Homework Help is Here – Start Your Trial Now!
Math
Geometry
In the figure below, sides DG and DE are congruent, and angles GDF and FDE are congruent as marked. Does this mean that triangles GDF and FDE are congruent? Tell what triangle congruence supports your answer. D F

In the figure below, sides DG and DE are congruent, and angles GDF and FDE are congruent as marked. Does this mean that triangles GDF and FDE are congruent? Tell what triangle congruence supports your answer. D F

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
See similar textbooks
icon
Related questions
Question
In the figure below, sides \( \overline{DG} \) and \( \overline{DE} \) are congruent, and angles \( \angle GDF \) and \( \angle FDE \) are congruent as marked. Does this mean that triangles \( \triangle GDF \) and \( \triangle FDE \) are congruent? Tell what triangle congruence supports your answer. **Diagram Explanation:** The diagram shows two triangles, \( \triangle GDF \) and \( \triangle FDE \), that share a common vertex \( D \). The sides \( \overline{DG} \) and \( \overline{DE} \) are marked as congruent with a single tick mark each, indicating they are equal in length. The angles \( \angle GDF \) and \( \angle FDE \) are shown with identical arcs, indicating they are also congruent. **Conclusion:** To determine if the triangles are congruent, the Angle-Side-Angle (ASA) Congruence Theorem can be applied. This theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. In this case: - \( \angle GDF \cong \angle FDE \) (given), - \( \overline{DG} \cong \overline{DE} \) (given), - \( \angle DGF \cong \angle DEF \) (since they are the remaining angles, they must be congruent by the Angle-Sum Property of triangles). Thus, \( \triangle GDF \cong \triangle FDE \) by ASA Congruence.
Transcribed Image Text:In the figure below, sides \( \overline{DG} \) and \( \overline{DE} \) are congruent, and angles \( \angle GDF \) and \( \angle FDE \) are congruent as marked. Does this mean that triangles \( \triangle GDF \) and \( \triangle FDE \) are congruent? Tell what triangle congruence supports your answer. **Diagram Explanation:** The diagram shows two triangles, \( \triangle GDF \) and \( \triangle FDE \), that share a common vertex \( D \). The sides \( \overline{DG} \) and \( \overline{DE} \) are marked as congruent with a single tick mark each, indicating they are equal in length. The angles \( \angle GDF \) and \( \angle FDE \) are shown with identical arcs, indicating they are also congruent. **Conclusion:** To determine if the triangles are congruent, the Angle-Side-Angle (ASA) Congruence Theorem can be applied. This theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. In this case: - \( \angle GDF \cong \angle FDE \) (given), - \( \overline{DG} \cong \overline{DE} \) (given), - \( \angle DGF \cong \angle DEF \) (since they are the remaining angles, they must be congruent by the Angle-Sum Property of triangles). Thus, \( \triangle GDF \cong \triangle FDE \) by ASA Congruence.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer