Question
For triangles Y Z X and C D B, sides Y Z, Z X, C D, and B D are 2.5 centimeters. Sides B C and Y X are 3.8 centimeters. Angles B, C, Y, X are 36 degrees. Angles Z and D are 108 degrees.
Which of the following congruency statements is correct?
ΔYZX ≅ ΔBCD
ΔYZX ≅ ΔDCB
ΔZXY ≅ ΔDCB
ΔYXZ ≅ ΔCDB
Which of the following congruency statements is correct?
ΔYZX ≅ ΔBCD
ΔYZX ≅ ΔDCB
ΔZXY ≅ ΔDCB
ΔYXZ ≅ ΔCDB
Answers
GPT-4o mini
Let's analyze the given triangles YZX and CDB to determine which congruency statement is correct.
Given:
- Sides: YZ = ZX = CD = BD = 2.5 cm, YX = BC = 3.8 cm
- Angles: ∠B = ∠C = ∠Y = ∠X = 36°, ∠Z = ∠D = 108°
To determine if the triangles are congruent, we can use the information provided.
1. **Side-Angle-Side (SAS) Congruence**:
- For triangles, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
- We can check:
- In triangle YZX:
- YZ = 2.5 cm, ZX = 2.5 cm, and ∠Z = 108°.
- In triangle CDB:
- CD = 2.5 cm, BD = 2.5 cm, and ∠D = 108°.
Since the angles and the two sides are equal in both triangles, we could suggest that:
ΔYZX ≅ ΔCDB using SAS because:
- YZ = CD (2.5 cm)
- ZX = BD (2.5 cm)
- ∠Z = ∠D (108°)
Now let's verify the other congruence statements:
- ΔYZX ≅ ΔBCD: This is incorrect because angles B and C are not equal to angles Y and Z respectively.
- ΔYZX ≅ ΔDCB: This is incorrect since angles Y and Z don't match angles D and C.
- ΔZXY ≅ ΔDCB: This can't be true because ΔZXY does not have matching angles and sides with ΔDCB.
Thus, the correct statement about the triangles is:
**ΔYZX ≅ ΔCDB**.
Given:
- Sides: YZ = ZX = CD = BD = 2.5 cm, YX = BC = 3.8 cm
- Angles: ∠B = ∠C = ∠Y = ∠X = 36°, ∠Z = ∠D = 108°
To determine if the triangles are congruent, we can use the information provided.
1. **Side-Angle-Side (SAS) Congruence**:
- For triangles, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
- We can check:
- In triangle YZX:
- YZ = 2.5 cm, ZX = 2.5 cm, and ∠Z = 108°.
- In triangle CDB:
- CD = 2.5 cm, BD = 2.5 cm, and ∠D = 108°.
Since the angles and the two sides are equal in both triangles, we could suggest that:
ΔYZX ≅ ΔCDB using SAS because:
- YZ = CD (2.5 cm)
- ZX = BD (2.5 cm)
- ∠Z = ∠D (108°)
Now let's verify the other congruence statements:
- ΔYZX ≅ ΔBCD: This is incorrect because angles B and C are not equal to angles Y and Z respectively.
- ΔYZX ≅ ΔDCB: This is incorrect since angles Y and Z don't match angles D and C.
- ΔZXY ≅ ΔDCB: This can't be true because ΔZXY does not have matching angles and sides with ΔDCB.
Thus, the correct statement about the triangles is:
**ΔYZX ≅ ΔCDB**.