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Math  /  Geometry

QuestionWhich would prove that △ABC−△XYZ\triangle A B C-\triangle X Y Z ? Select two options. BAYX=BCYZ=ACXZ\frac{B A}{Y X}=\frac{B C}{Y Z}=\frac{A C}{X Z} BAYX=BCYZ,∠C≥∠Z\frac{B A}{Y X}=\frac{B C}{Y Z}, \angle C \geq \angle Z ACxz=BAyx,∠A≅∠x\frac{A C}{x z}=\frac{B A}{y x}, \angle A \cong \angle x BAYX=ACYZ=BCXZ\frac{B A}{Y X}=\frac{A C}{Y Z}=\frac{B C}{X Z} BCXY=BAZX,∠C≈∠X\frac{B C}{X Y}=\frac{B A}{Z X}, \angle C \approx \angle X

Studdy Solution

STEP 1

1. We are looking for criteria that can prove the similarity of triangles △ABC\triangle ABC and △XYZ\triangle XYZ.
2. Similar triangles have corresponding angles that are equal and corresponding sides that are proportional.

STEP 2

1. Recall the criteria for triangle similarity.
2. Evaluate each option based on the similarity criteria.
3. Identify the options that satisfy the similarity criteria.

STEP 3

Recall the criteria for triangle similarity: - Angle-Angle (AA) Similarity: Two angles of one triangle are equal to two angles of another triangle. - Side-Angle-Side (SAS) Similarity: Two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal. - Side-Side-Side (SSS) Similarity: All three pairs of corresponding sides are proportional.

STEP 4

Evaluate each option:
Option 1: BAYX=BCYZ=ACXZ\frac{BA}{YX} = \frac{BC}{YZ} = \frac{AC}{XZ} - This satisfies the SSS Similarity criterion since all three pairs of corresponding sides are proportional.
Option 2: BAYX=BCYZ,∠C≥∠Z\frac{BA}{YX} = \frac{BC}{YZ}, \angle C \geq \angle Z - This does not satisfy any similarity criterion because it does not ensure equal angles or proportional sides with an included angle.
Option 3: ACXZ=BAYX,∠A≅∠X\frac{AC}{XZ} = \frac{BA}{YX}, \angle A \cong \angle X - This satisfies the SAS Similarity criterion since two pairs of sides are proportional and the included angle is equal.
Option 4: BAYX=ACYZ=BCXZ\frac{BA}{YX} = \frac{AC}{YZ} = \frac{BC}{XZ} - This satisfies the SSS Similarity criterion since all three pairs of corresponding sides are proportional.
Option 5: BCXY=BAZX,∠C≈∠X\frac{BC}{XY} = \frac{BA}{ZX}, \angle C \approx \angle X - This satisfies the SAS Similarity criterion since two pairs of sides are proportional and the included angle is equal.

STEP 5

Identify the options that satisfy the similarity criteria:
- Option 1: SSS Similarity - Option 3: SAS Similarity
The options that prove △ABC∼△XYZ\triangle ABC \sim \triangle XYZ are:
1. BAYX=BCYZ=ACXZ\frac{BA}{YX} = \frac{BC}{YZ} = \frac{AC}{XZ}
2. ACXZ=BAYX,∠A≅∠X\frac{AC}{XZ} = \frac{BA}{YX}, \angle A \cong \angle X

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