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Triangle DEF is similar to triangle ABC. What is the measure, in degrees, of ZF? Explain how you know.

Triangle DEF is similar to triangle ABC. What is the measure, in degrees, of ZF? Explain how you know.

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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### Transcription and Diagram Explanation for Educational Use #### Problem Statement Triangle \( DEF \) is similar to triangle \( ABC \). #### Diagram Description - **Triangle ABC**: - \( \angle A \) is labeled as 40°. - \( AB \) is the side adjacent to \( \angle A \). - \( BC \) is the base of the triangle, forming a right angle with \( AB \). - **Triangle DEF**: - This triangle is not labeled with any angles but is similar to triangle \( ABC \), suggesting that the corresponding angles are equal. #### Question What is the measure, in degrees, of \( \angle F \)? Explain how you know. #### Explanation Since triangle \( DEF \) is similar to triangle \( ABC \), the corresponding angles in similar triangles are equal. This means: - \( \angle F \) in triangle \( DEF \) is equal to \( \angle C \) in triangle \( ABC \). To find \( \angle F \): - The sum of angles in a triangle is always 180°. - Triangle \( ABC \) has a known angle of 40° (\( \angle A \)) and a right angle (\( \angle B \) as 90°), so: \[ \angle C = 180° - 90° - 40° = 50° \] Therefore, \( \angle F = 50° \). ### Conclusion The measure of \( \angle F \) is 50 degrees.
Transcribed Image Text:### Transcription and Diagram Explanation for Educational Use #### Problem Statement Triangle \( DEF \) is similar to triangle \( ABC \). #### Diagram Description - **Triangle ABC**: - \( \angle A \) is labeled as 40°. - \( AB \) is the side adjacent to \( \angle A \). - \( BC \) is the base of the triangle, forming a right angle with \( AB \). - **Triangle DEF**: - This triangle is not labeled with any angles but is similar to triangle \( ABC \), suggesting that the corresponding angles are equal. #### Question What is the measure, in degrees, of \( \angle F \)? Explain how you know. #### Explanation Since triangle \( DEF \) is similar to triangle \( ABC \), the corresponding angles in similar triangles are equal. This means: - \( \angle F \) in triangle \( DEF \) is equal to \( \angle C \) in triangle \( ABC \). To find \( \angle F \): - The sum of angles in a triangle is always 180°. - Triangle \( ABC \) has a known angle of 40° (\( \angle A \)) and a right angle (\( \angle B \) as 90°), so: \[ \angle C = 180° - 90° - 40° = 50° \] Therefore, \( \angle F = 50° \). ### Conclusion The measure of \( \angle F \) is 50 degrees.
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