theorem The sum of the measures of the interior angles of a triangle equals 180º.

Examples:
1. 180triangle1
Find mB
and mC.

Solution: mA + mB + mC = 180
38 + x + (x + 2) = 180
40 + 2x = 180
2x = 140
x = 70 = m∠B
x + 2 = 72 = m∠C

2. The angles in a triangle are represented by (4x - 6)º, (2x + 1)º and (x + 3)º. Is this a right triangle?

Solution:
(4x - 6) + (2x + 1) + (x + 3) = 180
7x - 2 = 180
7x = 182
x = 26
(4x - 6)º = 98º
(2x + 1)º = 53º
(x + 3)º = 29º
No. The triangle is obtuse.

3.180triangle3
mABC=mBCD
Find mACD.

Solution: mBCD = 56º.
In ΔABC, 85º + 56º + mBCA = 180
mBCA = 39º
m
ACD = 56º - 39º = 17º

4. The angles in a triangle are in the ratio of 1 : 2 : 3. Find the measure of the angles in the triangle.

Solution:
x + 2x + 3x = 180
6x = 180
x = 30
The angles are 30º, 60º, and 90º.

 

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In your previous studies of this theorem, you most likely saw how a triangle's angles can be cut off and rearranged to form a straight angle of 180º (as shown below).

cut up triangle

 

We now need to use more sophisticated ideas to establish that this theorem is actually true. Throughout history, several different methods of proof of this theorem have appeared.
Let's take a look at a few of these methods of proof.

Transformational Proofs:

Transformational Proof Using Translation
180proof1
[Also written as mA + mABC + mC = 180.]
180translate
Proof:
• Translate ΔABC so C' coincides with B forming a straight line from point C, through point B, to point B' (along vector vectorCB).
• Since translations are rigid transformations, we know that angle measure is preserved making m∠C = m∠A'BB'.
• A straight angle, <B'BC, whose measure by definition is 180º, exists giving
m∠A'BB' + m∠A'BA + m∠ABC
= 180.
• Angles ∠C and ∠A'BB' are congruent corresponding angles making parallel180. It can also be said that these sides are parallel because in a translation the corresponding segment sides of the pre-image and image are parallel.
• As a result, ∠A and ∠A'BA are congruent alternate interior angles. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent.
• By substitutions, m∠C + m∠A + m∠ABC = 180.



Transformational Proof Using Rotation
180rotategiven
[Or m∠ABC + m∠ACB + m∠BAC = 180.]

180rotate
Proof:
• Rotate ΔABC about the midpoint of BC.
• Then rotate ΔA'B'C' about the midpoint of A'C'.

• Since rotations are rigid transformations, angle measure is preserved and m∠ABC = m∠A'B'C' and m∠B'A'C' = m∠B''A''C''.

• These alternate interior angles will be congruent, making 180parallels
If 2 lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
Since both AB and B''B pass through point B and are parallel to A'B', we know the segment from A to B'' is straight (there is only one line through B parallel to A'B' - Parallel Postulate).
• A straight angle is an angle whose rays form a straight line, making ∠ABB'' a straight angle with a measure of 180º. Now, m∠ABC + m∠A'BB' + m∠B''BA' = 180.

• Rotations preserve angle measure, making m∠A'BB' = m∠ACB and m∠BAC = m∠BA'C = m∠A'BB''.
• By substitution, m∠ABC + m∠ACB + m∠BAC = 180.

 

 

Traditional (Classical) Proofs:

Proof Using An Auxiliary Parallel Line
180rotategiven

green180
Statements
Reasons
1. ΔABC
1. Given
2. Draw auxiliary line through B parallel to AC.
2. Through a point not on a line, only one line may be drawn parallel to a given line.
3. ∠DBE is a straight angle.
3. A straight line forms a straight angle.
4.  m∠DBE = 180º
4. A straight angle has a measure of 180º.
5.  m∠1 + m∠2 + m∠3 = m∠DBE
5. Angle Addition Postulate (whole quantity)
6.  SAN1
6. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent.
7.  m∠1 = mA;    m∠3 = mC
7. Congruent angles are angles of equal measure.
8.  m∠A + m∠2 + m∠C = 180º
8. Substitution



Slightly different version:
Proof Using An Auxiliary Parallel Line with an Extension
proof180Given

proof180B
Statements
Reasons
1. ΔABC
1. Given
2. Extend AC through C to E, and draw an auxiliary line through C parallel to AB.
2. Through a point not on a line, only one line may be drawn parallel to a given line.
3. ∠ACE is a straight angle.
3. A straight line forms a straight angle.
4.  m∠ACE = 180º
4. A straight angle has a measure of 180º.
5.  m∠1 + m∠2 + m∠3 = m∠ACE
5. Angle Addition Postulate (whole quantity)
6. SAN2
6. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent.
7. SAN3
7. If 2 parallel lines are cut by a transversal, the corresponding angles are congruent.
7.  mB = m∠2;    m∠3 = mA
7. Congruent angles are angles of equal measure.
8.  m∠1 + mB + m∠A = 180º
8. Substitution



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