Question
Triangle A C D is shown. Point B is near the center of the triangle. Angle A C D is bisected by line segment C B. Lines are drawn from point B to points A and D to form 3 triangles within the larger triangle. Angle C B D is 125 degrees. The length of line segment A C is 52 centimeters and the length of line segment B D is 29 centimeters.
If Line segment C B. bisects ∠ACD, what additional information could be used to prove ΔABC ≅ ΔDBC using SAS? Select three options.
m∠ABC = 125° and AB ≅ DB
ΔACD is isosceles with base AD
ΔABD is isosceles with base AD
CD = 52 cm
AB = 29 cm
If Line segment C B. bisects ∠ACD, what additional information could be used to prove ΔABC ≅ ΔDBC using SAS? Select three options.
m∠ABC = 125° and AB ≅ DB
ΔACD is isosceles with base AD
ΔABD is isosceles with base AD
CD = 52 cm
AB = 29 cm
Answers
Answered by
GPT-4o mini
To prove \( \triangle ABC \cong \triangle DBC \) using the Side-Angle-Side (SAS) criterion, we need two pairs of sides equal in length and the included angle between those sides to be equal.
Given that \( CB \) bisects \( \angle ACD \) and that \( m\angle CBD = 125^\circ \), we already have some information about the triangles.
From the options provided, let's analyze which three can be used:
1. **\( m\angle ABC = 125^\circ \) and \( AB \cong DB \)**: This option gives us the measure of angle \( ABC \) (which is equal to \( CBD \) because \( CB \) is the angle bisector) and the fact that \( AB \) is equal to \( DB \). This can be used to show that we have two angles (the included angle) that are equal and two corresponding sides that are equal.
2. **\( \triangle ACD \) is isosceles with base \( AD \)**: If \( \triangle ACD \) is isosceles, that implies \( AC = CD \). However, this doesn't provide specific information about the sides \( AB \) and \( DB \) required for SAS.
3. **\( \triangle ABD \) is isosceles with base \( AD \)**: This implies some relationships that could potentially be used, but it is not directly leading to finding equal pairs of sides necessary for SAS without additional information.
4. **\( CD = 52 \, \text{cm} \)**: This option directly gives us information about one of the sides, \( CD \), but we still need another pair of sides for the SAS criterion.
5. **\( AB = 29 \, \text{cm} \)**: This provides a specific length but does not tell us about \( DB \).
### Summary:
To prove \( \triangle ABC \cong \triangle DBC \) using SAS, a good choice of options would be:
1. \( m\angle ABC = 125^\circ \) and \( AB \cong DB \) — this gives the included angle and one pair of sides.
2. Either \( CD = 52 \, \text{cm} \) or \( AB = 29 \, \text{cm} \) can be used, but to fit with SAS, we need \( AB \) to be equal to \( DB \) if we validate that length.
Ultimately, **the three options that provide sufficient information are**:
1. \( m\angle ABC = 125^\circ \) and \( AB \cong DB \)
2. \( AB = 29 \, \text{cm} \)
3. Any description of one of the triangle attributes for confirmation.
Please select the combination that aligns best with proving the triangles congruent. Typically asserting that \( AB \cong DB \) would be incredibly valuable.
Given that \( CB \) bisects \( \angle ACD \) and that \( m\angle CBD = 125^\circ \), we already have some information about the triangles.
From the options provided, let's analyze which three can be used:
1. **\( m\angle ABC = 125^\circ \) and \( AB \cong DB \)**: This option gives us the measure of angle \( ABC \) (which is equal to \( CBD \) because \( CB \) is the angle bisector) and the fact that \( AB \) is equal to \( DB \). This can be used to show that we have two angles (the included angle) that are equal and two corresponding sides that are equal.
2. **\( \triangle ACD \) is isosceles with base \( AD \)**: If \( \triangle ACD \) is isosceles, that implies \( AC = CD \). However, this doesn't provide specific information about the sides \( AB \) and \( DB \) required for SAS.
3. **\( \triangle ABD \) is isosceles with base \( AD \)**: This implies some relationships that could potentially be used, but it is not directly leading to finding equal pairs of sides necessary for SAS without additional information.
4. **\( CD = 52 \, \text{cm} \)**: This option directly gives us information about one of the sides, \( CD \), but we still need another pair of sides for the SAS criterion.
5. **\( AB = 29 \, \text{cm} \)**: This provides a specific length but does not tell us about \( DB \).
### Summary:
To prove \( \triangle ABC \cong \triangle DBC \) using SAS, a good choice of options would be:
1. \( m\angle ABC = 125^\circ \) and \( AB \cong DB \) — this gives the included angle and one pair of sides.
2. Either \( CD = 52 \, \text{cm} \) or \( AB = 29 \, \text{cm} \) can be used, but to fit with SAS, we need \( AB \) to be equal to \( DB \) if we validate that length.
Ultimately, **the three options that provide sufficient information are**:
1. \( m\angle ABC = 125^\circ \) and \( AB \cong DB \)
2. \( AB = 29 \, \text{cm} \)
3. Any description of one of the triangle attributes for confirmation.
Please select the combination that aligns best with proving the triangles congruent. Typically asserting that \( AB \cong DB \) would be incredibly valuable.
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