Geometry for Elementary School/Congruence

From Wikibooks, open books for an open world
< Geometry for Elementary School
Jump to: navigation, search
Geometry for Elementary School
Congruence and similarity Congruence The Side-Side-Side congruence theorem

In this chapter, we will start the discussion of congruence and congruence theorems. We say the two figures are congruent if they have the same shape and size. Congruent figures have three things in common: corresponding sides (corr. sides), corresponding angles (corr. ∠s) and corresponding points (corr. points). We will only talk about congruent triangles.

Congruent triangles[edit]

The triangles \triangle ABC and \triangle DEF are congruent if and only if all the following conditions hold:

  1. The side \overline {AB} equals \overline {DE} . (Corresponding sides)
    Geom side congr 01.png


  2. The side \overline {BC} equals \overline {EF} . (Corresponding sides)
    Geom side congr 02.png


  3. The side \overline {AC} equals \overline {DF} . (Corresponding sides)
    Geom side congr 03.png


  4. The angle \angle ABC equals \angle DEF . (Corresponding angles)
    Geom side congr 04.png


  5. The angle \angle BCA equals \angle EFD . (Corresponding angles)
    Geom side congr 05.png


  6. The angle \angle CAB equals \angle FDE . (Corresponding angles)
    Geom side congr 06.png


Note that the order of vertices is important. It is possible that \triangle ABC and \triangle ACB are not congruent even though both refer to the same triangle. Remember that the place where corresponding points are must be the same on both triangles.

Congruence theorems give a set of the fewest conditions that are sufficient in order to show that two triangles are congruent. They are SSS, SAS, ASA, AAS and RHS. We will talk about them later on.

Finding the value of unknowns in triangles whose congruence is given[edit]

Let's say we have two triangles, \triangle ABC and \triangle DEF, and they are congruent. AB=3, ∠F=90° and ∠E=60°. We need to find DE and ∠A. Here's how:

\begin{align}\angle F + \angle E + \angle D &= 180^\circ \text{ (}\angle \text{ sum of } \triangle \text{)}\\
90^\circ + 60^\circ + \angle D &= 180^\circ \\
\angle D&= 180^\circ - 90^\circ - 60^\circ \\
&= 30^\circ \end{align}


\begin{align}
\because \triangle ABC &\cong \triangle DEF \text{ (given)}\\
\therefore DE&=AB \text{ (corr. sides, } \cong \triangle \text{s)}\\
DE&=3\\
\And \angle A &= \angle D \text{ (corr. } \angle \text{s, } \cong \triangle \text{s)}\\
\angle A &= 30^\circ
\end{align}