2.1 Proving Triangles Congruent (SSS and SAS)
Congruence, similarity, transformations, and invariance are not only some of the major concepts of geometry, but they are also powerful tools for discovering and establishing important, if not downright exciting, results. Two objects are similar if they have the same shape. Congruence adds a second condition — two objects are congruent if they have the same shape and the same size. In Chapter 1, we discussed congruent segments and angles. Since congruent segments have the same measure, as do congruent angles, we see that pairs of such objects are the same size. We extend that idea to other objects. If we imagine that we could move one figure on top of another, we have the notion that two objects are congruent when one of them fits precisely over the other as shown below.
We all know what a triangle is, but in this course we try to be precise about our use of words, so here are definitions concerning a triangle.
- Definitions:
Given three noncollinear points $A$, $B$, and $C$, triangle ABC (written $\triangle
ABC$) is
the figure formed by the three segments $\overline{AB}$,
$\overline{BC}$, and $\overline{CA}$.
The points $A$, $B$, and $C$ are the vertices of the triangle (each one is a vertex).
The segments $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ are the sides of the triangle.
When considering a particular triangle such as $\triangle XYZ$ shown below, we say that side $\overline{YZ}$ is opposite vertex $X$ and that sides $\overline{XY}$ and $\overline{XZ}$ include vertex $X$.
We represent congruence using the symbol $\cong$. To say that triangle $ABC$ is congruent to triangle $DEF$ we write $\triangle ABC\cong\triangle DEF$. It is important to note that a correspondence is implied here; i.e., angle $A$ is paired with angle $D$, $\angle B$ with $\angle E$, and $\angle C$ with $\angle F$, and each pair of angles has the same measure. Also, segment $\overline{AB}$ is paired with segment $\overline{DE}$, $\overline{BC}$ with $\overline{EF}$, and $\overline{AC}$ with $\overline{DF}$, and each pair of segments has the same length. This leads to the following definition of congruent triangles.
- Definition: $\triangle ABC \cong \triangle DEF$ if and only if $\angle A\cong \angle D$, $\angle B\cong \angle E$, $\angle C\cong \angle F$, $\overline {AB}\cong\overline{DE}$, $\overline {AC}\cong\overline{DF}$, and $\overline {BC}\cong\overline{EF}$.
To show that two triangles are congruent, we have to show that angles can be paired so that each pair has the same measure and that sides can be paired so that each pair has the same length. There must be an easier way to show that two triangles are congruent, for otherwise triangle congruence wouldn't be a particularly useful tool.
Try this experiment — take three sticks, drill holes in the ends, connect the ends by putting a nail or golf tee through pairs of holes, and you'll have a triangle. Note that the triangle is rigid, that its shape can't be changed. If you use four sticks you'll find that the shape can be changed. The implication is that given three lengths that form a triangle, only one triangle can be formed. Hence, two triangles with the same three sides must be congruent. We'll accept this as a postulate.
- Postulate: The
Side-Side-Side Postulate (SSS)
Given $\triangle ABC$ and $\triangle DEF$, if $\overline{AB}\cong \overline{DE}$, $\overline{BC}\cong\overline{EF}$, and $\overline{AC}\cong \overline{DF}$, then $\triangle ABC \cong \triangle DEF$.
Thus, instead of having to show that six pairs are congruent in order to establish the congruence of two triangles, we only need show that three pairs are congruent. We can then conclude that the other three parts, namely the angles, are congruent.
In similar fashion, take two sticks from the previous experiment and tighten the connection so that the angle is fixed. Can those two sticks with that fixed angle form a triangle with any other shape? Some experimentation ought to convince you that the triangle is unique. From this we could conclude that if two sides of one triangle and the angle formed by them (the included angle) are congruent to two sides of another triangle and the included angle formed by those sides, then the triangles are congruent. We'll state this as our next postulate.
- Postulate: The
Side-Angle-Side Postulate (SAS)
Given $\triangle ABC$ and $\triangle DEF$, if $\overline{AB}\cong \overline{DE}$, $\overline{AC} \cong \overline{DF}$, and $\angle BAC \cong \angle EDF$, then $\triangle ABC \cong \triangle DEF$.
We will use these postulates and others to prove things about triangles and other figures. Some students think that proof is auxiliary to mathematics; that, properly speaking, real mathematics is really about solving equations and algebraic manipulations. Nothing could be farther from the truth. Mathematics is mathematical reasoning and that is proof. Mathematics is a lot of other things but proof lies at its heart and distinguishes it from all other subjects. Using mathematics, we can describe reality theoretically, and reason to conclusions that are essentially discoveries. Consider this — how did we discover radio waves? Two hundred years ago, no one had touched them, seen them, tasted them, or even heard them. We were so far from knowing of their existence that no one had even thought of a scientific experiment to find them. We were absolutely clueless. Then along came James Clerk Maxwell who, in 1864, created mathematical equations that described electro-magnetism in such a way that the logical conclusion was that all sorts of waves, including radio waves, really existed. In 1887 Heinrich Hertz conducted experiments that confirmed the results of the theory. However, he didn't think that radio waves would be of any use — that discovery was left to Marconi and Tesla and it was their work that led to the age of the radio. Reasoning theoretically, i.e., mathematically, led to the conclusion that radio waves existed. That paved the way for their discovery via experimentation, and that eventually led to their utilization. You may think that this is all well and good, but that now nothing very dramatic could be gleaned from elementary geometry. You'd be wrong. When Edwin Catmull was a graduate student in computer graphics at the University of Utah, he was able to solve a fundamental problem in computer animation using high school geometry and that discovery enabled him to found Pixar and start producing Toy Story, Ratatouille, etc.
