2.3 Isosceles Triangles and Auxiliary Lines
Using congruent triangles, we can prove theorems about many geometric figures. Let's start with isosceles triangles.
- Definition
An isosceles triangle is a
triangle that has two of its sides congruent. The congruent sides
are called the legs of the
triangle.
To prove a theorem, in this case a theorem about isosceles triangles, we start with a general statement, interpret that statement with a diagram and then express the Given and Prove in terms of the diagram. The diagram can't be overly special. In other words, we can't use the diagram of an equilateral triangle if we're trying to establish a property of isosceles triangles. Here is an example of a proof of a theorem.
- Theorem 2.1
The Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Restatement:
Given: Isosceles
$\triangle ABC$ with $\overline{AB}\cong\overline{CB}$ Prove: $\angle A\cong \angle C$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
S& 1. \overline{AB}\cong\overline{CB} & 1. \text{ Given} \\
S& 2. \overline{CB}\cong\overline{AB} & 2. \text{ Given} \\
A& 3. \angle B\cong\angle B & 3. \text{ Reflexive}\\
&4. \triangle ABC \cong \triangle CBA & 4. \text{ SAS}\\
&5. \angle A\cong \angle C & 5. \text{ CPCTC}\\
\end{array}
This is a subtle proof since it sets up a correspondence between $\triangle ABC$ and $\triangle CBA$. In other words, we treat each of $\overline{AB}$ and $\overline{CB}$ as a side of two triangles that happen to be the same (though they are oriented differently).
The congruent angles of an isosceles triangle are known as the base angles. With two of the sides of an isosceles triangle congruent to each other, the third side is called the base; and with two of the angles of an isosceles triangle congruent to each other, the third angle is the vertex angle.
We should wonder if the converse of Theorem 2.1 is true; that is, if two angles of a triangle are congruent, are the sides opposite those angles congruent? To answer this question we introduce a problem-solving technique, the use of auxiliary lines.
Auxiliary Lines
Good geometric thinking is not necessarily deductive and linear; it does not always proceed from A to H in a straight-line fashion. There are powerful and imaginative ways of thinking in geometry that enable us to solve problems and develop proofs. As an example of imagination in action, consider the problem below. Try to solve it before reading the solution.
Example: A cylinder 10 units high and 4 units in diameter is partially filled with water. The cylinder is tilted so that the water level reaches 9 units up the side of the cylinder at the highest but only 3 units up at the lowest. Find the amount of water in the cylinder.
Solution: Imagine that the water in the tilted cylinder is frozen. Form a congruent solid and join the two to form a cylinder of diameter 4 and height 12. Its volume is $48\pi$, so half of $48\pi$ is the answer. |
In this section we will focus on the useful technique of drawing an additional line in the figure. This often works because it creates additional relationships that provide information that is key to the solution of the problem. But, and this is a big but, care must be taken in drawing the line. At this point we can connect two points by a line since the Line Postulate guarantees that there is exactly one line joining any two points. Thus, we can draw a median, the line connecting a vertex of a triangle to the midpoint of the opposite side. We can draw an angle bisector. Using the Ruler Postulate we can extend a line segment by a given amount; e.g., we can extend $\overline{AB}$ to $C$ such that $AB=BC$.
A major mistake is to ascribe too many properties to a given line. If, for example, one connects vertex $A$ of $\triangle ABC$ to the midpoint $M$ of $\overline{BC}$ and claims that $\overline{AM}$ is also perpendicular to $\overline{BC}$, one could actually prove that all triangles are isosceles!
Below are a few examples of how to incorporate auxiliary lines into proofs. The auxiliary lines are shown as dotted lines.
Connecting two points:
Statements |
Reasons |
|
1. Draw
$\overline{AC}$ |
1. The Line
Postulate
(Sec. 1.2): Through any two points there is exactly one line. |
Drawing a median:
Statements |
Reasons |
|
1. Let
$M$ be the midpoint of $\overline{AC}$. 2. Draw median $\overline{MB}$. |
1. Every
segment has exactly one midpoint. (Sec. 1.2) 2. The Line Postulate |
Drawing an angle bisector:
Statements |
Reasons |
|
1. Draw
angle bisector $\overline{AD}$ |
1. Every
angle has exactly one bisector. (Sec. 1.2) |
Extending a line segment:
Statements |
Reasons |
|
1.
Extend $\overline{DC}$ to $E$ such that $CE=BC$. |
1. The Ruler
Postulate. (Sec. 1.2) |
Consider now the correct use of an auxiliary line in the following proof.
- Theorem 2.2
The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Given: $\triangle
ABC$ with $\angle A \cong \angle C$. Prove: $\overline{AB}\cong \overline{BC}$. |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
A& 1. \angle A\cong \angle C & 1. \text{ Given} \\
& 2. \text{ Draw the angle bisector }\overline{BD}\text{ of }\angle ABC & 2. \text{ Every angle has an angle bisector.} \\
A& 3. \angle ABD\cong\angle CBD & 3. \text{ Definition of an angle bisector.}\\
S&4. \overline{BD}\cong\overline{BD} & 4. \text{ Reflexive}\\
&5. \triangle ABD \cong \triangle CBD & 5. \text{ AAS}\\
&6. \overline{AB}\cong\overline{CB} & 6. \text{ CPCTC}\\
\end{array}
Note: In proving that the sides opposite congruent angles of a triangle are congruent, we have in fact proved that the triangle is isosceles. The Converse of the Isosceles Triangle is sometimes phrased emphasizing that point: "If two angles of a triangle are congruent, then the triangle is isosceles." We could state the two theorems starting this section in one statement using the if and only if form: Two sides of a triangle are congruent if and only if the angles opposite those sides are congruent. In proofs, however, we will ask you to use specifically the Isosceles Triangle Theorem or its converse in order that you keep clearly in mind what is the hypothesis and what is the conclusion in a particular problem.
