This lesson is optional. It builds on the thinking that students did in the previous optional lesson. It is recommended to only do this lesson along with the previous optional lesson.
Many standardized assessments provide the special triangle relationships as a reference and ask students to reason about similar triangles. This lesson gives students an opportunity to apply their reasoning about similar figures and their skill with the Pythagorean Theorem to make sense of the relationship and understand how they might use a reference diagram to solve problems.
Students have an opportunity to reason abstractly and quantitatively (MP2) as they both measure lengths in, and solve analytically, 306090 triangles.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 3.1 Warmup: Notice and Wonder: Triangle Slices (5 minutes)
 3.2 Optional Activity: Decomposing Equilateral Triangles (15 minutes)

3.3 Optional Activity: Generalize Half Equilateral Triangles (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 3.4 Cooldown: Half of the Half (5 minutes)
Learning goals:
 Calculate side lengths in 306090 triangles.
 Generalize the properties of 306090 triangles including side lengths and ratios (using words and other representations).
Learning goals (student facing):
 Let’s investigate the properties of altitudes of equilateral triangles.
Learning targets (student facing):
 I can determine the side lengths of triangles with 30, 60, and 90 degree angles.
Required materials:
 Copies of blackline master
 Geometry toolkits (HS)
 Isometric dot paper
 Protractors
 Rulers
 Scientific calculators
Standards:
 This lesson builds on the standards: CCSS.8.G.B.7MS.8.G.7MO.8.GM.B.7CCSS.HSGCO.A.3MS.GCO.3MO.G.CO.A.3
 This lesson builds towards the standard: CCSS.HSGSRT.C.6MS.GSRT.6MO.G.SRT.C.5
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