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Math  /  Geometry

QuestionThe ratios of corresponding sides in the two triangles are equal.
What other information is needed to prove that △FGE\triangle F G E ∼△IJH\sim \triangle \mathrm{IJH} by the SAS similarity theorem? ∠F≅∠J\angle F \cong \angle J ∠I≅∠F\angle I \cong \angle F ∠E≅∠H\angle E \cong \angle H ∠G≅∠I\angle G \cong \angle I

Studdy Solution

STEP 1

1. We have two triangles, △FGE\triangle FGE and △IJH\triangle IJH.
2. The ratios of corresponding sides in the two triangles are equal.
3. We need to use the SAS (Side-Angle-Side) similarity theorem.

STEP 2

1. Recall the SAS similarity theorem.
2. Identify the given side ratios.
3. Determine the necessary angle congruence.
4. Choose the correct angle congruence option.

STEP 3

Recall the SAS similarity theorem:
The SAS similarity theorem states that two triangles are similar if two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent.

STEP 4

Identify the given side ratios:
Given that the side ratios are equal, we have: 1510=3020 \frac{15}{10} = \frac{30}{20}
These are the corresponding sides of △FGE\triangle FGE and △IJH\triangle IJH.

STEP 5

Determine the necessary angle congruence:
For the SAS similarity theorem, we need the angle between the two sides in each triangle to be congruent.

STEP 6

Choose the correct angle congruence option:
The sides 1515 and 3030 in △FGE\triangle FGE correspond to the sides 1010 and 2020 in △IJH\triangle IJH. Therefore, the included angle in △FGE\triangle FGE is ∠G\angle G, and the included angle in △IJH\triangle IJH is ∠I\angle I.
Thus, we need ∠G≅∠I\angle G \cong \angle I.
The necessary angle congruence is:
∠G≅∠I\angle G \cong \angle I

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