Lesson 5-2 Theorems

• Theorem 5.8 Exterior Angle Inequality Theorem– If an angle is an exterior angle of a triangle, then its measure is

greater than the measure of either of its corresponding remote interior angles

• Theorem 5.9– If one side of a triangle is longer than another side, then the

angle opposite the longer side has a greater measure than the angle opposite the shorter side

• Theorem 5.10– If one angle of a triangle has a greater measure than another

angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Determine which angle has the greatest measure.

Explore Compare the measure of 1 to the measures of 2, 3, 4, and 5.

Plan Use properties and theorems of real numbers to compare the angle measures.

Solve Compare m3 to m1.

By the Exterior Angle Theorem, m1 m3 m4. Since angle measures are positive numbers and from the definition of

inequality, m1 > m3.

Compare m4 to m1.

By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4.

Compare m5 to m1.

Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By

substitution, m1 > m5.

By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2.

Since we know that m1 > m5, by the Transitive Property, m1 > m2.

Compare m2 to m5.

Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure.

Answer: 1 has the greatest measure.

Determine which angle has the greatest measure.

Answer: 5 has the greatest measure.

Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14.

By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + m3. Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7. Answer: Thus, the measures of 4, 11, 9, 3, 2, 6, and 7 are all less than m14 .

Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5.

By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

a. all angles whose measures are less than m4

b. all angles whose measures are greater than m8

Answer: 5, 2, 8, 7

Answer: 4, 9, 5

Determine the relationship between the measures of RSU and SUR.

Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

Determine the relationship between the measures of TSV and STV.

Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

Determine the relationship between the measures of RSV and RUV.

Answer: mRSV > mRUV

mRSU > mSUR

mUSV > mSUV

mRSU + mUSV > mSUR + mSUV

mRSV > mRUV

Determine the relationship between the measures of the given angles.a. ABD, DAB

b. AED, EAD

c. EAB, EDB

Answer: ABD > DAB

Answer: AED > EAD

Answer: EAB < EDB

HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

Theorem 5.10 states that if one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Since X is opposite the longest side it has the greatest measure.

Answer: So, Ebony should tie the ends marked Y and Z.

KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing?

Answer: A and D