Question
Caroline is given this diagram and told that EG¯¯¯¯¯¯¯¯ is an angle bisector of ∠HEF. She looks carefully and finds one pair of congruent angles between two pairs of congruent sides. She determines that the triangles must be congruent by the SAS Congruent Theorem.
Match each pair of congruent corresponding parts with the reason Caroline knows that they are congruent
(1 point)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
EG¯¯¯¯¯¯¯¯ ≅ EG¯¯¯¯¯¯¯¯
Press enter key to drop response here.
∠HEG ≅ ∠FEG
Press enter key to drop response here.
HE¯¯¯¯¯¯¯¯ ≅ FE¯¯¯¯¯¯¯¯
Match each pair of congruent corresponding parts with the reason Caroline knows that they are congruent
(1 point)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
EG¯¯¯¯¯¯¯¯ ≅ EG¯¯¯¯¯¯¯¯
Press enter key to drop response here.
∠HEG ≅ ∠FEG
Press enter key to drop response here.
HE¯¯¯¯¯¯¯¯ ≅ FE¯¯¯¯¯¯¯¯
Answers
GPT-4o mini
To match each pair of congruent corresponding parts with the appropriate reason Caroline knows they are congruent, the following responses fit:
1. **EG̅̅̅̅̅̅ ≅ EG̅̅̅̅̅̅** - This represents that the segment EG is congruent to itself, which is based on the reflexive property of congruence.
2. **∠HEG ≅ ∠FEG** - This is due to the fact that EG̅̅̅̅̅̅ is an angle bisector of ∠HEF, meaning it divides the angle into two equal parts, thus these angles are congruent.
3. **HE̅̅̅̅̅̅ ≅ FE̅̅̅̅̅̅** - This is because of the Side-Side (SS) congruence, derived from the fact that they are given in the problem that the sides HE and FE are congruent.
So the matching would look like this:
- **EG̅̅̅̅̅̅ ≅ EG̅̅̅̅̅̅** - Reflexive Property
- **∠HEG ≅ ∠FEG** - Angle Bisector
- **HE̅̅̅̅̅̅ ≅ FE̅̅̅̅̅̅** - Given (or sufficient information to conclude)
Remember that the SAS Congruent Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Caroline concludes the triangles are congruent using the SAS congruence theorem with these corresponding parts.
1. **EG̅̅̅̅̅̅ ≅ EG̅̅̅̅̅̅** - This represents that the segment EG is congruent to itself, which is based on the reflexive property of congruence.
2. **∠HEG ≅ ∠FEG** - This is due to the fact that EG̅̅̅̅̅̅ is an angle bisector of ∠HEF, meaning it divides the angle into two equal parts, thus these angles are congruent.
3. **HE̅̅̅̅̅̅ ≅ FE̅̅̅̅̅̅** - This is because of the Side-Side (SS) congruence, derived from the fact that they are given in the problem that the sides HE and FE are congruent.
So the matching would look like this:
- **EG̅̅̅̅̅̅ ≅ EG̅̅̅̅̅̅** - Reflexive Property
- **∠HEG ≅ ∠FEG** - Angle Bisector
- **HE̅̅̅̅̅̅ ≅ FE̅̅̅̅̅̅** - Given (or sufficient information to conclude)
Remember that the SAS Congruent Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Caroline concludes the triangles are congruent using the SAS congruence theorem with these corresponding parts.