Math  /  Geometry

QuestionDetermine if triangle EFGE F G and triangle HIJH I J are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)
Answer Attempt 1 out of 2 \square SSS: three sides proportipnate SSS: three sides congruent SAS: two sides proportionate, included angle congruent SAS: two sides + included angle congruent AA: two angles congruent

Studdy Solution

STEP 1

1. We are given two triangles, EFG \triangle EFG and HIJ \triangle HIJ .
2. The side lengths of EFG \triangle EFG are EF=14 EF = 14 , FG=18 FG = 18 , and EG=14 EG = 14 .
3. The side lengths of HIJ \triangle HIJ are HI=21 HI = 21 , IJ=27 IJ = 27 , and HJ=21 HJ = 21 .
4. The angle F \angle F in EFG \triangle EFG is 49 49^\circ .
5. The angle I \angle I in HIJ \triangle HIJ is 49 49^\circ .

STEP 2

1. Check for similarity using the Side-Side-Side (SSS) criterion.
2. Check for similarity using the Side-Angle-Side (SAS) criterion.
3. Conclude if the triangles are similar and state the reason.

STEP 3

Check for similarity using the Side-Side-Side (SSS) criterion:
- Compare the ratios of corresponding sides: \frac{EF}{HI} = \frac{14}{21} = \frac{2}{3} \] \frac{FG}{IJ} = \frac{18}{27} = \frac{2}{3} \] $ \frac{EG}{HJ} = \frac{14}{21} = \frac{2}{3} \]
Since all three ratios are equal, the triangles are similar by the SSS criterion.

STEP 4

Check for similarity using the Side-Angle-Side (SAS) criterion:
- Compare two sides and the included angle: - F=I=49 \angle F = \angle I = 49^\circ - Ratios of the sides: \frac{EF}{HI} = \frac{14}{21} = \frac{2}{3} \] \frac{EG}{HJ} = \frac{14}{21} = \frac{2}{3} \]
Since the included angles are congruent and the sides are proportional, the triangles are similar by the SAS criterion.

STEP 5

Conclude if the triangles are similar and state the reason:
The triangles EFG \triangle EFG and HIJ \triangle HIJ are similar by both the SSS and SAS criteria.
The triangles are similar because their corresponding sides are proportional and their included angles are congruent.

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