# NCTM Standards: 2, 3, 6, 8, 9, 10. (Only one is possible) These properties can also be applied to the measures of angles & segments.

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NCTM Standards: 2, 3, 6, 8, 9, 10

(Only one is possible) These properties can also be applied to the measures of angles & segments.

Example 2-1a Determine which angle has the greatest measure. ExploreCompare the measure of 1 to the measures of 2, 3, 4, and 5. PlanUse properties and theorems of real numbers to compare the angle measures.

Example 2-1a Solve Compare m 3 to m 1. By the Exterior Angle Theorem, m 1 m 3 m 4. Since angle measures are positive numbers and from the definition of inequality, m 1 > m 3. Compare m 4 to m 1. By the Exterior Angle Theorem, m 1 m 3 m 4. By the definition of inequality, m 1 > m 4. Compare m 5 to m 1. Since all right angles are congruent, 4 5. By the definition of congruent angles, m 4 m 5. By substitution, m 1 > m 5.

By the Exterior Angle Theorem, m 5 m 2 m 3. By the definition of inequality, m 5 > m 2. Since we know that m 1 > m 5, by the Transitive Property, m 1 > m 2. Example 2-1a Compare m 2 to m 5. ExamineThe results on the previous slides show that m 1 > m 2, m 1 > m 3, m 1 > m 4, and m 1 > m 5. Therefore, 1 has the greatest measure. Answer: 1 has the greatest measure.

Determine which angle has the greatest measure. 1. 1 2. 2 3. 3 4. 4 5. 5

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.

Example 2-2a Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m 14. By the Exterior Angle Inequality Theorem, m 14 > m 4, m 14 > m 11, m 14 > m 2, and m 14 > m 4 + m 3. Since 11 and 9 are vertical angles, they have equal measure, so m 14 > m 9. m 9 > m 6 and m 9 > m 7, so m 14 > m 6 and m 14 > m 7. Answer: Thus, the measures of 4, 11, 9, 3, 2, 6, and 7 are all less than m 14.

Example 2-2b Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m 5. By the Exterior Angle Inequality Theorem, m 10 > m 5, and m 16 > m 10, so m 16 > m 5, m 17 > m 5 + m 6, m 15 > m 12, and m 12 > m 5, so m 15 > m 5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m 5.

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 m 4 b. all angles whose measures are greater than m 8 Example 2-2c Answer: 5, 2, 8, 7 Answer: 4, 9, 5

Theorem 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. If RQ is the longest side, Angle P is the largest angle. If PQ is the shortest side, Angle R is the smallest angle.

Example 2-3a Determine the relationship between the measures of RSU and SUR. Answer: The side opposite RSU is longer than the side opposite SUR, so m RSU > m SUR.

Example 2-3b Determine the relationship between the measures of TSV and STV. Answer: The side opposite TSV is shorter than the side opposite STV, so m TSV < m STV.

Example 2-3c Determine the relationship between the measures of RSV and RUV. Answer: m RSV > m RUV m RSU > m SUR m USV > m SUV m RSU + m USV > m SUR + m SUV m RSV > m RUV

Example 2-3d 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

Theorem 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. If Angle P is the largest angle, RQ is the longest side. If, Angle R is the smallest angle, PQ is the shortest side.

Example 2-4a 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?

Example 2-4a 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.

Example 2-4b 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

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