We'll use geometry to introduce you to proof, that is, to theoretical mathematics. To be certain that we make our reasoning transparent, we'll continue to organize our proofs as we did in Chapter 1, separating our conclusions and reasons by writing the conclusions in the left-hand column and the reasons in the right-hand column. We'll number our conclusions and reasons so that we know which belongs to which, and we'll put S or A to the left of a step to highlight the main points as well as to make sure we have properly justified claims of congruence.
Example:
Given: $\overline{AB}
\cong \overline{BC}$ and $\overline{AD} \cong \overline{CD}$
Prove: $\triangle BAD \cong \triangle BCD$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
S,\;S& 1. \overline{AB}\cong\overline{BC} \text{ and }\overline{AD}\cong\overline{CD}& 1. \text{ Given} \\
S& 2. \overline{BD}\cong \overline{BD} & 2. \text{ Reflexive Property} \\
& 3. \triangle BAD\cong\triangle BCD & 3. \text{ SSS}\\
\end{array}
If the problem had been to prove that $\angle ABD\cong\angle CBD$, we would have added a fourth step, the statement that $\angle ABD\cong\angle CBD$ and the reason would have been the Definition of Congruent Triangles. We will abbreviate the definition using the acronym CPCTC which stands for "Corresponding Parts of Congruent Triangles are Congruent."
Here's a problem that uses the Side-Angle-Side Postulate.
Example:
Given: $E$ is the
midpoint of $\overline{AC}$ and $\overline{DB}$. Prove: $\overline{AB}\cong\overline{CD}$ and $\angle B\cong \angle D$. |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. E\text{ is the midpoint of }\overline{AC}\text{ ,}\overline{DB}& 1. \text{ Given} \\
S,\;S& 2. \overline{AE}\cong \overline{CE}\text{ and }\overline{DE}\cong\overline{BE} & 2. \text{ Definition of midpoint} \\
A & 3. \angle AEB\cong\angle CED & 3. \text{ Vertical angles are congruent}\\
& 4. \triangle AEB\cong\triangle CED&4. \text{ SAS}\\
& 5. \overline{AB}\cong \overline{CD}\text{ and }\angle B\cong \angle D& 5. \text{ CPCTC}\\
\end{array}
Note that we must be careful to get the correspondence right in step 4. We cannot be sloppy. It would be incorrect to write $\triangle AEB\cong\triangle DEC$ because that implies that $\angle A\cong \angle D$.
In some problems it is necessary to prove two pairs of triangles congruent in order to achieve the desired result. Here's an example. We'll use subscripted S's and A's to distinguish between the steps used to prove the congruence of the triangles.
Example:
Given:
$\overline{AB}\cong\overline{CB}$ and $\overline{AE}\cong
\overline{CE}$ Prove: $\angle 1\cong \angle 2$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
S_1,\;S_1& 1. \overline{AB}\cong\overline{CB} \text{ and } \overline{AE}\cong \overline{CE} & 1. \text{ Given} \\
S_1& 2. \overline{BE}\cong\overline{BE}& 2. \text{ Reflexive}\\
& 3. \triangle ABE\cong\triangle CBE &3. \text{ SSS}\\
& 4. \angle 3 \cong \angle 4 &4. \text{ CPCTC}\\
& 5. m\angle 3+m\angle AED = 180 &5. \text{ Angle Addition Postulate}\\
&\quad m\angle 4 +m\angle CED = 180 &\\
& 6. \angle 3 \text{ and }\angle AED\text{ are supplements.} &6.\text{ Definition of supplements.}\\
&\quad\angle 4\text{ and }\angle CED\text{ are supplements.}&\\
A_2& 7. \angle AED\cong\angle CED &7.\text{ Supplements of $\cong$ angles are $\cong$.}\\
S_2& 8. \overline{ED}\cong\overline{ED} &8.\text{ Reflexive}\\
S_2& 9. \overline{AE}\cong\overline{CE} &9.\text{ Given}\\
& 10. \triangle AED\cong\triangle CED &10.\text{ SAS}\\
& 11. \angle 1\cong\angle 2 &11.\text{ CPCTC}\\
\end{array}
Note that we restated $\overline{AE}\cong \overline{CE}$ in step 9. While not logically necessary since we mentioned this congruence in step 1, repeating this information in step 9 is useful because it clarifies the argument; we use that congruence to establish first the congruence of triangles $ABE$ and $CBE$ and then again the congruence of triangles $AED$ and $CED$.
There are some special segments associated with a triangle. An altitude is a segment drawn from a vertex of the triangle perpendicular to the line that contains the opposite side; the endpoints of an altitude are one of the vertices of the triangle and a point on the opposite side (extended, if necessary). In a right triangle, two of the altitudes are actually sides of the triangle. In a triangle that has an obtuse angle, two of the altitudes lie outside the triangle!
A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. We also use the expression angle bisector of an angle of a triangle in a special way. Whereas an angle bisector is normally thought of as a ray, when we have the bisector of an angle of a triangle, we usually are considering a segment bisecting an angle of the triangle and with its endpoints at a vertex of the triangle and at a point on the opposite side.