Lesson 11

Side-Side-Angle (Sometimes) Congruence

  • Let’s explore triangle congruence criteria that are ambiguous.

11.1: Notice and Wonder: Congruence Fail

What do you notice? What do you wonder?

In triangles \(GBD\) and \(KHI\):

  • Angle \(GBD\) is congruent to angle \(KHI\).
  • Segment \(BD\) is congruent to segment \(HI\).
  • Segment \(DG\) is congruent to segment \(IK\).

11.2: Dare to Be (Even More) Different

Use technology to make a triangle using the given angle and side lengths so that the given angle is not between the 2 given sides. Try to make your triangle different from the triangles created by the other people in your group.

  • Angle: \(40^\circ\)
  • Side length: 6 cm
  • Side length: 8 cm

11.3: Ambiguously Ambiguous?

Your teacher will give you some sets of information.

  • For each set of information, make a triangle using that information.
  • If you think you can make more than one triangle, make more than one triangle.
  • If you think you can’t make any triangle, note that.

When you are confident they are accurate, create a visual display.

Triangle \(ABC\) is shown. Use your straightedge and compass to construct a new point \(D\) on line \(AC\) so that the length of segment \(BD\) is the same as the length of segment \(BC\).

Triangle A B C. Angle A is 51 degrees. Side A B is 9. Side B C is 8.

Now use the straightedge and compass to construct the midpoint of \(CD\). Label that midpoint \(M\).

  1. Explain why triangle \(ABM\) is a right triangle.
  2. Explain why knowing the angle at \(A\) and the side lengths of \(AB\) and \(BC\) was not enough to define a unique triangle, but knowing the angle at \(A\) and the side lengths of \(AB\) and \(BM\) would be enough to define a unique triangle.


Imagine we know triangles have 2 pairs of corresponding, congruent side lengths, and a pair of corresponding, congruent angles that is not between the given sides. What can we conclude?

Sometimes this is not enough information to determine that the triangles made with those measurements are congruent. These triangles have 2 pairs of congruent sides and a pair of congruent angles, but they are not congruent triangles. 

Triangles that share a marked angle and a long side. Adjacent to the long side, each triangle has a short side with 1 tick mark. The short sides extend at different angles, creating the 2 triangles.

If the longer of the 2 given sides is opposite the given angle though, that does guarantee congruent triangles. In a right triangle, the longest side is always the hypotenuse. If we know the hypotenuse and the leg of a right triangle, we can be confident they are congruent.

2 right triangles. Sides across from the right angle have double tick marks. Sides adjacent to the right angle have a single tick mark.

Glossary Entries

  • auxiliary line

    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.

  • converse

    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."

  • corresponding

    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 \(EFD\).

  • parallelogram

    A quadrilateral in which pairs of opposite sides are parallel.