Math by First Principles
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Triangle Congruence and SSS: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered
Question 1 of 12
  1. 1

    Two scalene triangles have

    AB=5BC=9CA=11PQ=11QR=5RP=9.\begin{aligned} AB&=5 \\ BC&=9 \\ CA&=11 \\ PQ&=11 \\ QR&=5 \\ RP&=9. \end{aligned}

    Which ordered statement follows from these side pairs?

    Use the six given side labelsThe first triangle has AB 5, BC 9, and CA 11. The second has PQ 11, QR 5, and RP 9. Only these given lengths and vertex names are shown; no ordered congruence statement is printed.ABC5911QRP5911Schematic; use the stated givens.
  2. 2

    Triangles ABCABC and DEFDEF have corresponding lengths

    AB=x+4DE=2yBC=x+yEF=17.\begin{aligned} AB&=x+4 \\ DE&=2y \\ BC&=x+y \\ EF&=17. \end{aligned}

    The third side pair is already equal. Which values complete SSS?

    Keep both variable pairs in the same orderTriangles ABC and DEF have side labels AB x plus 4, DE 2y, BC x plus y, and EF 17. Their third sides AC and DF have matching ticks, as given. No values of x or y are shown.ABCx + 4x + yDEF2y17Schematic; use the stated givens.
  3. 3

    After equal bases are aligned, a third vertex must lie on a radius-12 circle about the first endpoint and a radius-17 circle about the second. Why is the intersection condition useful?

    Meet both prescribed circle conditionsPortions of a radius-12 circle centered at A and a radius-17 circle centered at B intersect at C. The radii are listed beside their circle names. Triangle sides have no derived numerical labels.ABCA-circle radius: 12B-circle radius: 17The base length is not specified.
  4. 4

    Which argument completes the reflected case of SSS without changing the matched base?

    Keep the common base fixedTwo candidate triangles share base AB with third vertices on opposite sides of its line. Only the base and candidate positions are drawn; no motion instruction is printed.ABCC′Schematic; use the stated givens.
  5. 5

    Triangles KLMKLM and RSTRST are claimed congruent in that order. The given side data are:

    KL=RSKL=16LM=STLM=18KM=20\begin{aligned} KL&=RS \\ KL&=16 \\ LM&=ST \\ LM&=18 \\ KM&=20 \end{aligned}

    Which value of RTRT makes SSS apply?

    Complete the unmarked corresponding sideTriangles KLM and RST have given labels KL 16, LM 18, KM 20, RS 16, and ST 18. RT has no numerical label or matching tick.KLM161820RST1618Schematic; use the stated givens.
  6. 6

    A construction uses fixed base UVUV and two circles centered at UU and VV with fixed positive radii. The circles meet at WW above the base line and W′W' below. Which reason shows that these are two placements of one triangle shape up to congruence?

    Compare both candidate third verticesTwo circles centered at U and V meet at W above the base and W-prime below. Both candidate triangles are outlined with their shared base. No motion name or congruence verdict is shown.UVWW′Schematic; use the stated givens.
  7. 7

    The given ordered match is △MNP≅△XYZ\triangle MNP\cong\triangle XYZ.

    The side lengths in centimeters are:

    MN=14NP=19MP=22\begin{aligned} MN&=14 \\ NP&=19 \\ MP&=22 \end{aligned}

    What is the perimeter of △YZX\triangle YZX?

    Read lengths in a reordered triangle nameCongruent triangles MNP and XYZ are shown in different orientations. Only MN 14, NP 19, and MP 22 centimeters are labeled; no perimeter is computed.MNP14 cm19 cm22 cmXYZSchematic; use the stated givens.
  8. 8

    A triangle and a four-sided frame each have all their side lengths fixed. Which conclusion is justified?

  9. 9

    Two actual triangles have matching side lengths. Which extra measurement is required before SSS can establish congruence?

  10. 10

    In the SSS justification, the two equations are

    x2+y2=r2.(x−d)2+y2=s2.\begin{aligned} x^2+y^2&=r^2. \\ (x-d)^2+y^2&=s^2. \end{aligned}

    Subtracting the first from the second gives

    d2−2dx=s2−r2.\begin{aligned} d^2-2dx&=s^2-r^2. \end{aligned}

    Why must the base length dd be nonzero?

    Read the legs in the squared-distance equationsThe third vertex lies above a horizontal base in this pictured case. Its distances to the base endpoints are r and s. A dashed perpendicular to the base has a right-angle marker at the foot. The full base length is d.(x,y)(0,0)(x,0)(d,0)xydrs
  11. 11

    Suppose a side-matching calculation gives

    a=4.\begin{aligned} a&=4. \end{aligned}

    The two matching expressions are

    First expression: 3a+2.Second expression: 5a−6.\begin{aligned} \text{First expression: }&3a+2. \\ \text{Second expression: }&5a-6. \end{aligned}

    What check confirms this side pair?

  12. 12

    Which statement accurately limits the conclusion of SSS?

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