a. Answer Numbered angles are congruent because their corresponding arcs are also congruent. Explanation Giveninformation: A regular polygon with 7 sides is given as shown in below figure, Concept used: Use following properties of a circle , (i) By theorem 9.4, congruent chords have congruent arcs. (ii) In a circle, congruent arcs subtend congruent inscribed angles on their remaining arcs. To prove: In given figure, all number angles are congruent. Key steps of proof of case III: Circumscribe a circle along given regular polygon with 7 sides. Thus, congruent sides of regular polygon are congruent chords of this circle. By above (i) concept, their corresponding arcs are also congruent. Now as these congruent arcs make numbered inscribed angles, by above concept (ii), these numbered angles will also be congruent. Conclusion : Thus,It concludes that all numbered angles are congruent as their corresponding arcs of circumscribed circle are also congruent. b. To determine To find: If angles inscribed by arcs of a regular polygon are congruent, irrespective of its total sides in it. Answer In a regular polygon of any number of sides, the inscribed angles by its arcs are always congruent. Explanation Giveninformation: A regular polygon with 7 sides is given as shown in below figure, Concept used: Use following properties of a circle, (i) By theorem 9.4, congruent chords have congruent arcs. (ii) In a circle, congruent arcs subtend congruent inscribed angles on their remaining arcs. To prove: If result of part (a) is applicable to a regular polygon with any number of its sides. Key steps of proof of case III: Circumscribe a circle along a regular polygon with any number of sides. Thus, congruent sides of regular polygon are congruent chords of this circle. By above (i) concept, their corresponding arcs are also congruent. Now as these congruent arcs make numbered inscribed angles, by above concept (ii), these numbered angles will also be congruent. Conclusion : Thus, It concludes that all numbered angles are congruent as their corresponding arcs of circumscribed circle are also congruent. And it does not depend on number of sides of the regular polygon.

McDougal Littell Jurgensen Geometry: Student Edition Geometry

5th Edition

ISBN: 9780395977279

Author: Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Publisher: Houghton Mifflin Company College Division

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Question

Chapter 9.5, Problem 24WE

To determine

To find:The reason thatthe numbered angles are all congruent in a given regular polygon.

Expert Solution

Answer to Problem 24WE

Numbered angles are congruent because their corresponding arcs are also congruent.

Explanation of Solution

Giveninformation: A regular polygon with 7 sides is given as shown in below figure,

McDougal Littell Jurgensen Geometry: Student Edition Geometry, Chapter 9.5, Problem 24WE , additional homework tip 1

Concept used:Use following properties of a circle,

(i) By theorem 9.4, congruent chords have congruent arcs.

(ii) In a circle, congruent arcs subtend congruent inscribed angles on their remaining arcs.

To prove: In given figure, all number angles are congruent.

Key steps of proof of case III:

Circumscribe a circle along given regular polygon with 7 sides.
Thus, congruent sides of regular polygon are congruent chords of this circle.
By above (i) concept, their corresponding arcs are also congruent.
Now as these congruent arcs make numbered inscribed angles, by above concept (ii), these numbered angles will also be congruent.

Conclusion: Thus,It concludes that all numbered angles are congruent as their corresponding arcs of circumscribed circle are also congruent.

To determine

To find:If angles inscribed by arcs of a regular polygon are congruent, irrespective of its total sides in it.

Expert Solution

Answer to Problem 24WE

In a regular polygon of any number of sides, the inscribed angles by its arcs are always congruent.

Explanation of Solution

Giveninformation: A regular polygon with 7 sides is given as shown in below figure,

McDougal Littell Jurgensen Geometry: Student Edition Geometry, Chapter 9.5, Problem 24WE , additional homework tip 2

Concept used:Use following properties of a circle,

(i) By theorem 9.4, congruent chords have congruent arcs.

(ii) In a circle, congruent arcs subtend congruent inscribed angles on their remaining arcs.

To prove: If result of part (a) is applicable to a regular polygon with any number of its sides.

Key steps of proof of case III:

Circumscribe a circle along a regular polygon with any number of sides.
Thus, congruent sides of regular polygon are congruent chords of this circle.
By above (i) concept, their corresponding arcs are also congruent.
Now as these congruent arcs make numbered inscribed angles, by above concept (ii), these numbered angles will also be congruent.

