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Prove △TUX and △TVW Similarity with AA Criterion | Geometry Practice Problem
Geometry
Grade 10 (Junior High School)
Question Content
△TUX and △TVW are shown below. (Diagram shows right angles at U and V, with points T, U, V colinear, and points X, W on the same side of line TV, forming △TUX and △TVW). Which statement is true? Option 1: △TUX is similar to △TVW. Option 2: △TUX is not similar to △TVW. Option 3: There is not enough information to determine whether the triangles are similar.
Correct Answer
△TUX is similar to △TVW.
Detailed Solution Steps
1
Step 1: Identify shared angles. Both △TUX and △TVW share ∠T, so ∠T ≅ ∠T by the reflexive property of congruence.
2
Step 2: Identify right angles. The diagram shows ∠TUX and ∠TVW are right angles (marked with square symbols), so ∠TUX ≅ ∠TVW = 90°.
3
Step 3: Apply the Angle-Angle (AA) Similarity Criterion. Since two pairs of corresponding angles are congruent (∠T ≅ ∠T and ∠TUX ≅ ∠TVW), the AA Similarity Criterion confirms that △TUX ~ △TVW.
Knowledge Points Involved
1
Angle-Angle (AA) Similarity Criterion
If two pairs of corresponding angles of two triangles are congruent, then the two triangles are similar. This is a foundational similarity rule for triangles, used to prove similarity without needing to check side lengths.
2
Reflexive Property of Congruence
An angle (or line segment, shape) is congruent to itself. This is used to identify shared angles in overlapping or adjacent figures, like the common ∠T in this problem.
3
Right Angle Congruence
All right angles are congruent, meaning they all measure exactly 90°. This is a basic angle property that simplifies identifying congruent angle pairs in geometric figures.
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