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 non-right 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.
- Generate examples and counter-examples of Side-Side-Angle triangle congruence (using words and other representations).
- Let’s explore triangle congruence criteria that are ambiguous.
- I know Side-Side-Angle does not guarantee triangles are congruent.
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