Conclusion: Thus, It concludes that all numbered angles are congruent as their corresponding arcs of circumscribed circle are also congruent. And it does not depend on number of sides of the regular polygon.

Chapter 9.5, Problem 23WE

Chapter 9.5, Problem 25WE

Chapter 9 Solutions

McDougal Littell Jurgensen Geometry: Student Edition Geometry

Ch. 9.1 - Prob. 1CE Ch. 9.1 - Prob. 2CE Ch. 9.1 - Prob. 3CE Ch. 9.1 - Prob. 4CE Ch. 9.1 - Prob. 5CE Ch. 9.1 - Prob. 6CE Ch. 9.1 - Prob. 7CE Ch. 9.1 - Prob. 8CE Ch. 9.1 - Prob. 9CE Ch. 9.1 - Prob. 10CE

Ch. 9.1 - Prob. 11CE Ch. 9.1 - Prob. 1WE Ch. 9.1 - Prob. 2WE Ch. 9.1 - Prob. 3WE Ch. 9.1 - Prob. 4WE Ch. 9.1 - Prob. 5WE Ch. 9.1 - Prob. 6WE Ch. 9.1 - Prob. 7WE Ch. 9.1 - Prob. 8WE Ch. 9.1 - Prob. 9WE Ch. 9.1 - Prob. 10WE Ch. 9.1 - Prob. 11WE Ch. 9.1 - Prob. 12WE Ch. 9.1 - Prob. 13WE Ch. 9.1 - Prob. 14WE Ch. 9.1 - Prob. 15WE Ch. 9.1 - Prob. 16WE Ch. 9.1 - Prob. 17WE Ch. 9.1 - Prob. 18WE Ch. 9.1 - Prob. 19WE Ch. 9.1 - Prob. 20WE Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.2 - Prob. 1CE Ch. 9.2 - Prob. 2CE Ch. 9.2 - Prob. 3CE Ch. 9.2 - Prob. 4CE Ch. 9.2 - Prob. 5CE Ch. 9.2 - Prob. 1WE Ch. 9.2 - Prob. 2WE Ch. 9.2 - Prob. 3WE Ch. 9.2 - Prob. 4WE Ch. 9.2 - Prob. 5WE Ch. 9.2 - Prob. 6WE Ch. 9.2 - Prob. 7WE Ch. 9.2 - Prob. 8WE Ch. 9.2 - Prob. 9WE Ch. 9.2 - Prob. 10WE Ch. 9.2 - Prob. 11WE Ch. 9.2 - Prob. 12WE Ch. 9.2 - Prob. 13WE Ch. 9.2 - Prob. 14WE Ch. 9.2 - Prob. 15WE Ch. 9.2 - Prob. 16WE Ch. 9.2 - Prob. 17WE Ch. 9.2 - Prob. 18WE Ch. 9.2 - Prob. 19WE Ch. 9.2 - Prob. 20WE Ch. 9.2 - Prob. 21WE Ch. 9.2 - Prob. 22WE Ch. 9.2 - Prob. 23WE Ch. 9.2 - Prob. 1MRE Ch. 9.2 - Prob. 2MRE Ch. 9.2 - Prob. 3MRE Ch. 9.3 - Prob. 1CE Ch. 9.3 - Prob. 2CE Ch. 9.3 - Prob. 3CE Ch. 9.3 - Prob. 4CE Ch. 9.3 - Prob. 5CE Ch. 9.3 - Prob. 6CE Ch. 9.3 - Prob. 7CE Ch. 9.3 - Prob. 8CE Ch. 9.3 - Prob. 9CE Ch. 9.3 - Prob. 10CE Ch. 9.3 - Prob. 11CE Ch. 9.3 - Prob. 12CE Ch. 9.3 - Prob. 13CE Ch. 9.3 - Prob. 1WE Ch. 9.3 - Prob. 2WE Ch. 9.3 - Prob. 3WE Ch. 9.3 - Prob. 4WE