Determine whether the triangles are similar. If so, what is the similarity statement and the postulate or theorem used.
a. ΔDGH ~ ΔDFE, SSS ~
b. ΔDGH ~ ΔDFE, SAS ~
c. ΔDGH ~ ΔFEG, SAS ~
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ΔABC ~
ΔDFE, AA ~ (angles are corresponding)
To determine similarity between triangles, we need to check if their corresponding angles are congruent and their corresponding sides are proportional.
a. ΔDGH ~ ΔDFE, SSS ~ (Side-Side-Side similarity theorem)
- In this case, we need to check if the ratios of corresponding sides are equal. We can compare the side lengths:
- DG/DF = 3/2, GH/FE = 5/4, DH/DE = 4/3
- The ratios are all equal, so the triangles are similar.
- The similarity statement is: ΔDGH ~ ΔDFE
b. ΔDGH ~ ΔDFE, SAS ~ (Side-Angle-Side similarity theorem)
- In this case, we need to check if two pairs of corresponding sides are proportional and the included angles are congruent. We can compare the ratios:
- DG/DF = 3/2, GH/FE = 5/4
- The ratios of the corresponding sides are equal, but we don't have information about the included angles, so we can't determine similarity based on SAS.
- The triangles are not similar.
c. ΔDGH ~ ΔFEG, SAS ~ (Side-Angle-Side similarity theorem)
- In this case, we need to check if two pairs of corresponding sides are proportional and the included angles are congruent. We can compare the ratios:
- DG/FE = 3/4, GH/EG = 5/5
- The ratios of the corresponding sides are equal, and we also have angle G congruent to angle G, so the triangles are similar.
- The similarity statement is: ΔDGH ~ ΔFEG