Theorems about Triangles

In the triangle below, point D is the midpoint of AC¯¯¯¯¯¯¯¯ and point E is the midpoint of BC¯¯¯¯¯¯¯¯. Make a conjecture about how DE¯¯¯¯¯¯¯¯ relates to AB¯¯¯¯¯¯¯¯.

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°, then it is an acute triangle. If the measure of one angle is greater than 90°, 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.

Let's take a look at some example problems. 

1. Prove that the sum of the interior angles of a triangle is 180°.

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 AC¯¯¯¯¯¯¯¯ through B. Draw this line.

∠DBA≅∠A because they are alternate interior angles and alternate interior angles are congruent when lines are parallel. Therefore, m∠DBA=m∠A. Similarly, ∠EBC≅∠C because they are also alternate interior angles, and so m∠EBC=m∠C. m∠DBA+m∠ABC+m∠EBC=180° because these three angles form a straight line. By substitution, m∠A+m∠ABC+m∠C=180°.

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°” is a theorem. Now that it has been proven, you can use it in future proofs without proving it again.

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

The base angles of an isosceles triangle are the angles opposite the congruent sides. Below, the base angles are marked for isosceles ΔABC.

Your job is to prove that ∠B≅∠C given that AB¯¯¯¯¯¯¯¯≅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.

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.

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

An exterior angle of a triangle is an angle outside of a triangle created by extending one of the sides of the triangles. Below, ∠ACD is an exterior angle. For exterior angle ∠ACD, the angles ∠A and ∠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.

In the interactive below, move the red points to change the shape of the triangle. Move the blue points to compare angles ∠1,∠2, and ∠3.

Notice that because they are vertical angles, the angle pairs m∠1=m∠4,m∠2=m∠5, and m∠3=m∠6.

m∠1+m∠2+m∠3m∠4+m∠5+m∠6=360°=360°

Example 1

Earlier, you were asked to make a conjecture how DE¯¯¯¯¯¯¯¯ relates to AB¯¯¯¯¯¯¯¯.

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. DE¯¯¯¯¯¯¯¯ ll AB¯¯¯¯¯¯¯¯
2. The length of DE¯¯¯¯¯¯¯¯ is half the length of AB¯¯¯¯¯¯¯¯

Both of these conjectures will be proved in the examples and Review questions. 

Consider the picture below. The following questions will guide you through proving that DE¯¯¯¯¯¯¯¯ ll AB¯¯¯¯¯¯¯¯.

Example 2

Prove that ΔFEB≅ΔDEC. 

Example 3

Continue your proof from #1 to prove that BF¯¯¯¯¯¯¯¯ ll AC¯¯¯¯¯¯¯¯.

Example 4

Continue your proof from #2 to prove that ΔADB≅ΔFBD.

Example 5

Continue your proof from #3 to prove that DE¯¯¯¯¯¯¯¯ ll AB¯¯¯¯¯¯¯¯.

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.

1. In Example A you proved that the sum of the interior angles of a triangle is 180° 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 ΔADB≅ΔFBD as proved in the Guided Practice questions, prove that DE=12AB.

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.

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

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

12. Given that ΔABC≅ΔBAD, prove that ΔAEB is isosceles.

13. Given the markings in the picture below, explain why CD¯¯¯¯¯¯¯¯ is the perpendicular bisector of AB¯¯¯¯¯¯¯¯.

14. In the picture below, ΔABC is isosceles with AC¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯. E is the midpoint of AC¯¯¯¯¯¯¯¯ and D is the midpoint of CB¯¯¯¯¯¯¯¯. Prove that ΔEAB≅ΔDBA.

15. Explain why knowing that ΔABC is isosceles is not enough information to prove that ΔABD≅ΔCBD.

16 Given: ∠CBD≅∠EFD; CB¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯ 

Prove: ∠DBF≅∠DFB

 17. Given: BC¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯; CF¯¯¯¯¯¯¯¯≅EB¯¯¯¯¯¯¯¯

Prove: ΔBDF is isosceles

 18. Given: DE¯¯¯¯¯¯¯¯ midsegment of ΔABC ; GF¯¯¯¯¯¯¯¯≅JI¯¯¯¯¯¯; FH¯¯¯¯¯¯¯¯≅IK¯¯¯¯¯¯¯ 

Prove: ΔGFH≅ΔJIK