Ch. 9.3 - Prob. 5WE Ch. 9.3 - Prob. 6WE Ch. 9.3 - Prob. 7WE Ch. 9.3 - Prob. 8WE Ch. 9.3 - Prob. 9WE Ch. 9.3 - Prob. 10WE Ch. 9.3 - Prob. 11WE Ch. 9.3 - Prob. 12WE Ch. 9.3 - Prob. 13WE Ch. 9.3 - Prob. 14WE Ch. 9.3 - Prob. 15WE Ch. 9.3 - Prob. 16WE Ch. 9.3 - Prob. 17WE Ch. 9.3 - Prob. 18WE Ch. 9.3 - Prob. 19WE Ch. 9.3 - Prob. 20WE Ch. 9.3 - Prob. 21WE Ch. 9.3 - Prob. 22WE Ch. 9.3 - Prob. 23WE Ch. 9.3 - Prob. 24WE Ch. 9.4 - Prob. 1CE Ch. 9.4 - Prob. 2CE Ch. 9.4 - Prob. 3CE Ch. 9.4 - Prob. 4CE Ch. 9.4 - Prob. 5CE Ch. 9.4 - Prob. 6CE Ch. 9.4 - Prob. 7CE Ch. 9.4 - Prob. 1WE Ch. 9.4 - Prob. 2WE Ch. 9.4 - Prob. 3WE Ch. 9.4 - Prob. 4WE Ch. 9.4 - Prob. 5WE Ch. 9.4 - Prob. 6WE Ch. 9.4 - Prob. 7WE Ch. 9.4 - Prob. 8WE Ch. 9.4 - Prob. 9WE Ch. 9.4 - Prob. 10WE Ch. 9.4 - Prob. 11WE Ch. 9.4 - Prob. 12WE Ch. 9.4 - Prob. 13WE Ch. 9.4 - Prob. 14WE Ch. 9.4 - Prob. 15WE Ch. 9.4 - Prob. 16WE Ch. 9.4 - Prob. 17WE Ch. 9.4 - Prob. 18WE Ch. 9.4 - Prob. 19WE Ch. 9.4 - Prob. 20WE Ch. 9.4 - Prob. 21WE Ch. 9.4 - Prob. 22WE Ch. 9.4 - Prob. 23WE Ch. 9.4 - Prob. 24WE Ch. 9.4 - Prob. 25WE Ch. 9.4 - Prob. 26WE Ch. 9.4 - Prob. 27WE Ch. 9.4 - Prob. 1ST1 Ch. 9.4 - Prob. 2ST1 Ch. 9.4 - Prob. 3ST1 Ch. 9.4 - Prob. 4ST1 Ch. 9.4 - Prob. 5ST1 Ch. 9.4 - Prob. 6ST1 Ch. 9.5 - Prob. 1CE Ch. 9.5 - Prob. 2CE Ch. 9.5 - Prob. 3CE Ch. 9.5 - Prob. 4CE Ch. 9.5 - Prob. 5CE Ch. 9.5 - Prob. 6CE Ch. 9.5 - Prob. 7CE Ch. 9.5 - Prob. 8CE Ch. 9.5 - Prob. 9CE Ch. 9.5 - Prob. 10CE Ch. 9.5 - Prob. 11CE Ch. 9.5 - Prob. 12CE Ch. 9.5 - Prob. 13CE Ch. 9.5 - Prob. 14CE Ch. 9.5 - Prob. 1WE Ch. 9.5 - Prob. 2WE Ch. 9.5 - Prob. 3WE Ch. 9.5 - Prob. 4WE Ch. 9.5 - Prob. 5WE Ch. 9.5 - Prob. 6WE Ch. 9.5 - Prob. 7WE Ch. 9.5 - Prob. 8WE Ch. 9.5 - Prob. 9WE Ch. 9.5 - Prob. 10WE Ch. 9.5 - Prob. 11WE Ch. 9.5 - Prob. 12WE Ch. 9.5 - Prob. 13WE Ch. 9.5 - Prob. 14WE Ch. 9.5 - Prob. 15WE Ch. 9.5 - Prob. 16WE Ch. 9.5 - Prob. 17WE Ch. 9.5 - Prob. 18WE Ch. 9.5 - Prob. 19WE Ch. 9.5 - Prob. 20WE Ch. 9.5 - Prob. 21WE Ch. 9.5 - Prob. 22WE Ch. 9.5 - Prob. 23WE Ch. 9.5 - Prob. 24WE Ch. 9.5 - Prob. 25WE Ch. 9.5 - Prob. 26WE Ch. 9.5 - Prob. 27WE Ch. 9.5 - Prob. 28WE Ch. 9.5 - Prob. 29WE Ch. 9.5 - Prob. 1MRE Ch. 9.5 - Prob. 2MRE Ch. 9.5 - Prob. 3MRE Ch. 9.5 - Prob. 4MRE Ch. 9.5 - Prob. 5MRE Ch. 9.5 - Prob. 6MRE Ch. 9.6 - Prob. 1CE Ch. 9.6 - Prob. 2CE Ch. 9.6 - Prob. 3CE Ch. 9.6 - Prob. 4CE Ch. 9.6 - Prob. 5CE Ch. 9.6 - Prob. 6CE Ch. 9.6 - Prob. 7CE Ch. 9.6 - Prob. 8CE Ch. 9.6 - Prob. 9CE Ch. 9.6 - Prob. 10CE Ch. 9.6 - Prob. 1WE Ch. 9.6 - Prob. 2WE Ch. 9.6 - Prob. 3WE Ch. 9.6 - Prob. 4WE Ch. 9.6 - Prob. 5WE Ch. 9.6 - Prob. 6WE Ch. 9.6 - Prob. 7WE Ch. 9.6 - Prob. 8WE Ch. 9.6 - Prob. 9WE Ch. 9.6 - Prob. 10WE Ch. 9.6 - Prob. 11WE Ch. 9.6 - Prob. 12WE Ch. 9.6 - Prob. 13WE Ch. 9.6 - Prob. 14WE Ch. 9.6 - Prob. 15WE Ch. 9.6 - Prob. 16WE Ch. 9.6 - Prob. 17WE Ch. 9.6 - Prob. 18WE Ch. 9.6 - Prob. 19WE Ch. 9.6 - Prob. 20WE Ch. 9.6 - Prob. 21WE Ch. 9.6 - Prob. 22WE Ch. 9.6 - Prob. 23WE Ch. 9.6 - Prob. 24WE Ch. 9.6 - Prob. 25WE Ch. 9.6 - Prob. 26WE Ch. 9.6 - Prob. 27WE Ch. 9.6 - Prob. 28WE Ch. 9.6 - Prob. 29WE Ch. 9.6 - Prob. 30WE Ch. 9.6 - Prob. 31WE Ch. 9.6 - Prob. 32WE Ch. 9.6 - Prob. 1EX Ch. 9.6 - Prob. 2EX Ch. 9.6 - Prob. 3EX Ch. 9.7 - Prob. 1CE Ch. 9.7 - Prob. 2CE Ch. 9.7 - Prob. 3CE Ch. 9.7 - Prob. 4CE Ch. 9.7 - Prob. 5CE Ch. 9.7 - Prob. 6CE Ch. 9.7 - Prob. 7CE Ch. 9.7 - Prob. 1WE Ch. 9.7 - Prob. 2WE Ch. 9.7 - Prob. 3WE Ch. 9.7 - Prob. 4WE Ch. 9.7 - Prob. 5WE Ch. 9.7 - Prob. 6WE Ch. 9.7 - Prob. 7WE Ch. 9.7 - Prob. 8WE Ch. 9.7 - Prob. 9WE Ch. 9.7 - Prob. 10WE Ch. 9.7 - Prob. 11WE Ch. 9.7 - Prob. 12WE Ch. 9.7 - Prob. 13WE Ch. 9.7 - Prob. 14WE Ch. 9.7 - Prob. 15WE Ch. 9.7 - Prob. 16WE Ch. 9.7 - Prob. 17WE Ch. 9.7 - Prob. 18WE Ch. 9.7 - Prob. 19WE Ch. 9.7 - Prob. 20WE Ch. 9.7 - Prob. 21WE Ch. 9.7 - Prob. 22WE Ch. 9.7 - Prob. 23WE Ch. 9.7 - Prob. 24WE Ch. 9.7 - Prob. 25WE Ch. 9.7 - Prob. 26WE Ch. 9.7 - Prob. 27WE Ch. 9.7 - Prob. 28WE Ch. 9.7 - Prob. 1ST2 Ch. 9.7 - Prob. 2ST2 Ch. 9.7 - Prob. 3ST2 Ch. 9.7 - Prob. 4ST2 Ch. 9.7 - Prob. 5ST2 Ch. 9.7 - Prob. 6ST2 