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Triangle Relationships

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Theorems about Triangles
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In the triangle below, point  D is the midpoint of  \overline{AC} and point  E is the midpoint of \overline{BC} . Make a conjecture about how  \overline{DE} relates to \overline{AB} .

Guidance

Recall that a triangle is a shape with exactly three sides. Triangles can be classified by their sides and by their angles

When classifying a triangle by its sides, you should look to see if any of the sides are the same length. If no sides are the same length, then it is a scalene triangle . If two sides are the same length, then it is an isosceles triangle . If all three sides are the same length, then it is an equilateral triangle.

When classifying a triangle by its angles, you should look at the size of the angles. If there is a right angle, then it is a right triangle . If the measures of all angles are less than 90^\circ , then it is an acute triangle . If the measure of one angle is greater than 90^\circ , then it is an obtuse triangle . Additionally, if all angles of a triangle are the same, the triangle is equiangular.

In the examples and practice, you will learn how to prove many different properties of triangles.

Example A

Prove that the sum of the interior angles of a triangle is 180^\circ .

Solution: This is a property of triangles that you have heard and used before, but you may not have ever seen a proof for why it is true. Here is a proof in the paragraph format, that relies on parallel lines and alternate interior angles. 

Consider the generic triangle below.

By the parallel postulate, there exists exactly one line parallel to \overline{AC}  through B . Draw this line.

\angle DBA \cong \angle A  because they are alternate interior angles and alternate interior angles are congruent when lines are parallel. Therefore, m \angle DBA=m \angle A . Similarly, \angle EBC \cong \angle C  because they are also alternate interior angles, and so m \angle EBC=m \angle Cm \angle DBA+m \angle ABC+m \angle EBC=180^\circ because these three angles form a straight line. By substitution, m \angle A+m \angle ABC+m \angle C=180^\circ .

The picture below uses color coding to show the angles that are congruent, referenced in the above proof.

The statement “ the sum of the measures of the interior angles of a triangle is 180^\circ ” is a theorem . Now that it has been proven, you can use it in future proofs without proving it again.

Example B

Prove that the base angles of an isosceles triangle are congruent.

Solution: The base angles of an isosceles triangle are the angles opposite the congruent sides. Below, the base angles are marked for isosceles \Delta ABC .

Your job is to prove that  \angle B \cong \angle C given that  \overline{AB} \cong \overline{AC} . Here is a proof in the two-column format, that relies on angle bisectors and congruent triangles. The proof will reference the picture below.

Statements

Reasons

Isosceles \Delta ABC

Given

\overline{AB} \cong \overline{AC}

Definition of isosceles triangle

Construct \overleftrightarrow{A D} , the angle bisector of \angle A , with  F the intersection of  \overline{BC} and \overleftrightarrow{A D}

An angle has only one angle bisector

\overline{AF} \cong \overline{AF}

Reflexive Property

\angle BAF \cong \angle CAF

Definition of angle bisector

  \Delta ABF \cong \Delta ACF

SAS \cong

\angle B \cong \angle C

CPCTC

The statement “ the base angles of an isosceles triangle are congruent ” is a theorem . Now that it has been proven, you can use it in future proofs without proving it again.

Example C

Prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Solution: An exterior angle of a triangle is an angle outside of a triangle created by extending one of the sides of the triangles. Below, \angle ACD  is an exterior angle. For exterior angle \angle ACD , the angles \angle A  and \angle B  are the remote interior angles , because they are the interior angles that are not adjacent to the exterior angle.

Here is a flow diagram proof of this theorem.

The statement “the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles ” is a theorem . Now that it has been proven, you can use it in future proofs without proving it again.

Concept Problem Revisited

A conjecture is a guess about something that might be true. After making a conjecture, usually you will try to prove it. Two possible conjectures are:

  1. \overline{DE} \ \| \ \overline{AB}
  2. The length of  \overline{DE} is half the length of  \overline{AB}

Both of these conjectures will be proved in the Guided Practice and Practice questions.

Vocabulary

A scalene triangle has no sides that are congruent.

An isosceles triangle has two sides that are congruent.

