This document covers theorems and proofs related to triangle inequalities. It discusses the side-angle theorem, angle-side theorem, exterior angle inequality theorem, hinge theorem, and converse of the hinge theorem. Examples are provided to show ordering angles and sides of a triangle based on these theorems. Two two-column proofs are also presented, one using the exterior angle inequality theorem and one using triangle inequality theorem 1.

1. Lesson 2
APPLIES THEOREMS
AND
PROVES TRIANGLE INEQUALITIES

2. Inequalities in One Triangle
Triangle Inequality Theorem 1 SsAa or (Side Angle Theorem)
states that if one side of a triangle is longer than a second side, then the angle opposite
the first side is larger than the angle opposite the second side.

3. Inequalities in One Triangle
Triangle Inequality Theorem 1 SsAa or (Side Angle Theorem)
states that if one side of a triangle is longer than a second side, then the angle opposite
the first side is larger than the angle opposite the second side.
For triangle ABC, order its angle measures from least to greatest.
Least to greatest:
∠C, ∠A, ∠B
∴ ∠C < ∠A
∴ ∠A < ∠B
∴ ∠C < ∠B
∴ ∠B > ∠A
∴ ∠A > ∠C
A
C
B
15
12
10

4. Inequalities in One Triangle
Triangle Inequality Theorem 1 SsAa or (Side Angle Theorem)
states that if one side of a triangle is longer than a second side, then the angle opposite
the first side is larger than the angle opposite the second side.
For triangle ABC, order its angle measures from least to greatest.
Least to greatest:
∠C, ∠A, ∠B
∴ ∠C < ∠A
∴ ∠A < ∠B
∴ ∠C < ∠B
∴ ∠B > ∠A
∴ ∠A > ∠C
∴ ∠B > ∠C
A
C
B
15
12
10

5. Triangle Inequality Theorem 2 AaSs or (Angle Side Theorem)
states that if one angle of a triangle is larger than a second angle, then the side opposite
the first angle is longer than the side opposite the second angle.
For triangle ABC, order its side lengths from shortest to longest.
Inequalities in One Triangle
Shortest to longest:
BC, AB, AC
∴ BC < AB
∴ AB < AC
∴ BC < AC
∴ AC > AB
∴ AB > BC
∴ AC > BC
A
C
B
30°
100°
50°

6. Inequalities in One Triangle
Exterior Angle Inequality Theorem
states that the measure of any exterior angle of a triangle is greater than either of the
opposite interior angles.
Consider the figure below, fill in each space to complete true inequality
statements:
If m∠c = 40° and m∠d = 29°,
then m∠a > 40° and m∠a > 29°
If m∠b = 54° and m∠d = 38°,
then m∠a > 54° and m∠a + m∠d = 180°

7. Hinge Theorem
states that if two triangles have two congruent sides (sides of equal length), then the
triangle with the larger angle between those sides will have a longer third side.
Inequalities in Two Triangles

8. Hinge Theorem
states that if two triangles have two congruent sides (sides of equal length), then the
triangle with the larger angle between those sides will have a longer third side.
From the inequalities in the triangles shown, a conclusion can be reached using
the hinge theorem. What would be the last statement?
Inequalities in Two Triangles
25° 35°
10 10
8
∠GOD < ∠SOD,
∴ GD < SD
G
O
D
S

9. Converse of Hinge Theorem
states that if two triangles have two congruent sides, then the triangle with the longer
third side will have a larger angle opposite that third side.
Inequalities in Two Triangles

10. Use the symbol <, > or = to complete the statements about the figure shown.
Justify your answer.
Applying Triangle Inequality Theorems
= Hinge Theorem
> Converse of Hinge Theorem
< Hinge Theorem
> Converse of Hinge Theorem

11. Complete the two-column proof.
Given: ΔMON with exterior angle ∠ONP
Prove: m∠ONP > m∠MON
Construction: Midpoint Q on ON such that OQ ≅ NQ. MR through Q such that MQ ≅ QR
Proving Triangle Inequalities
Statement Reason
1. OQ ≅ NQ; MQ ≅ QR By construction
2. ∠3 ≅ ∠4 Vertical Angles Theorem
3. ΔMQO ≅ ΔRQN SAS Postulate
4. m∠MON ≅ m∠1 CPCTC Theorem
5. m∠ONP ≅ m∠1 + m∠2 Angle Addition Postulate (AAP)
6. ∴m∠ONP > m∠MON Exterior Angle Inequality theorem

12. Complete the two-column proof.
Given: AV = EV, AW > EW
Prove: m∠WVE < m∠WVA
Proving Triangle Inequalities
Statement Reason
1. AV = EV Given
2. AW > EW Given
3. WV ≅ WV Reflexive Property of Equality
4. ∴m∠WVE < m∠WVA Triangle Inequality Theorem 1