Math by First Principles
Level 2 · Intermediate ← Back to lesson

AAS, HL, and Invalid Tests: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered
Question 1 of 12
  1. 1

    Triangles ABC and DEF have the following data:

    m∠A=49∘m∠D=49∘m∠B=58∘m∠E=58∘BC=EF\begin{aligned} m\angle A&=49^\circ \\ m\angle D&=49^\circ \\ m\angle B&=58^\circ \\ m\angle E&=58^\circ \\ BC&=EF \end{aligned}

    Which criterion directly describes the given data?

    The recorded triangle partsTriangles ABC and DEF have matching single arcs at A and D, each labeled 49 degrees, and matching double arcs at B and E, each labeled 58 degrees. BC and EF have matching single ticks.49°58°ABC49°58°DEF
  2. 2

    In triangles KLM and RST, angles K and R are congruent and angles L and S are congruent. Which equality would allow AAS?

    The recorded triangle partsTriangles KLM and RST have matching single arcs at K and R, and matching double arcs at L and S. All sides are unmarked.KLMRST
  3. 3

    Two AAS-congruent triangles have corresponding sides of lengths 4x+34x+3 and 7x−187x-18. Find x.

  4. 4

    Triangles ABC and DEF are right at B and E. Which side equality must be the hypotenuse equality for HL?

    The recorded triangle partsTriangles ABC and DEF have right angle squares at B and E. All sides are unmarked.BCAEFD
  5. 5

    Right triangles ABC and DEF are right at A and D, with

    AB=DEBC=EF\begin{aligned} AB&=DE \\ BC&=EF \end{aligned}

    If

    AC=5t−6DF=2t+15\begin{aligned} AC&=5t-6 \\ DF&=2t+15 \end{aligned}

    what is t?

    The recorded triangle partsTriangles ABC and DEF have right angle squares at A and D. AB and DE have matching single ticks; BC and EF have matching double ticks. AC is labeled 5t minus 6, and DF is labeled 2t plus 15.5t−6ABC2t+15DEF
  6. 6

    Two right triangles each have one leg of length 11 and their other legs of length 20. Which criterion can establish their congruence without first finding the hypotenuse?

    The recorded triangle partsTriangles ABC and DEF have right angle squares at A and D. AB and DE each have one tick and a label of 11. AC and DF each have two ticks and a label of 20. BC and EF are unmarked.1120ABC1120DEF
  7. 7

    Triangles PQR and XYZ have angles

    m∠P=32∘m∠X=32∘m∠Q=97∘m∠Y=97∘\begin{aligned} m\angle P&=32^\circ \\ m\angle X&=32^\circ \\ m\angle Q&=97^\circ \\ m\angle Y&=97^\circ \end{aligned}

    If

    PQ=XYPQ=XY

    which criterion fits the given side?

    The recorded triangle partsTriangles PQR and XYZ have matching single arcs at P and X, each labeled 32 degrees, and matching double arcs at Q and Y, each labeled 97 degrees. PQ and XY have matching single ticks.32°97°PQR32°97°XYZ
  8. 8

    Two equiangular triangles have perimeters 36 and 69 centimeters. Which statement is correct?

  9. 9

    A compass construction fixes AC, a ray from A, and a circle centered at C. The circle meets the ray at distinct B1 and B2, both beyond A. Why can AB1 and AB2 not be equal?

    Two triangles fit the same SSA measurements A ray from A meets a circle centered at C at B1 and B2, both beyond A. AC is labeled 10, and CB1 and CB2 are each labeled 7. The angle at A has an arc and a label of 30 degrees. AB₁B₂C1077 30° at A
  10. 10

    For two triangles, the following are the only givens:

    AB=DE.AC=DF.∠B≅∠E.\begin{aligned} AB&=DE. \\ AC&=DF. \\ \angle B&\cong\angle E. \end{aligned}

    What conclusion is justified?

    The recorded triangle partsTriangles ABC and DEF have matching single ticks on AB and DE, and matching double ticks on AC and DF. Angles B and E have matching single arcs. No other angles are marked.ABCDEF
  11. 11

    In the HL overlay, angles BCF and CFB are congruent because

    BC=BFBC=BF

    What fact then makes the two small angles at B congruent?

    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
  12. 12

    Two triangles are proposed with a matching corresponding side pair and angles of 118 and 62 degrees in each triangle. Can these measurements describe nondegenerate triangles for an AAS comparison?

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