2.2 Proving Triangles Congruent (ASA and AAS)
There are other ways to prove that two triangles are congruent. As shown in the figure below, fixed angles placed at either end of a line segment will intersect in a unique point.
This suggests that any two triangles with a pair of congruent angles and a congruent segment between them, called the included side, are necessarily the same size and shape. We take this as a postulate.
- Postulate: The
Angle-Side-Angle Postulate (ASA)
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
We can use ASA in proofs such as the following.
Example:
Given:
$\overline{AB}\cong\overline{AE}$, $\angle 1\cong \angle 2$, $\angle
B\angle E$ Prove: $\angle 3\cong\angle 4$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
S,\;A,\;A& 1. \overline{AB}\cong\overline{AE},\; \angle 1\cong \angle 2,\; \angle B\angle E & 1. \text{ Given} \\
& 2. \triangle ABD \cong \triangle AED & 2. \text{ ASA} \\
& 3. \angle 3\cong\angle 4 & 3. \text{ CPCTC}\\
\end{array}
Note that by using the Angle Addition Postulate, we could have shown in addition that $\triangle BAD\cong\triangle EAC$. Combining this with $\overline{AB}\cong\overline{AE}$ and $\angle B\cong\angle C$ would enable us to prove that $\triangle BAD\cong\triangle EAC$ by Angle-Side-Angle again. Or, by using the Segment Addition Postulate, we could have shown that $\overline{BD}\cong\overline{EC}$. Combining this with $\overline{AB}\cong\overline{AE}$ and $\angle B\cong\angle E$ would enable us to prove that $\triangle BAD\cong\triangle EAC$ by Side-Angle-Side.
If we had the Parallel Postulate at this point in the course we could show that the sum of the measures of the angles of a triangle equals $180^{\circ}$ and this would enable us to develop a fourth way to prove triangles congruent. But we don't have that postulate yet, so we can't establish the fourth method by proof. However, we can take it as a postulate in order to quickly enhance our ability to prove statements.
- Postulate: The
Angle-Angle-Side Postulate (AAS)
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
Here's an exaple of a proof:
Example:
Given:
$\overline{BC}\cong\overline{ED},\;\angle 1\cong\angle 2,\;\angle
3\cong\angle 4$ Prove: $\overline{AC}\cong\overline{AD}$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
S,\;A,\;A& 1. \overline{BC}\cong\overline{ED},\;\angle 1\cong\angle 2,\;\angle 3\cong\angle 4 & 1. \text{ Given} \\
& 2. \triangle ABC \cong \triangle AED & 2. \text{ AAS} \\
& 3. \overline{AC}\cong\overline{AD} & 3. \text{ CPCTC}\\
\end{array}
We could extend this problem by asking for proof that $\triangle ABD\cong\triangle AEC$. Using the Segment Addition Postulate, we could prove that $\overline{BD}\cong\overline{EC}$ and then complete the proof using SSS.
Although we have stressed writing proofs in two-column form, proofs may also be written in paragraph form. No articles in mathematical journals are written in two-column form, but for beginning geometry students, the two-column form has distinct advantages. It forces students to write a coherent, logical sequence of steps with explicit reasons and justifications. By contrast, students' first paragraph proofs often skip steps and reasons, ending up as a set of dogmatic statements rather than a sequence of reasoned statements. One of the goals of this course is to enable students to write coherent paragraph proofs as well as to give clear and effective oral proofs, but until students assimilate and become quite comfortable with presenting a tightly reasoned proof, we will delay the use of paragraph proofs. However, by the end of the course we want students to be able to give clear and logical explanations no matter what form they use. A reader of a good proof should respond this way: yes, . . . yes that's right, . . . , very good, . . . , very nice, . . . , clever, . . . bravo. A good proof should be compelling. Too often, however, readers find themselves responding in this way: huh, . . . , how does that follow, . . . where's the third side, . . . what on earth, . . . , #&@%. Students should conscientiously avoid provoking this response.
Proof Strategies
At the level of a research mathematician, a good proof is often the result of hours, if not days, if not months of sustained work. The reason is that at that level of mathematics, one must either combine ideas in a way that no one has thought of before or develop a new way of thinking. Creativity, imagination, and persistence are certainly called for. At all levels of mathematics, however, there are certain strategies that have proved remarkably useful over the centuries. In Section 2.3 we will introduce auxiliary lines. Drawing the right auxiliary line will bring into play additional relationships that prove key to discovering a proof. In this section we introduce the technique of working backwards. Simply put, assume that you've solved the problem. What is the step that would have led immediately to the desired conclusion? Then look for the step that would have led to that step, and so on. In many cases working backwards brings a focus and coherence to one's thinking. Confusion is swept away. Here's an example.
Example:
Given: $\angle
1\cong\angle 2,\ \angle 3\cong\angle 4\text{, and }$ $\angle BAD\cong\angle BCD$ Prove: $\angle 5\cong\angle 6$ |
How can you prove that $\angle 5\cong\angle 6$? You could accomplish this in one of several ways. If you prove that $\triangle ABD \cong\triangle CBD$, then $\angle 5\cong \angle 6$ follows by CPCTC. Since it is given that $\angle 1 \cong \angle 2$, $\angle BAD \cong \angle BCD$, and we know that $\overline{BD}\cong \overline{BD}$ by the reflexive property, then $\triangle ABD\cong \triangle CBD$. To get this you've got $\overline{ED}\cong \overline{ED}$; then you need $\overline{AE}\cong \overline{CE}$ and $\angle AED\cong\angle CED$ in order to prove the triangles are congruent by SAS. You can get $\overline{AE}\cong\overline{CE}$ by showing $\triangle ABE\cong \triangle CBE$. That's doable by AAS since it is given that $\angle 1\cong \angle 2$, $\angle 3\cong \angle 4$, and $\overline{BE}\cong\overline{BE}$ by the reflexive property. Since $\angle AEB\cong\angle CEB$ by CPCTC you can also obtain $\angle AED\cong \angle CED$ using the Supplement Theorem. That will make $\triangle AED\cong \triangle CED$ by SAS and once again you've figured out how to proceed to write the proof. Here's the proof by the first appraoch.
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
A,\;A& 1. \angle 1\cong\angle 2,\;\angle BAD\cong\angle BCD & 1. \text{ Given} \\
S& 2. \overline{BD}\cong\overline{BD} & 2. \text{ Reflexive} \\
& 3. \triangle ABD\cong \triangle CBD & 3. \text{ AAS}\\
& 4. \angle 5\cong\angle 6 &4. \text{ CPCTC}\\
\end{array}