel. The measure of $$\angle PAX$$ is $${31^{\circ}}$$, and the measure of $$\angle PBY$$ is $${54^{\circ}}$$.
What is the measure of $$\angle APB$$?
Find the Missing Angle, accessed on Oct. 13, 2017, 4:22 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.
Determine the angle measures of each interior angle in the triangle.
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 diagram below, $$L1$$ is parallel to $$L2$$ and $$L3$$ is parallel to $$L4$$.
a. What is the measure of ∠5? Show or explain your reasoning.
b. What other angles have the same measure as ∠5? Show or explain your reasoning.
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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Define and use the exterior angle theorem for triangles.
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
Define and identify alternate interior and alternate exterior angles in parallel line diagrams. Find missing angles in parallel line diagrams.
Solve for missing angle measures in parallel line diagrams using equations.
8.G.A.5
Define and use the angle-angle criterion for similar triangles.
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