Math by First Principles
Level 3 · Challenge ← Back to lesson

AAS, HL, and Invalid Tests: Practice

12 multiple-choice questions, progressively harder.

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

    In the HL overlay, why must C, A, and F be collinear after the third vertices are put on opposite sides of AB?

    Aligned right triangles with equal hypotenusesTriangles ABC and ABF share side AB, with C above AB and F below it. Both have right angle squares at A. BC and BF have matching single ticks. No other side or angle equalities are marked.ABCF
  2. 2

    Which statement would make the isosceles-based HL proof circular?

    Triangles on an aligned baseTriangles ABC and ABF share side AB, with C above AB and F below it. Each triangle has a right angle square at A. BC and BF have matching single ticks; AC and AF are unmarked.ABCF
  3. 3

    Angles A and B in triangle ABC match angles D and E in triangle DEF. A record also says

    AC=EFAC=EF

    Why does this record not by itself give AAS under the stated matching?

    The recorded triangle partsTriangles ABC and DEF have matching single arcs at A and D, and matching double arcs at B and E. The matching single side ticks are on AC in the first triangle and EF in the second. All other sides are unmarked.ABCDEF
  4. 4

    Two right triangles are right at Q and S. The given hypotenuses satisfy

    PR=RT.\begin{aligned} PR&=RT. \end{aligned}

    The given legs satisfy

    PQ=RS.\begin{aligned} PQ&=RS. \end{aligned}

    Which ordered congruence statement follows?

    The supplied hypotenuse and leg pairsTriangles PQR and RST meet at R. Right angle squares mark Q and S. PQ and RS have matching single ticks; PR and RT have matching double ticks. No other side or angle equalities are marked.PQRST
  5. 5

    Triangles ABC and DEF have the following data:

    m∠A=37∘m∠D=37∘m∠B=84∘m∠E=84∘AC=DF\begin{aligned} m\angle A&=37^\circ \\ m\angle D&=37^\circ \\ m\angle B&=84^\circ \\ m\angle E&=84^\circ \\ AC&=DF \end{aligned}

    Which pair of angle vertices can be used with the equal side in an ASA proof?

    The recorded triangle partsTriangles ABC and DEF have matching single arcs at A and D, each labeled 37 degrees, and matching double arcs at B and E, each labeled 84 degrees. AC and DF have matching single ticks.37°84°ABC37°84°DEF
  6. 6

    A construction yields two triangles with

    AC=14CB1=CB2CB1=10\begin{aligned} AC&=14 \\ CB_1&=CB_2 \\ CB_1&=10 \end{aligned}

    and the same angle at A. Their bases AB1 and AB2 are unequal. What is the most direct reason these triangles are noncongruent under the shared-vertex correspondence?

    Two recorded third verticesA ray from A meets a circle centered at C at B1 and B2, both beyond A. AC is labeled 14, and CB1 and CB2 are each labeled 10. The shared angle at A has an arc without a numerical label.AB₁B₂C141010
  7. 7

    A right triangle has legs 12 and 16. Another right triangle has a leg of 12, but its hypotenuse length is not given. Is the one equal leg pair enough to prove congruence?

  8. 8

    In the HL proof, once both small angles at B have been shown equal, which side pairs complete SAS?

    Aligned right triangles with equal hypotenusesTriangles ABC and ABF share side AB, with C above AB and F below it. Both have right angle squares at A. BC and BF have matching single ticks. No other side or angle equalities are marked.ABCF
  9. 9

    Which extra fact is sufficient to turn an AAA comparison into a congruence proof when all three angle pairs have a fixed correspondence?

  10. 10

    For fixed SSA data, a construction circle touches the given ray at exactly one point beyond its endpoint. How many nondegenerate triangles does that intersection supply?

  11. 11

    Could triangles satisfying unrestricted SSA measurements still be congruent?

  12. 12

    Two AAS-congruent triangles have corresponding angles (6u−7)∘(6u-7)^\circ and (3u+20)∘(3u+20)^\circ. What is that common angle?

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