A collection of theorems has a certain structure to it — there are the major theorems and then there are two kinds of minor theorems. One is called a lemma and it serves to help prove a major theorem; it is a precursor to the main result. The other is called a corollary, and it follows quite directly, usually in just a couple of steps, from the major theorem. Here are two corollaries to the Isosceles Triangle Theorem and the Converse of the Isosceles Triangle Theorem whose proofs are called for in the exercises. They refer to an equilateral triangle (a triangle with all three sides congruent) and an equiangular triangle (a triangle with three congruent angles).
- Corollary 2.3: Every equilateral triangle is equiangular.
- Corollary 2.4:
Every equiangular triangle is equilateral
Appearing in the exercises for this section are some other important results about isosceles triangles. We state these theorems here (so that they're not buried in the exercises).
- Theorem 2.5: The bisector of the vertex angle of an isosceles triangle is perpendicular to the base.
- Theorem 2.6: The median drawn from the vertex angle of an isosceles triangle is the angle bisector of the vertex angle.
- Theorem 2.7: The medians drawn to the legs of an isosceles triangle are congruent.
- Theorem 2.8: The angle bisectors of the base angles of an isosceles triangle are congruent.
- Theorem 2.9: The altitudes drawn to the legs of an isosceles triangle are congruent.
We finish this section with two timesaving theorems, each of which we illustrate with an example.
- Theorem 2.10:
Halves of congruent angles are congruent.
Given: $\angle ABC \cong
\angle WXY$; $\overline{BD}$ bisects $\angle ABC$; $\overline{XZ}$ bisects $\angle WXY$. Prove: $\angle ABD \cong \angle WXZ$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \angle ABC \cong \angle WXY & 1. \text{ Given} \\
& 2. \frac{1}{2}m\angle ABC=\frac{1}{2}m\angle WXY & 2. \text{ Multiplication property of equality.} \\
& 3. \overline{BD}\text{ bisects }\angle ABC & 3. \text{ Given}\\
&\quad\overline{XZ}\text{ bisects }\angle WXY& \\
&4. m\angle ABD=\frac{1}{2}\angle ABC & 4. \text{ Angle Bisector Theorem}\\
&\quad m\angle WXZ=\frac{1}{2}\angle WXY& \\
&5. m\angle ABD = m\angle WXZ & 5. \text{ Substitution}\\
&6. \angle ABD \cong \angle WXZ & 6. \text{ Definition of congruent angles}\\
\end{array}
Example:
Given: In $\triangle
ABC$, $\overline{AB}\cong\overline{AC}$. $\overline{BP}$
and
$\overline{CP}$ bisect $\angle ABC$ and $\angle ACB$, respectively. Prove: $\overline{BP}\cong\overline{CP}$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \overline{AB}\cong\overline{AC} & 1. \text{ Given} \\
& 2. \angle ABC\cong\angle ACB & 2. \text{ Isosceles Triangle Theorem}\\
&3. \overline{BP}\text{ bisects }\angle ABC& 3. \text{ Given}\\
&\quad\overline{CP}\text{ bisects }\angle ACB& \\
&4. \angle PBC\cong\angle PCB & 4. \text{ Halves of congruent angles are congruent.}\\
&5. \overline{BP}\cong\overline{CP} & 5. \text{ Converse of the Isosceles Triangle Theorem.}\\
\end{array}
Very much like the last theorem is the next one.
- Theorem 2.11:
Halves of congruent segments are congruent.
Given:
$\overline{AC}\cong\overline{XZ}$;
$B$ is the midpoint of $\overline{AC}$;
$Y$ is the midpoint of $\overline{XZ}$.
Prove: $\overline{AB}\cong\overline{XY}$
$B$ is the midpoint of $\overline{AC}$;
$Y$ is the midpoint of $\overline{XZ}$.
Prove: $\overline{AB}\cong\overline{XY}$
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \overline{AC}\cong\overline{XZ} & 1. \text{ Given} \\
&2. \frac{1}{2}AC=\frac{1}{2}XZ & 2. \text{ Multiplication property of equality}\\
&3. B \text{ is the midpoint of }\overline{AC} & 3. \text{ Given}\\
&\quad Y\text{ is the midpoint of } \overline{XZ} & \\
&4. AB=\frac{1}{2}AC & 4. \text{ Midpoint Theorem}\\
&\quad XY=\frac{1}{2}XZ & \\
&5. AB=XY & 5. \text{ Substitution}\\
&6. \overline{AB}\cong\overline{XY} & 6.\text{ Definition of congruent segments}\\
\end{array}
Example:
Given:
$\overline{AB}\cong\overline{AC}$; $\overline{BM}$ and $\overline{CN}$
are medians. Prove: $\overline{BM}\cong\overline{CN}$ |
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \overline{AB}\cong\overline{AC} & 1. \text{ Given} \\
&\quad \overline{BM}\text{ and }\overline{CN}\text{ are medians.} &\\
&2. M\text{ is the midpoint of }\overline{AC} & 2. \text{ Definition of median}\\
&\quad N\text{ is the midpoint of }\overline{AB} & \\
S&3. \overline{BN}\cong\overline{CM} & 3. \text{ Halves of congruent segments are congruent.}\\
A&4. \angle ABC\cong \angle ACB & 4. \text{ Isosceles Triangle theorem}\\
S&5. \overline{BC}\cong \overline{BC} & 5. \text{ Reflexive}\\
&6. \triangle NBC \cong \triangle MCB & 6. \text{ SAS}\\
&7. \overline{BM}\cong \overline{CN} & 7. \text{ CPCTC}\\
\end{array}