In a triangle, we can often use information about the lengths of sides to gain information
about the measures of angles.
Or, we can use information about the measures of angles to gain information
about the lengths of sides.
This section explores notation and results that relate the angles and sides in triangles.
First, recall the biconditional statement:
to prove a sentence of the form
$\,A\Longleftrightarrow B\,$,
one can prove both the forward direction
$\,A\Rightarrow B\,$
and the reverse direction
$\,A\Leftarrow B\,$
(that is, $\,B\Rightarrow A\,$).
Together, this proves the equivalence.
The justification is the truth table below,
which shows that
$\,A\Longleftrightarrow B\,$
is equivalent to
$(A\Rightarrow B) \text{ and } (B\Rightarrow A)\,$.
Notice that the last two columns are identical!
$A$  $B$  $A\Rightarrow B$  $B\Rightarrow A$  $(A\Rightarrow B)\text{ and } (B\Rightarrow A)$  $A\Longleftrightarrow B$ 
T  T  T  T  T  T 
T  F  F  T  F  F 
F  T  T  F  F  F 
F  F  T  T  T  T 
The biconditional statement is used below to prove our first result relating sides and angles:
if two sides in a triangle have the same length,
then the opposite angles have the same measure.
And, if two angles in a triangle have the same measure,
then the opposite sides have the same lengths.
Notice what a beautiful application of SAS and ASA congruence the proof is.
It's a simple and clever idea—showing that a triangle is congruent to a ‘flipped’ copy of itself.
GIVEN: $AB=AC$ PROVE: $m\angle B=m\angle C$ 

PROOF #1:  
STATEMENTS  REASONS 
1. $\,AB=AC\,$  given 
2. $\,AC=AB\,$  given 
3. $\,m\angle A=m\angle A\,$  reflexive property (equality is an equivalence relation on the set of real numbers) 
4. $\,\Delta ABC\cong \Delta ACB\,$  SAS 
5. $\,m\angle B=m\angle C\,$  CPCTC 
GIVEN: $m\angle B=m\angle C$ PROVE: $AB=AC$ 

PROOF #2:  
STATEMENTS  REASONS 
1. $\,m\angle B=m\angle C\,$  given 
2. $\,m\angle C=m\angle B\,$  given 
3. $\,BC=CB\,$  reflexive property (equality is an equivalence relation on the set of real numbers) 
4. $\,\Delta ABC\cong \Delta ACB\,$  ASA 
5. $\,AB=AC\,$  CPCTC 
Together, we have:
Proof:
‘$\,\Rightarrow \,$’
See Proof #1 above.
‘$\,\Leftarrow \,$’
See Proof #2 above.
Q.E.D.
Note:
‘Q.E.D’
is an abbreviation for the Latin phrase
‘quod erat demonstrandum’
which means ‘that which was to be demonstrated (proved).’
It is often used to mark the end of a proof.
Recall that since the angles in a triangle sum to $\,180^{\circ}\,$, a triangle can have at most one obtuse angle.
The name theorem in mathematics is usually reserved for important results.
Things that don't seem quite worthy of being called ‘theorems’ are often given other names.
In particular, a corollary is usually an interesting consequence of a theorem.
Here's a corollary to the previous theorem:
The proof is left to the reader.
Finally, while we're on the subject of angles and sides in a triangle,
here's an interesting and useful theorem:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
