This lesson is optional.
In this lesson, students study the ambiguous case of triangle congruence. Students know that two pairs of corresponding sides are congruent and a pair of corresponding angles not between the two sides are congruent. They create triangles with this ambiguous information and notice that multiple triangles can be produced with the same information. They then study the case in which the longer side is known to be across from the given angle, which is not ambiguous. Finally, students practice recognizing situations in which they can and can’t determine if two triangles are congruent, given information about two pairs of corresponding sides and one pair of corresponding angles. Students are looking for structure both as they build cases and as they apply their reasoning to new problems (MP7).
While studying the ambiguous case is optional, it can help students better understand in which situations knowing two sides and an angle not between them defines a unique triangle. For students who plan to study trigonometry in greater depth, this lesson prepares them to understand when the law of sines and law of cosines might give ambiguous results when solving nonright 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
 11.1 Warmup: Notice and Wonder: Congruence Fail (5 minutes)

11.2 Optional Activity: Dare to Be (Even More) Different (10 minutes)
 Note that two slightly different versions of task statements appear on the cards in this activity depending on whether students will be using physical materials to construct triangles or digital, geometry technology.

11.3 Optional Activity: Ambiguously Ambiguous? (20 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 11.4 Cooldown: Are They Ambiguous? (5 minutes)
Learning goals:
 Generate examples and counterexamples of SideSideAngle triangle congruence (using words and other representations).
Learning goals (student facing):
 Let’s explore triangle congruence criteria that are ambiguous.
Learning targets (student facing):
 I know SideSideAngle does not guarantee triangles are congruent.
Required materials:
 1inch strips cut from card stock with evenlyspaced holes
 Dried linguine pasta
 Geometry toolkits
 Metal paper fasteners
 Preprinted slips, cut from copies of the blackline master
 Tools for creating a visual display
Standards:
 This lesson builds on the standard: CCSS.HSGCO.B.8MS.GCO.8MO.G.CO.B.7
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