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# Third Angle Theorem

## Third angles are the same if the other two sets are congruent.

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Third Angle Theorem

What if you were given FGH\begin{align*}\triangle FGH\end{align*} and XYZ\begin{align*}\triangle XYZ\end{align*} and you were told that FX\begin{align*}\angle{F} \cong \angle{X}\end{align*} and GY\begin{align*}\angle{G} \cong \angle{Y}\end{align*}? What conclusion could you draw about H\begin{align*}\angle{H}\end{align*} and Z\begin{align*}\angle{Z}\end{align*}

### Third Angle Theorem

Find mC\begin{align*}m \angle C\end{align*} and mJ\begin{align*}m \angle J\end{align*}.

The sum of the angles in each triangle is 180\begin{align*}180^\circ\end{align*} by the Triangle Sum Theorem. So, for ABC,35+88+mC=180\begin{align*}\triangle ABC, 35^\circ + 88^\circ + m \angle C = 180^\circ\end{align*} and mC=57\begin{align*}m \angle C = 57^\circ\end{align*}. For HIJ\begin{align*}\triangle HIJ\end{align*}, 35+88+mJ=180\begin{align*}35^\circ + 88^\circ + m \angle J = 180^\circ\end{align*} and mJ\begin{align*}m \angle J\end{align*} is also 57\begin{align*}57^\circ\end{align*}.

Notice that we were given that mA=mH\begin{align*}m \angle A = m \angle H\end{align*} and mB=mI\begin{align*}m \angle B = m \angle I\end{align*} and we found out that mC=mJ\begin{align*}m \angle C = m \angle J\end{align*}. This can be generalized into the Third Angle Theorem.

Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent.

In other words, for triangles ABC\begin{align*}\triangle ABC\end{align*} and DEF\begin{align*}\triangle DEF\end{align*}, if AD\begin{align*} \angle A \cong \angle D\end{align*} and BE\begin{align*}\angle B \cong \angle E\end{align*}, then CF\begin{align*}\angle C \cong \angle F\end{align*}.

Notice that this theorem does not state that the triangles are congruent. That is because if two sets of angles are congruent, the sides could be different lengths. See the picture below.

#### Measuring Missing Angles

Determine the measure of the missing angles.

From the markings, we know that AD\begin{align*}\angle A \cong D\end{align*} and EB\begin{align*}\angle E \cong \angle B\end{align*}. Therefore, the Third Angle Theorem tells us that CF\begin{align*}\angle C \cong \angle F\end{align*}. So,

mA+mB+mCmD+mB+mC42+83+mCmC=180=180=180=55=mF\begin{align*}m \angle A+m \angle B+m \angle C &= 180^\circ\\ m \angle D+m \angle B+m \angle C &= 180^\circ\\ 42^\circ+83^\circ+m \angle C &= 180^\circ\\ m \angle C &= 55^\circ=m \angle F\end{align*}

#### Understanding the Third Angle Theorem

The Third Angle Theorem states that if two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. What additional information would you need to know in order to be able to determine that the triangles are congruent?

In order for the triangles to be congruent, you need some information about the sides. If you know two pairs of angles are congruent and at least one pair of corresponding sides are congruent, then the triangles will be congruent.

#### Measuring Angles

Determine the measure of all the angles in the triangle:

First we can see that mDCA=15\begin{align*} m \angle DCA=15^\circ\end{align*}. This means that mBAC=15\begin{align*}m\angle BAC =15^\circ\end{align*} also because they are alternate interior angles. mABC=153\begin{align*} m\angle ABC=153^\circ\end{align*} was given. This means by the Triangle Sum Theorem that mBCA=12\begin{align*} m \angle BCA=12^\circ\end{align*}. This means that mCAD=12\begin{align*} m\angle CAD=12^\circ\end{align*} also because they are alternate interior angles. Finally, mADC=153\begin{align*} m\angle ADC =153^\circ\end{align*} by the Triangle Sum Theorem.

#### Earlier Problem Revisited

For two given triangles FGH\begin{align*} \triangle FGH\end{align*} and XYZ\begin{align*} \triangle XYZ\end{align*}, you were told that FX\begin{align*} \angle F \cong \angle X\end{align*} and GY\begin{align*} \angle G \cong \angle Y\end{align*}.

By the Third Angle Theorem, HZ\begin{align*} \angle H \cong \angle Z \end{align*}.

### Examples

Determine the measure of all the angles in the each triangle.

#### Example 1

mA=86\begin{align*}m\angle A=86\end{align*}, mC=42\begin{align*}m\angle C = 42\end{align*} and by the Triangle Sum Theorem mB=52\begin{align*}m\angle B=52\end{align*}.

mY=42\begin{align*}m\angle Y=42\end{align*}, mX=86\begin{align*} m\angle X = 86 \end{align*} and by the Triangle Sum Theorem, mZ=52\begin{align*}m\angle Z = 52\end{align*}.

#### Example 2

mC=mA=mY=mZ=35\begin{align*} m \angle C = m \angle A=m \angle Y=m \angle Z =35\end{align*}. By the Triangle Sum Theorem mB=mX=110\begin{align*} m\angle B= m \angle X =110\end{align*}.

#### Example 3

mA=28\begin{align*} m \angle A=28\end{align*}, mABE=90\begin{align*}m \angle ABE = 90\end{align*} and by the Triangle Sum Theorem, mE=62\begin{align*}m \angle E = 62\end{align*}. mD=mE=62\begin{align*} m\angle D= m\angle E=62\end{align*} because they are alternate interior angles and the lines are parallel. mC=mA=28\begin{align*} m\angle C= m\angle A=28\end{align*} because they are alternate interior angles and the lines are parallel. \begin{align*} m \angle DBC = m\angle ABE = 90\end{align*} because they are vertical angles.

### Review

Determine the measures of the unknown angles.

1. \begin{align*}\angle XYZ\end{align*}
2. \begin{align*}\angle ZXY\end{align*}
3. \begin{align*}\angle LNM\end{align*}
4. \begin{align*}\angle MLN\end{align*}

1. \begin{align*}\angle CED\end{align*}
2. \begin{align*}\angle GFH\end{align*}
3. \begin{align*}\angle FHG\end{align*}

1. \begin{align*}\angle ACB\end{align*}
2. \begin{align*}\angle HIJ\end{align*}
3. \begin{align*}\angle HJI\end{align*}
4. \begin{align*}\angle IHJ\end{align*}

1. \begin{align*}\angle RQS\end{align*}
2. \begin{align*}\angle SRQ\end{align*}
3. \begin{align*}\angle TSU\end{align*}
4. \begin{align*}\angle TUS\end{align*}

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