Relationships Between Angles and Sides in Triangles

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

RELATIONSHIPS BETWEEN ANGLES AND SIDES IN TRIANGLES

Jump right to the exercises!

To prove a sentence of the form A⇔B ,
one often proves the forward direction A⇒B
and the reverse direction B⇒A .
Together, this proves the equivalence.
The justification is the truth table below, which shows that
A⇔B is equivalent to ( A⇒B ) and ( B⇒A ).
(Notice that the last two columns are identical!)

A	B	A⇒B	B⇒A	(A⇒B) and (B⇒A)	A⇔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

Notice how this form of proof is used below.
Also 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∠B=m∠C
PROOF #1:
STATEMENTS	REASONS
1. AB=AC	given
2. AC=AB	given
3. m∠A=m∠A	reflexive property
4. ΔABC≅ ΔACB	SAS
5. m∠B=m∠C	CPCTC

GIVEN: m∠B=m∠C PROVE: AB=AC
PROOF #2:
STATEMENTS	REASONS
1. m∠B=m∠C	given
2. m∠C=m∠B	given
3. BC=CB	reflexive property
4. ΔABC≅ ΔACB	ASA
5. AB=AC	CPCTC

Together, we have:

THEOREM:
Two sides of a triangle have equal lengths ⇔ the angles opposite them have equal measures.

Proof:
"⇒" See Proof #1 above.
"⇐" 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.

DEFINITIONS: Triangle Classifications according to Lengths of Sides
A triangle is   equilateral   if and only if   all its sides are equal.
A triangle is   isosceles   if and only if   it has at least two equal sides.
A triangle is   scalene   if and only if   all its sides have different lengths.

DEFINITIONS: Triangle Classifications according to Sizes of Angles
A triangle is   equiangular   if and only if   all its angles are equal.
A triangle is   obtuse   if and only if   it has an obtuse angle.
A triangle is   acute   if and only if   all its angles are acute.

Recall that since the angles in a triangle sum to 180°, 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:

COROLLARY:
Every equilateral triangle is equiangular.
Every equiangular triangle is equilateral.

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 theorem:

THEOREM:
In a scalene triangle, the longest side is opposite the biggest angle;
the medium side is opposite the medium angle; and
the shortest side is opposite the smallest angle.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.