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Third Angle Theorem

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Third Angle Theorem

What if you were given \triangle FGH and \triangle XYZ and you were told that \angle{F} \cong \angle{X} and \angle{G} \cong \angle{Y} ? What conclusion could you draw about \angle{H} and \angle{Z} ? After completing this Concept, you'll be able to make such a conclusion.

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CK-12 Foundation: Chapter4TheThirdAngleTheoremA

Guidance

Find m \angle C and m \angle J .

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

Notice that we were given that m \angle A = m \angle H and m \angle B = m \angle I and we found out that m \angle C = m \angle J . 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 \triangle ABC and \triangle DEF , if   \angle A \cong \angle D and \angle B \cong \angle E , then \angle C \cong \angle F .

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.

Example A

Determine the measure of the missing angles.

From the markings, we know that \angle A \cong D and \angle E \cong  \angle B . Therefore, the Third Angle Theorem tells us that \angle C \cong \angle F . So,

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

Example B

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.

Example C

Determine the measure of all the angles in the triangle:

First we can see that  m \angle DCA=15^\circ . This means that m\angle BAC =15^\circ also because they are alternate interior angles.  m\angle ABC=153^\circ was given. This means by the Triangle Sum Theorem that  m \angle BCA=12^\circ . This means that  m\angle CAD=12^\circ also because they are alternate interior angles. Finally,  m\angle ADC =153^\circ by the Triangle Sum Theorem.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter4TheThirdAngleTheoremB

Concept Problem Revisited

For two given triangles  \triangle FGH and  \triangle XYZ , you were told that  \angle F \cong \angle X and  \angle G \cong \angle Y .

By the Third Angle Theorem,  \angle H \cong \angle Z .

Vocabulary

Two figures are congruent if they have exactly the same size and shape. Two triangles are congruent if the three corresponding angles and sides are congruent. The Triangle Sum Theorem states that the measure of the three interior angles of any triangle will add up to 180^\circ . 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.

Guided Practice

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

1.

2.

3.

Answers:

1. m\angle A=86 , m\angle C = 42 and by the Triangle Sum Theorem m\angle B=52 .

m\angle Y=42 ,  m\angle X = 86 and by the Triangle Sum Theorem, m\angle Z = 52 .

2.  m \angle C = m \angle A=m \angle Y=m \angle Z =35 . By the Triangle Sum Theorem  m\angle B= m \angle X =110 .

3.  m \angle A=28 , m \angle ABE = 90 and by the Triangle Sum Theorem, m \angle E = 62 .  m\angle D= m\angle E=62 because they are alternate interior angles and the lines are parallel.  m\angle C= m\angle A=28 because they are alternate interior angles and the lines are parallel.  m \angle DBC = m\angle ABE = 90 because they are vertical angles.

Explore More

Determine the measures of the unknown angles.

  1. \angle XYZ
  2. \angle ZXY
  3. \angle LNM
  4. \angle MLN

  1. \angle CED
  2. \angle GFH
  3. \angle FHG

  1. \angle ACB
  2. \angle HIJ
  3. \angle HJI
  4. \angle IHJ

  1. \angle RQS
  2. \angle SRQ
  3. \angle TSU
  4. \angle TUS

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