2.5 The Hypotenuse-Leg and Angle Bisector Characterization Theorems
We have identified four methods of proving triangles congruent: SSS, SAS, ASA, AAS. Are there others? Consider AAA. Sometimes it is true that if three angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are congruent. Compare triangles ABC and DEF. But often they aren't as shown by $\triangle KLM$.
For a method for proving triangles congruent to be valid, it must work in all cases. The conclusion must necessarily follow. Now consider SSA. In the diagram below, start with triangle LMN and swing out an arc with center $M$ and length $MN$. The arc intersects $\overleftrightarrow{QR}$ at $P$. Clearly, $\overline{MN}\cong\overline{MP}$ and since $\overline{ML}\cong\overline{ML}$ and $\angle L\cong\angle L$, here we have two triangles MLN and MLP with two pairs of corresponding sides congruent, and a pair of corresponding congruent angles that are not included by the sides. As the figure shows, the triangles are not congruent.
Since SSA doesn't necessarily guarantee congruence, it can't be used to prove congruence.
Notice that the formation of two different triangles relied upon $\overline{MN}$ rotating around $M$ with $P$ and $N$ lying on $\overleftrightarrow{QR}$ in two different places. If $\overline{MN}$ were perpendicular to $\overline{QR}$, there would be a unique position for $N$ and therefore a unique triangle. This leads to the following theorem.
- Theorem 2.15:
The Hypotenuse-Leg Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Restatement:
Given: Right triangles $ABC$ and $DEF$ with $\overline{AB}\cong\overline{DE}$ and $\overline{AC}\cong\overline{DF}$. Prove: $\triangle{ABC}\cong\triangle{DEF}$ |
\begin{array}{ c l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \triangle ABC \text{ and }\triangle DEF\text{ have right angles at $C$ and $F$} & 1. \text{ Given} \\
S&2. \text{ Extend $\overline{EF}$ so that }BC=FG & 2. \text{ Ruler Posutlate}\\
&3. \text{ Draw $\overline{DG}$} & 3. \text{ Two points determine a line.}\\
S&4. \overline{AC}\cong\overline{DF} & 4. \text{ Given}\\
A&5. \angle ACB\cong\angle DFG & 5. \text{ All right angles are congruent.}\\
&6. \triangle ABC\cong\triangle DGF & 6. \text{ SAS}\\
&7. \overline{AB}\cong\overline{DG} & 7. \text{ CPCTC}\\
&8. \overline{AB}\cong\overline{DE} & 8. \text{ Given}\\
S&9. \overline{DE}\cong\overline{DG} & 9. \text{ Transitive}\\
A&10. \angle E\cong \angle G & 10. \text{ Isosceles Triangle Theorem}\\
A&11. \angle DFE\cong\angle DFG & 11. \text{ All right angles are congruent.}\\
&12. \triangle DEF\cong\triangle DGF & 12. \text{ AAS}\\
&13. \triangle ABC\cong\triangle DEF & 13. \text{ Transitive Prop. of Congruent triangles.}\\
\end{array}
When we measure the distance between two points, the Ruler Postulate gives us a scheme. It's different when we measure the distance from a point to a line. In this case, the question is: What point on the line should we measure the distance to? If you think about how far you are from one of the walls of the room you're in, you would probably think in terms of going to the closest point on the wall; in fact, that point is found by drawing a line from your location perpendicular to the wall. When we talk about the distance from a point to a line, that is the way we measure the distance — by taking a perpendicular to the line from the point.
The Angle-Angle-Side Postulate and the Hypotenuse-Leg Theorem enable us to prove a fundamental theorem about points on an angle bisector. This theorem, like the Perpendicular Bisector Characterization Theorem, has two parts. The proofs are exercises 7 and 8.
- Theorem 2.16:
The Angle Bisector Characterization Theorem (ABC)
The bisector of an angle is the set of all points in the interior of the angle that are equidistant from the sides of the angle.