Download 160927-proofs-if-then-perp-bis-and-iso-tri

Name
September 27, 2016
Math 2 class work and homework
page 1
Proofs of if-then statements
(perpendicular bisectors and isosceles triangles)
Proof-writing objective: Write proofs of statements written in if-then form.
Geometry objective: Prove key theorems about perpendicular bisectors and about isosceles
triangles.
Many mathematical facts are written in a form “If ______________ then ________________.”
To prove statements of this form, first we must extract from the sentence what information is
given (generally that’s in the “if” part of the sentence) and what needs to be proved (generally
that’s what follows “then” in the sentence). Also it’s usually necessary to draw a picture
if one has not been provided.
Perpendicular bisectors
1. Write a proof of this theorem.
Perpendicular Bisector Theorem: If a point P is on the perpendicular bisector
of segment AB then it is equally distant from the endpoints A and B.
Given:
Draw a picture:
Prove:
Proof (in your choice of flowchart format or paragraph format):
Name
September 27, 2016
Math 2 class work and homework
page 2
2. Write a proof of this theorem.
If a point P is equally distant from points A and B then it must be on the
perpendicular bisector of segment AB.
Given:
Draw a picture:
Prove:
Proof (in your choice of flowchart format or paragraph format):
Reflection: Compare the theorem you proved on page 1 with the theorem you proved on page 2.
How are they related? How are they different?
Name
September 27, 2016
Math 2 class work and homework
page 3
Isosceles triangles
Here you’ll prove a key theorem about isosceles triangles, the Isosceles Triangle Theorem.
3. a. What is an isosceles triangle? Draw a picture of one.
b. If a triangle is isosceles, what do you think is true about its angles?
c. Construct the bisector of the vertex angle. Label the point where the bisector intersects
the base of the triangle.
d. Now prove that the base angles are congruent.
Name
September 27, 2016
Math 2 class work and homework
page 4
If you did the previous page correctly, you have proved this theorem.
Isosceles Triangle Theorem: If a triangle has two congruent sides
then the angles opposite those sides are congruent.
Homework
Below is another theorem to prove. Note that it involves reversing the “if’ and “then” parts of the
previous theorem, which is called forming the converse of a statement.
Converse Isosceles Triangle Theorem: If a triangle has two congruent angles
then the sides opposite those angles are congruent.
A converse statement is logically different from the original statement, because what’s given and
what needs to be proved have been reversed. (In fact, there are many examples of true statements
for which the converse statement is false.) Thus, a separate proof is needed.
4. Write a proof of this theorem.
Converse Isosceles Triangle Theorem: If a triangle has two congruent angles
then the sides opposite those angles are congruent.
Given:
Prove:
Proof (in any format of your choice):
Draw a picture:
Name
September 27, 2016
Math 2 class work and homework
page 5
Definitions needed for the following problems:



An equilateral triangle is a triangle where all three sides are congruent to each other.
An equiangular triangle is a triangle where all three angles are congruent to each other.
A square is a quadrilateral where all four sides are congruent and all four angles are congruent.
5. Write a proof of this theorem. Start by drawing a picture then writing what’s given and
what’s to be proved.
If a triangle is equilateral then it is equiangular.
Hint: Start by proving two of the angles are congruent, then work on the third angle.
Name
September 27, 2016
Math 2 class work and homework
page 6
6. Use the diagram for page 494 exercise 7.
Part b is the same question asked in the book; part a is a hint to help you get there.
a. Given: FACG and DABE are squares.
Prove: FAB  CAD.
b. Given: FACG and DABE are squares.
Prove: FAB  CAD.
Name
September 27, 2016
Math 2 class work and homework
page 7
7. Write a proof of this theorem. Start by drawing a picture then writing what’s given and
what’s to be proved.
If a triangle is equiangular then it is equilateral.
8. This question calls for an answer that you won’t be able to prove yet: When a triangle is
equiangular, how many degrees is each of the angles? Briefly explain why you think so.