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.
An extra line drawn in a figure to reveal hidden structure.
For example, the line shown in the isosceles triangle is a line of symmetry, and the lines shown in the parallelogram suggest a way of rearranging it into a rectangle.
The converse of an if-then statement is the statement that interchanges the hypothesis and the conclusion. For example, the converse of "if it's Tuesday, then this must be Belgium" is "if this is Belgium, then it must be Tuesday."
For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.
In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).
A quadrilateral in which pairs of opposite sides are parallel.
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