An equilateral triangle has three sides that are congruent.

A right triangle has one right angle.

An obtuse triangle has one angle with a measure that is greater than 90^\circ .

An acute triangle has all three angles with measures less than 90^\circ .

An equiangular triangle has three congruent angles.

Guided Practice

Consider the picture below. The following questions will guide you through proving that \overline{DE} \ \| \ \overline{AB} .

1. Prove that \Delta FEB \cong \Delta DEC

2. Continue your proof from #1 to prove that \overline{BF} \ \| \ \overline{AC} .

3. Continue your proof from #2 to prove that \Delta ADB \cong \Delta FBD .

4. Continue your proof from #3 to prove that \overline{DE} \ \| \ \overline{AB} .

Answers:

 

Statements

Reasons

1.

\overline{DC} \cong \overline{AD} , \overline{CE} \cong \overline{EB} , \overline{DE} \cong \overline{EF}

Given

 

\angle CED \cong \angle FEB

Vertical angles are congruent

 

\Delta FEB \cong \Delta DEC SAS \cong

2.

\angle FBE \cong \angle ECD

CPCTC

 

\overline{BF} \ \| \ \overline{AC}

If alternate interior angles are congruent then lines are parallel.

3.

\angle ADB \cong \angle DBF

If lines are parallel then alternate interior angles are congruent.

 

\overline{DB} \cong \overline{DB}

Reflexive property

 

\overline{BF} \cong \overline{DC}

CPCTC

 

\overline{BF} \cong \overline{AD}

Substitution

 

\Delta ADB \cong \Delta FBD SAS \cong

4. 

\angle ABD \cong \angle FDB

CPCTC

 

\overline{DE} \ \| \ \overline{AB}

If alternate interior angles are congruent then lines are parallel.

Note that there are other ways to prove that the two segments are parallel. One method relies on similar triangles, which will be explored in another concept.

Practice

1. In Example A you proved that the sum of the interior angles of a triangle is  180^\circ using a paragraph proof. Now, rewrite this proof in the two-column format.

2. Rewrite the proof from #1 again in the flow diagram format.

3. In Example B you proved that the base angles of an isosceles triangle are congruent using a two-column proof. Now, rewrite this proof in the paragraph format.

4. Rewrite the proof from #3 again in the flow diagram format.

5. In Example C you proved that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles using the flow diagram format. Now, rewrite this proof in the paragraph format.

6. Rewrite the proof from #5 again in the two-column format.

7. In the Guided Practice questions you proved that the segment joining midpoints of two sides of a triangle is parallel to the third side of the triangle. Given the diagram below and that \Delta ADB \cong \Delta FBD  as proved in the Guided Practice questions, prove that DE=\frac{1}{2}AB .

8. In Example B you proved that “if a triangle is isosceles, the base angles are congruent”. What is the converse of this statement? Do you think the converse is also true?

9. Prove that if two angles of a triangle are congruent, then the triangle is isosceles. Use the diagram and two-column proof below and fill in the blanks to complete the proof.

Statements

Reasons

\angle B \cong \angle C

________

Construct  \overleftrightarrow{A F} , the angle bisector of \angle A , with  F the intersection of \overline{BC} and \overleftrightarrow{A F}

An angle has only one angle bisector

________

Definition of angle bisector

________

Reflexive Property

\Delta ABF \cong \Delta ACF ________
________

CPCTC

10. Rewrite the proof from #9 in the flow diagram format.

11. Rewrite the proof from #9 in the paragraph format.

12. Given that  \Delta ABC \cong \Delta BAD , prove that \Delta AEB  is isosceles.

13. Given the markings in the picture below, explain why  \overline{CD} is the perpendicular bisector of \overline{AB} .

14. In the picture below, \Delta ABC  is isosceles with \overline{AC} \cong \overline{CB} . E is the midpoint of  \overline{AC} and  D is the midpoint of \overline{CB} . Prove that \Delta EAB \cong \Delta DBA .

15. Explain why knowing that \Delta ABC  is isosceles is not enough information to prove that \Delta ABD \cong \Delta CBD .

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