Curriculum / Math / 8th Grade / Unit 3: Transformations and Angle Relationships / Lesson 22
Math
Unit 3
8th Grade
Lesson 22 of 22
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Define and use the angle-angle criterion for similar 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.1 — Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
In investigating the angle-angle criterion for similar triangles, students draw on concepts learned from throughout the unit, including rigid transformations, dilations, and the interior angle sum theorem. This lesson is a good opportunity to review these concepts from earlier in the unit.Â
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Two triangles share two pairs of equal angles, as marked in the diagram.
Mari thinks that $${\angle x}$$ and $${\angle y}$$ will always be equal, regardless of what the other angle measurements are. Do you agree with Mari’s reasoning? Why or why not?
Triangle $${{{ABC}}}$$ and triangle $${{AB'C'}}$$ share a common angle, $${\angle A}$$, and $${\overline{BC}}$$ is parallel to $${\overline{B'C'}}$$.
a. Is triangle $${{{ABC}}}$$ similar to triangle $${{AB'C'}}$$? Explain how you know.
b. If the measure of $${\angle B}$$ is $${40^{\circ}}$$, then what is the measure of $${\angle C'}$$? Explain how you know.
c. Draw a line in the diagram above that creates a new triangle that is similar to triangle $${{{ABC}}}$$.
Triangles $${ABC}$$ and $${PQR}$$ below share two pairs of congruent angles as marked:
a. Explain, using dilations, translations, reflections, and/or rotations, why $$\triangle {PQR}$$ is similar to $$\triangle {ABC}$$.
b. Are angles $$C$$ and $$R$$ congruent? Explain your reasoning.
c. Suppose $${{DEF}}$$ and $${{KLM}}$$ are two triangles with $${m(\angle D)=m(\angle K )}$$ and $${m(\angle E)=m(\angle L )}$$. Are triangles $${{DEF}}$$ and $${{KLM}}$$ similar?
Similar Triangles II, accessed on Oct. 13, 2017, 4:25 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.
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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
Triangle $${{HIJ}}$$ and triangle $${{PQR}}$$ are shown below.
Are triangles $${{HIJ}}$$ and $${{PQR}}$$ similar? Justify your answer.
An example response to the Target Task at the level of detail expected of the students.
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 21
Topic A: Congruence and Rigid Transformations
Understand the rigid transformations that move figures in the plane (translation, reflection, rotation).
8.G.A.1.A 8.G.A.1.B 8.G.A.1.C 8.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.A 8.G.A.1.B 8.G.A.1.C 8.G.A.2 8.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.2 8.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.3 8.G.A.4
Determine and informally prove or disprove if two figures are similar or congruent using transformations.
8.G.A.2 8.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.2 8.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 interior angle sum theorem for triangles.
Define and use the exterior angle theorem for triangles.
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