Ch. 9.7 - Prob. 7ST2 Ch. 9.7 - Prob. 8ST2 Ch. 9.7 - Prob. 1E Ch. 9.7 - Prob. 2E Ch. 9.7 - Prob. 3E Ch. 9.7 - Prob. 4E Ch. 9.7 - Prob. 5E Ch. 9 - Prob. 1CR Ch. 9 - Prob. 2CR Ch. 9 - Prob. 3CR Ch. 9 - Prob. 4CR Ch. 9 - Prob. 5CR Ch. 9 - Prob. 6CR Ch. 9 - Prob. 7CR Ch. 9 - Prob. 8CR Ch. 9 - Prob. 9CR Ch. 9 - Prob. 10CR Ch. 9 - Prob. 11CR Ch. 9 - Prob. 12CR Ch. 9 - Prob. 13CR Ch. 9 - Prob. 14CR Ch. 9 - Prob. 15CR Ch. 9 - Prob. 16CR Ch. 9 - Prob. 17CR Ch. 9 - Prob. 18CR Ch. 9 - Prob. 19CR Ch. 9 - Prob. 20CR Ch. 9 - Prob. 21CR Ch. 9 - Prob. 22CR Ch. 9 - Prob. 23CR Ch. 9 - Prob. 24CR Ch. 9 - Prob. 1CT Ch. 9 - Prob. 2CT Ch. 9 - Prob. 3CT Ch. 9 - Prob. 4CT Ch. 9 - Prob. 5CT Ch. 9 - Prob. 6CT Ch. 9 - Prob. 7CT Ch. 9 - Prob. 8CT Ch. 9 - Prob. 9CT Ch. 9 - Prob. 10CT Ch. 9 - Prob. 11CT Ch. 9 - Prob. 12CT Ch. 9 - Prob. 13CT Ch. 9 - Prob. 14CT Ch. 9 - Prob. 15CT Ch. 9 - Prob. 16CT Ch. 9 - Prob. 17CT Ch. 9 - Prob. 18CT Ch. 9 - Prob. 1CUR Ch. 9 - Prob. 2CUR Ch. 9 - Prob. 3CUR Ch. 9 - Prob. 4CUR Ch. 9 - Prob. 5CUR Ch. 9 - Prob. 6CUR Ch. 9 - Prob. 7CUR Ch. 9 - Prob. 8CUR Ch. 9 - Prob. 9CUR Ch. 9 - Prob. 10CUR Ch. 9 - Prob. 11CUR Ch. 9 - Prob. 12CUR Ch. 9 - Prob. 13CUR Ch. 9 - Prob. 14CUR Ch. 9 - Prob. 15CUR Ch. 9 - Prob. 16CUR Ch. 9 - Prob. 17CUR Ch. 9 - Prob. 18CUR Ch. 9 - Prob. 19CUR Ch. 9 - Prob. 20CUR Ch. 9 - Prob. 21CUR Ch. 9 - Prob. 22CUR Ch. 9 - Prob. 23CUR Ch. 9 - Prob. 24CUR Ch. 9 - Prob. 25CUR Ch. 9 - Prob. 26CUR Ch. 9 - Prob. 27CUR Ch. 9 - Prob. 28CUR Ch. 9 - Prob. 29CUR Ch. 9 - Prob. 30CUR

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The reason thatthe numbered angles are all congruent in a given regular polygon .

Videos

To find:The reason thatthe numbered angles are all congruent in a given regular polygon.

Answer to Problem 24WE

Explanation of Solution

To find:If angles inscribed by arcs of a regular polygon are congruent, irrespective of its total sides in it.

Answer to Problem 24WE

Explanation of Solution

Chapter 9 Solutions

Additional Math Textbook Solutions