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# Isosceles Triangles

## Properties of triangles with two equal sides.

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Isosceles Triangles

### Isosceles Triangles

An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle are called the legs. The other side is called the base and the angles between the base and the congruent sides are called base angles. The angle made by the two legs of the isosceles triangle is called the vertex angle.

#### Investigation: Isosceles Triangle Construction

Tools Needed: pencil, paper, compass, ruler, protractor

1. Using your compass and ruler, draw an isosceles triangle with sides of 3 in, 5 in and 5 in. Draw the 3 in side (the base) horizontally 6 inches from the top of the page.
2. Now that you have an isosceles triangle, use your protractor to measure the base angles and the vertex angle.

The base angles should each be 72.5\begin{align*}72.5^\circ\end{align*} and the vertex angle should be 35\begin{align*}35^\circ\end{align*}.

Watch the first part of this video.

We can generalize this investigation into the Base Angles Theorem.

Base Angles Theorem: The base angles of an isosceles triangle are congruent.

To prove the Base Angles Theorem, we will construct the angle bisector through the vertex angle of an isosceles triangle.

Given: Isosceles triangle DEF\begin{align*}\triangle DEF\end{align*} with DE¯¯¯¯¯¯¯¯EF¯¯¯¯¯¯¯¯\begin{align*}\overline{DE} \cong \overline{EF}\end{align*}

Prove: DF\begin{align*}\angle D \cong \angle F\end{align*}

Statement Reason
1. Isosceles triangle DEF\begin{align*}\triangle DEF\end{align*} with DE¯¯¯¯¯¯¯¯EF¯¯¯¯¯¯¯¯\begin{align*}\overline{DE} \cong \overline{EF}\end{align*} Given

2. Construct angle bisector EG¯¯¯¯¯¯¯¯\begin{align*}\overline{EG}\end{align*} for E\begin{align*}\angle E\end{align*}

Every angle has one angle bisector
3. DEGFEG\begin{align*}\angle DEG \cong \angle FEG\end{align*} Definition of an angle bisector
4. EG¯¯¯¯¯¯¯¯EG¯¯¯¯¯¯¯¯\begin{align*}\overline{EG} \cong \overline{EG}\end{align*} Reflexive PoC
5. DEGFEG\begin{align*}\triangle DEG \cong \triangle FEG\end{align*} SAS
6. DF\begin{align*}\angle D \cong \angle F\end{align*} CPCTC

By constructing the angle bisector, EG¯¯¯¯¯¯¯¯\begin{align*}\overline{EG}\end{align*}, we designed two congruent triangles and then used CPCTC to show that the base angles are congruent. Now that we have proven the Base Angles Theorem, you do not have to construct the angle bisector every time. It can now be assumed that base angles of any isosceles triangle are always equal. Let’s further analyze the picture from step 2 of our proof.

Because DEGFEG\begin{align*}\triangle DEG \cong \triangle FEG\end{align*}, we know that EGDEGF\begin{align*}\angle EGD \cong \angle EGF\end{align*} by CPCTC. Thes two angles are also a linear pair, so they are congruent supplements, or 90\begin{align*}90^\circ\end{align*} each. Therefore, EG¯¯¯¯¯¯¯¯DF¯¯¯¯¯¯¯¯\begin{align*}\overline{EG} \bot \overline{DF}\end{align*}. Additionally, DG¯¯¯¯¯¯¯¯GF¯¯¯¯¯¯¯¯\begin{align*}\overline{DG} \cong \overline{GF}\end{align*} by CPCTC, so G\begin{align*}G\end{align*} is the midpoint of DF¯¯¯¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}. This means that EG¯¯¯¯¯¯¯¯\begin{align*}\overline{EG}\end{align*} is the perpendicular bisector of DF¯¯¯¯¯¯¯¯\begin{align*}\overline{DF}\end{align*}, in addition to being the angle bisector of DEF\begin{align*}\angle DEF\end{align*}.

Isosceles Triangle Theorem: The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector to the base.

The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.

Base Angles Theorem Converse: If two angles in a triangle are congruent, then the opposite sides are also congruent.

So, for a triangle ABC\begin{align*}\triangle ABC\end{align*}, if AB\begin{align*}\angle A \cong \angle B\end{align*}, then CB¯¯¯¯¯¯¯¯CA¯¯¯¯¯¯¯¯\begin{align*}\overline{CB} \cong \overline{CA}\end{align*}. C\begin{align*}\angle C\end{align*} would be the vertex angle.

Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.

In other words, if ABC\begin{align*}\triangle ABC\end{align*} is isosceles, AD¯¯¯¯¯¯¯¯CB¯¯¯¯¯¯¯¯\begin{align*}\overline{AD} \bot \overline{CB}\end{align*} and CD¯¯¯¯¯¯¯¯DB¯¯¯¯¯¯¯¯\begin{align*}\overline{CD} \cong \overline{DB}\end{align*}, then CADBAD\begin{align*}\angle CAD \cong \angle BAD\end{align*}.

#### Recognizing Congruent Angles

Which two angles are congruent?

This is an isosceles triangle. The congruent angles, are opposite the congruent sides.

From the arrows we see that SU\begin{align*}\angle S \cong \angle U\end{align*}.

#### Measuring Vertex Angles

If an isosceles triangle has base angles with measures of 47\begin{align*}47^\circ\end{align*}, what is the measure of the vertex angle?

Draw a picture and set up an equation to solve for the vertex angle, v\begin{align*}v\end{align*}.

47+47+vvv=180=1804747=86\begin{align*}47^\circ+47^\circ+v &= 180^\circ\\ v &= 180^\circ-47^\circ-47^\circ\\ v &= 86^\circ\end{align*}

#### Measuring Base Angles

If an isosceles triangle has a vertex angle with a measure of 116\begin{align*}116^\circ\end{align*}, what is the measure of each base angle?

Draw a picture and set up and equation to solve for the base angles, b\begin{align*}b\end{align*}. Recall that the base angles are equal.

116+b+b2bb=180=64=32\begin{align*}116^\circ+b+b &= 180^\circ\\ 2b &= 64^\circ\\ b &= 32^\circ\end{align*}

### Examples

#### Example 1

Find the value of x\begin{align*}x\end{align*} and the measure of each angle.

Set the angles equal to each other and solve for x\begin{align*}x\end{align*}.

(4x+12)15=(5x3)=x\begin{align*}(4x+12)^\circ &= (5x-3)^\circ\\ 15^\circ &= x\end{align*}

If x=15\begin{align*}x = 15^\circ\end{align*}, then the base angles are 4(15)+12\begin{align*}4(15^\circ) +12^\circ\end{align*}, or 72\begin{align*}72^\circ\end{align*}. The vertex angle is 1807272=36\begin{align*}180^\circ-72^\circ-72^\circ=36^\circ\end{align*}.

#### Example 2

Find the measure of \begin{align*}x\end{align*}.

The two sides are equal, so set them equal to each other and solve for \begin{align*}x\end{align*}.

\begin{align*}2x-9 & = x+5\\ x & = 14 \end{align*}

#### Example 3

True or false: Base angles of an isosceles triangle can be right angles.

This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right (\begin{align*}90^\circ\end{align*}) angles. The Triangle Sum Theorem states that the sum of the three angles in a triangle is \begin{align*}180^\circ\end{align*}. If two of the angles in a triangle are right angles, then the third angle must be \begin{align*}0^\circ\end{align*} and the shape is no longer a triangle.

### Review

Find the measures of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

Determine if the following statements are true or false.

1. Base angles of an isosceles triangle are congruent.
2. Base angles of an isosceles triangle are complementary.
3. Base angles of an isosceles triangle can be equal to the vertex angle.
4. Base angles of an isosceles triangle are acute.

Complete the proofs below.

1. Given: Isosceles \begin{align*}\triangle CIS\end{align*}, with base angles \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*}
1. Given: Isosceles \begin{align*}\triangle ICS\end{align*} with \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*}

On the \begin{align*}x-y\end{align*} plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.

1. (-2, 1), (1, -2), (-5, -2)
2. (-2, 5), (2, 4), (0, -1)
3. (6, 9), (12, 3), (3, -6)
4. (-10, -5), (-8, 5), (2, 3)
5. (-1, 2), (7, 2), (3, 9)

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### Vocabulary Language: English

Base Angles

The base angles of an isosceles triangle are the angles formed by the base and one leg of the triangle.

Base Angles Theorem Converse

The base angles theorem converse states if two angles in a triangle are congruent, then the sides opposite those angles are also congruent.

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that the perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

Vertical Angles

Vertical angles are a pair of opposite angles created by intersecting lines.