In the triangle below, point is the midpoint of and point is the midpoint of . Make a conjecture about how relates to .
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 , then it is an acute triangle. If the measure of one angle is greater than , 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 .
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 through . Draw this line.
because they are alternate interior angles and alternate interior angles are congruent when lines are parallel. Therefore, . Similarly, because they are also alternate interior angles, and so . because these three angles form a straight line. By substitution, .
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 ” 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 .
Your job is to prove that given that . Here is a proof in the twocolumn format, that relies on angle bisectors and congruent triangles. The proof will reference the picture below.
Statements 
Reasons 
Isosceles 
Given 
Definition of isosceles triangle 

Construct , the angle bisector of , with the intersection of and 
An angle has only one angle bisector 
Reflexive Property 

Definition of angle bisector 



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, is an exterior angle. For exterior angle , the angles and 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:
 The length of is half the length of
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 .
An acute triangle has all three angles with measures less than .
An equiangular triangle has three congruent angles.
Guided Practice
Consider the picture below. The following questions will guide you through proving that .
1. Prove that .
2. Continue your proof from #1 to prove that .
3. Continue your proof from #2 to prove that .
4. Continue your proof from #3 to prove that .
Answers:

Statements 
Reasons 
1. 
, , 
Given 

Vertical angles are congruent 



2. 
CPCTC 


If alternate interior angles are congruent then lines are parallel. 

3. 
If lines are parallel then alternate interior angles are congruent. 


Reflexive property 


CPCTC 


Substitution 



4. 
CPCTC 


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 using a paragraph proof. Now, rewrite this proof in the twocolumn 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 twocolumn 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 twocolumn 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 as proved in the Guided Practice questions, prove that .
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 twocolumn proof below and fill in the blanks to complete the proof.
Statements 
Reasons 
________ 

Construct , the angle bisector of , with the intersection of and 
An angle has only one angle bisector 
________ 
Definition of angle bisector 
________ 
Reflexive Property 
________  
________ 
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 , prove that is isosceles.
13. Given the markings in the picture below, explain why is the perpendicular bisector of .
14. In the picture below, is isosceles with . is the midpoint of and is the midpoint of . Prove that .
15. Explain why knowing that is isosceles is not enough information to prove that .