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B. E A 83°

B. E A 83°

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Understanding Circle Geometry** In the given diagram, we see a circle with various intersecting lines and marked angles. The points and angles labeled on the diagram are as follows: - Points A, B, C, D, E, G, H, and J are marked on the circle. - There is a line passing through points B, C, and E such that it intersects the circle. - There is a line passing through points H and G and another through points D and J. - Angle ∠BAJ is marked as 85°. - Angle ∠CDE is marked as 83°. - Angle ∠GHE is marked as 48°. - Angle ∠EGH is marked as 25°. ### Explanation of the Diagram: 1. **Points and Lines:** - Points A, B, H, and G are significant for understanding the geometry inside the circle. - Lines intersecting at these points help to form various angles within the circle. 2. **Angles:** - ∠BAJ (the angle between the line segments BA and AJ) is 85°. - ∠CDE (the angle formed within the circle) is 83°. - ∠GHE (formed between points G, H, and E) is 48°. - ∠EGH (formed between points E, G, and H) is 25°. ### Task: **Find the measure of an angle or arc that is not already labeled.** To solve for an unknown angle or arc measurement in this geometric setup, you can apply various circle theorem properties, such as the Inscribed Angle Theorem or the Alternate Segment Theorem, alongside known geometric rules (the sum of angles in a triangle, etc.). For example, if you want to find the measurement of the unknown angle ∠HGJ, you can use the interior and exterior angle properties along with the given angles. ### Key Principles to Recall: - **Inscribed Angle Theorem**: An inscribed angle is equal to half the measure of the intercepted arc. - **Exterior Angle Theorem**: The exterior angle is equal to the sum of the two interior opposite angles. - **Sum of Angles in a Triangle**: The sum of all internal angles in a triangle is always 180°. With these principles, you can establish relationships between the known and unknown angles and solve for the unknowns in the
Transcribed Image Text:**Understanding Circle Geometry** In the given diagram, we see a circle with various intersecting lines and marked angles. The points and angles labeled on the diagram are as follows: - Points A, B, C, D, E, G, H, and J are marked on the circle. - There is a line passing through points B, C, and E such that it intersects the circle. - There is a line passing through points H and G and another through points D and J. - Angle ∠BAJ is marked as 85°. - Angle ∠CDE is marked as 83°. - Angle ∠GHE is marked as 48°. - Angle ∠EGH is marked as 25°. ### Explanation of the Diagram: 1. **Points and Lines:** - Points A, B, H, and G are significant for understanding the geometry inside the circle. - Lines intersecting at these points help to form various angles within the circle. 2. **Angles:** - ∠BAJ (the angle between the line segments BA and AJ) is 85°. - ∠CDE (the angle formed within the circle) is 83°. - ∠GHE (formed between points G, H, and E) is 48°. - ∠EGH (formed between points E, G, and H) is 25°. ### Task: **Find the measure of an angle or arc that is not already labeled.** To solve for an unknown angle or arc measurement in this geometric setup, you can apply various circle theorem properties, such as the Inscribed Angle Theorem or the Alternate Segment Theorem, alongside known geometric rules (the sum of angles in a triangle, etc.). For example, if you want to find the measurement of the unknown angle ∠HGJ, you can use the interior and exterior angle properties along with the given angles. ### Key Principles to Recall: - **Inscribed Angle Theorem**: An inscribed angle is equal to half the measure of the intercepted arc. - **Exterior Angle Theorem**: The exterior angle is equal to the sum of the two interior opposite angles. - **Sum of Angles in a Triangle**: The sum of all internal angles in a triangle is always 180°. With these principles, you can establish relationships between the known and unknown angles and solve for the unknowns in the
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