Lesson 5-5 Inequalities involving two triangles

• Theorem 5.13 SAS Inequality/Hinge Theorem– If two sides of a triangle are congruent to two sides of

another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle.

• Theorem 5.14 SSS Inequality– If two sides of a triangle are congruent to two sides of

another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle.

Write a two-column proof.

Given:

Prove:

Proof:

Statements Reasons

1. 1. Given

2. 2. Alternate interior angles are congruent.

3. 3. Substitution

4. 4. Subtraction Property

5. 5. Given6. 6. Reflexive Property

7. 7. SAS Inequality

Prove: AD < AB

Given: m1 < m3E is the midpoint of

Write a two-column proof.

Proof:

Statements1.

2.3.4.5.6.7.

Reasons1. Given

2. Definition of midpoint3. Reflexive Property4. Given5. Definition of vertical angles 6. Substitution7. SAS Inequality

E is the midpointof

Given:

Prove:

Proof:

Statements Reasons

1. 1. Given2. 2. Reflexive Property3. 3. Given

4. 4. Given

5. 5. Substitution

6. 6. SSS Inequality

Given: X is the midpoint ofMCX is isosceles.CB > CM

Prove:

Proof:

Statements1.2.3.4.

5.6.7.

Reasons1. Given2. Definition of midpoint3. Given4. Definition of isosceles triangle5. Given6. Substitution7. SSS Inequality

X is the midpoint of

MCX is isosceles.

Write an inequality comparing mLDM and mMDN using the information in the figure.

The SSS Inequality allows us to conclude that

Answer:

Write an inequality finding the range of values containing a using the information in the figure.

By the SSS Inequality,

SSS Inequality

Substitution

Subtract 15 from each side.

Divide each side by 9.

Also, recall that the measure of any angle is always greater than 0.

Subtract 15 from each side.

Divide each side by 9.

The two inequalities can be written as the compound inequality

Answer:

Write an inequality using the information in the figure.a.

b. Find the range of values containing n.

Answer:

Answer: 6 < n < 25