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Wize High School Geometry Textbook (Common Core) > Congruent & Similar Triangles

Congruent Triangles (SAS)

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Congruent Triangles using SAS




Even though two congruent triangles have many corresponding sides and angles that must be congruent, we do not need to immediately need show all of them are congruent.

In many cases there are theorems which allow us to only show specific sides and angels are congruent. Once these are shown, then the triangles are considered congruent via the theorem.

Side Angle Side (SAS) Theorem

If two triangles have a pair of congruent sides, and the angle between the sides is also congruent, then the triangles are congruent.

More precisely if we have
  • CA‾≅FD‾,∠A≅∠D, and AB‾≅DE‾\overline{CA} \cong \overline{FD}, \angle{A} \cong \angle{D}, \text{ and } \overline{AB} \cong \overline{DE}
Then
  • △ABC≅△FDE\triangle{ABC} \cong \triangle{FDE}

Example:
Which pair of triangles can be considered congruent?

ANSWER:
The first two tringles labeled A and B are congruent. Even though all of the sides are not marked out, we have a pair of congruent sides, along with the angle between the side. By the SAS theorem, these triangles are congruent.

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Construct a copy of a triangle (SAS)

Our goal in this construction is to copy a given triangle using two sides and the angle in between them. This is know as side-angle-side or simply SAS.

Using Technology


Begin with a triangle △ABC\triangle ABC
  1. Draw a line A′D↔\overleftrightarrow{A'D}where you want to build the new triangle.
  2. Use the compass tool to draw a circle with radius AB‾\overline{AB}centered at A′A'. Label the intersection of the line and the circle as point B′B'.
  3. Construct a new angle with side A′B′‾\overline{A'B'}congruent angle ∠CAB\angle CAB.
  4. Use the compass tool to draw a circle with radius AC‾\overline{AC}centered at A′A'. Label the intersection of the line and the circle as point C′C'.
  5. Draw the line segment C′B′‾\overline{C'B'}.
For this we now have that △ABC≅△A′B′C′\triangle{ABC} \cong \triangle A'B'C', using side-angle-side.

Congruent Triangles

While getting ready to mail a letter, Beth noticed that the envelope she was using made a pair of lines. These lines intersected the middle of each other, as seen in the diagram.


Are the two triangles formed (in green) congruent or is there not enough information to determine this?


Congruent Triangles

In the shop, Jack is working on making a sail for a tiny toy boat as seen in the diagram.
If the bottom of each triangle is exactly the same length, can he conclude the triangles are congruent?