Similarity and Congruence Proofs Quiz

Grade 10 · Math · Quiz

Identify congruence and similarity criteria and apply them in proof-based geometry questions.

Questions

  1. Which set of information is NOT sufficient to prove two triangles congruent?

    1. Two sides and a non-included angle (SSA)
    2. Two sides and the included angle (SAS)
    3. Two angles and the included side (ASA)
    4. All three sides (SSS)
    Show answer

    Answer: Two sides and a non-included angle (SSA)

    SSA is the ambiguous case — it can produce 0, 1, or 2 valid triangles, so it is not a valid congruence criterion.

  2. Triangle PQR ~ Triangle XYZ with PQ = 8 and XY = 12. If QR = 6, what is YZ?

    1. 9
    2. 4
    3. 8
    4. 12
    Show answer

    Answer: 9

    Scale factor = 12/8 = 3/2. YZ = QR × (3/2) = 6 × 3/2 = 9.

  3. In the diagram, DE is parallel to BC in triangle ABC. Which similarity criterion proves triangle ADE ~ triangle ABC?

    1. AA
    2. SAS
    3. SSS
    4. RHS
    Show answer

    Answer: AA

    DE ∥ BC creates equal corresponding angles (angle ADE = angle ABC and angle AED = angle ACB), so AA similarity applies.

  4. All equilateral triangles are similar to each other.

    Show answer

    Answer: True

    All equilateral triangles have three 60° angles; by AA, any two equilateral triangles are similar.

  5. Two similar solids have a surface area ratio of 25:16. What is their volume ratio?

    1. 125:64
    2. 25:16
    3. 5:4
    4. 625:256
    Show answer

    Answer: 125:64

    Surface area ratio k² = 25:16, so k = 5:4. Volume ratio = k³ = 125:64.

  6. In a proof, what property justifies 'angle A = angle A' when the same angle appears in two triangles sharing vertex A?

    1. Reflexive Property
    2. Symmetric Property
    3. Transitive Property
    4. CPCTC
    Show answer

    Answer: Reflexive Property

    The reflexive property states that any geometric figure is congruent (or equal) to itself.

  7. Triangle ABC has AB = 5, BC = 12, AC = 13. Triangle DEF has DE = 10, EF = 24, DF = 26. Which criterion proves similarity?

    1. SSS~
    2. AA~
    3. SAS~
    4. RHS~
    Show answer

    Answer: SSS~

    All sides are in ratio 10:5 = 24:12 = 26:13 = 2:1, so the triangles are similar by SSS~.

  8. If two similar triangles have a scale factor of 4:1, the ratio of their perimeters is ______.

    Show answer

    Answer: 4:1

    The ratio of perimeters equals the scale factor: 4:1. (Area ratio would be 16:1; volume ratio 64:1.)

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