NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-6 Study Guide and Intervention
Inequalities in Two Triangles
Hinge Theorem The following theorem and its converse involve the relationship between the sides of two triangles and
an angle in each triangle.
Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle and the included angle of the first is
larger than the included angle of the second, then the
third side of the first triangle is longer than the third side
of the second triangle.
Converse of the
Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle, and the third side in the first is longer
than the third side in the second, then the included angle
in the first triangle is greater than the included angle in
the second triangle.
Example 1 : Compare the measures of ̅̅̅̅
𝑮𝑭 and ̅̅̅̅
𝑭𝑬 .
Example 2 : Compare the measures
of ∠ ABD and ∠ CBD.
Two sides of △HGF are congruent to two sides of
△HEF, and m∠ GHF > m∠ EHF. By the Hinge
Theorem, GF > FE.
Two sides of △ABD are congruent to two sides of
△CBD, and AD > CD. By the Converse of the Hinge
Theorem, m∠ ABD > m∠ CBD.
Exercises
Compare the given measures.
1. MR and RP
2. AD and CD
3. m∠ C and m∠ Z
4. m∠ XYW and m∠ WYZ
Write an inequality for the range of values of x.
5.
Chapter 5
6.
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Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-6 Study Guide and Intervention (continued)
Inequalities Involving Two Triangles
PROVE RELATIONSHIPS IN TWO TRIANGLES You can use the Hinge Theorem and its converse to prove
relationships in two triangles.
Example :
Given: RX = XS
m∠ SXT = 97
Prove: ST > RT
Proof:
Statements
Reasons
1. ∠ SXT and ∠ RXT are
supplementary.
1. Def of linear pair.
2. m ∠ SXT + m∠ RXT = 180
3. m∠ SXT = 97
2. Def of supplementary.
4. 97 + m∠ RXT = 180
4. Substitution
5. m∠ RXT = 83
5. Subtraction
6. 97 > 83
7. m∠ SXT > m∠ RXT
6. Inequality
3. Given
8. RX = XS
7. Substitution
8. Given
9. TX = TX
9. Reflexive Property
10. ST > RT
10. Hinge Theorem
Exercises
Complete the proof.
Given: rectangle AFBC
ED = DC
Prove: AE > FB
Proof:
Statements
Reasons
1. rectangle AFBC, ED = DC
1. Given
2. AD = AD
2. Reflexive Property
3. m∠ EDA > m∠ ADC
3. Exterior Angle Inequality
4.
5.
6. AE > FB
4. Hinge Theorem
Chapter 5
5. Opp sides in rectangle are ≅.
6. Substitution
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Glencoe Geometry