Math by First Principles
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SAS and ASA Congruence: Practice

12 multiple-choice questions, progressively harder.

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

    In triangle ABCABC, an exterior angle at A measures 130∘130^\circ. The other givens are:

    ∠B=72∘AB=13 cm\begin{aligned} \angle B&=72^\circ \\ AB&=13\text{ cm} \end{aligned}

    Triangle DEFDEF has these givens:

    ∠D=50∘∠E=72∘DE=13 cm\begin{aligned} \angle D&=50^\circ \\ \angle E&=72^\circ \\ DE&=13\text{ cm} \end{aligned}

    Which criterion establishes congruence in the listed order?

    The recorded triangle partsTriangle ABC is above triangle DEF. Sides AB and DE have single ticks and are each labeled 13 centimeters. Double arcs at B and E are labeled 72 degrees. BA extends past A, where an exterior angle is labeled 130 degrees. Angle D has a single arc labeled 50 degrees. No interior angle at A is marked.13 cm72°ABC13 cm50°72°DEF130°
  2. 2

    Two triangles have these given matching parts:

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

    The included angles measure (4x−9)∘(4x-9)^\circ and (2x+17)∘(2x+17)^\circ. What xx makes the given information an SAS match?

  3. 3

    For SAS, one side pair already matches and the included angles both measure 75∘75^\circ. The remaining side lengths are 2x+72x+7 cm and 3x−43x-4 cm. Which result completes the criterion?

  4. 4

    The intended ordered conclusion is △ABC≅△DEF\triangle ABC\cong\triangle DEF.

    Given AB=DEAB=DE, which set of additional facts directly completes an SAS argument?

    The recorded triangle partsTriangle ABC is above triangle DEF. Only sides AB and DE have matching single ticks. No angles or other sides are marked.ABCDEF
  5. 5

    Triangle ABCABC has these angle measures:

    ∠A=(x+14)∘∠B=(2x+10)∘\begin{aligned} \angle A&=(x+14)^\circ \\ \angle B&=(2x+10)^\circ \end{aligned}

    An exterior angle at C measures 132∘132^\circ. Triangle DEFDEF has these givens:

    ∠D=50∘∠E=82∘DE=AB\begin{aligned} \angle D&=50^\circ \\ \angle E&=82^\circ \\ DE&=AB \end{aligned}

    What value of xx makes the ASA data match in the listed order?

  6. 6

    A positive base is to have endpoint angles (2t+10)∘(2t+10)^\circ and (t+20)∘(t+20)^\circ, where t>0t>0. For which values of tt do these data construct a triangle?

  7. 7

    Use this proposed correspondence:

    A↔DB↔EC↔F\begin{aligned} A&\leftrightarrow D \\ B&\leftrightarrow E \\ C&\leftrightarrow F \end{aligned}

    The report gives:

    AB=DEAC=DF∠C≅∠F\begin{aligned} AB&=DE \\ AC&=DF \\ \angle C&\cong\angle F \end{aligned}

    Which assessment of a direct SAS argument is correct?

    The recorded triangle partsTriangle ABC is above triangle DEF. Sides AB and DE have single ticks. Sides AC and DF have double ticks. Single arcs mark angles C and F.ABCDEF
  8. 8

    In the ASA motion proof, the bases have been aligned, but the third vertices are in opposite half-planes. Why must the possible reflection be considered before claiming the angle rays coincide?

    Third vertices on opposite sidesTwo triangles share base AB, with C prime above it and F below it. Single arcs mark both angles at A. Double arcs mark both angles at B. No side ticks or motion arrows are shown.ABC′F
  9. 9

    Triangle ABCABC has vertices A(0,0)A(0,0), B(4,0)B(4,0), and C(0,7)C(0,7). Triangle DEFDEF has vertices D(2,1)D(2,1), E(2,5)E(2,5), and F(−5,1)F(-5,1). Which argument establishes congruence in that order without finding the slanted side lengths?

    Read the supplied coordinatesOn an x and y coordinate grid, triangle ABC has A at (0, 0), B at (4, 0), and C at (0, 7). Triangle DEF has D at (2, 1), E at (2, 5), and F at (negative 5, 1). No lengths, equality ticks, or angle arcs are marked.-5-4-3-2-112341234567ABCDEFxy
  10. 10

    The given angle measures are:

    ∠J=40∘∠M=40∘∠K=80∘∠N=80∘\begin{aligned} \angle J&=40^\circ \\ \angle M&=40^\circ \\ \angle K&=80^\circ \\ \angle N&=80^\circ \end{aligned}

    The given side lengths are:

    JK=5 cmMN=6 cm\begin{aligned} JK&=5\text{ cm} \\ MN&=6\text{ cm} \end{aligned}

    Use this proposed correspondence:

    J↔MK↔NL↔P\begin{aligned} J&\leftrightarrow M \\ K&\leftrightarrow N \\ L&\leftrightarrow P \end{aligned}

    Can triangles JKL and MNP be congruent under that correspondence?

    The recorded triangle partsTriangle JKL is above triangle MNP. Side JK is labeled 5 centimeters, and side MN is labeled 6 centimeters. Neither side has an equality tick. Single arcs at J and M are labeled 40 degrees. Double arcs at K and N are labeled 80 degrees.5 cm40°80°JKL6 cm40°80°MNP
  11. 11

    Two successful constructions share a base of length 88 cm. Both use a second side of 55 cm from the left endpoint and an included angle of 42∘42^\circ, but their third vertices lie on opposite sides of the base. How do the results compare?

    Two completed placementsTriangles ABC and ABC prime share base AB, labeled 8 centimeters. C lies above AB, and C prime lies below it. Sides AC and AC prime have single ticks and are each labeled 5 centimeters. Both angles at A have single arcs labeled 42 degrees. No motion arrow is shown.5 cm42°5 cm42°ABCC′AB = 8 cm
  12. 12

    The known matching parts are:

    AB=DE∠A≅∠D\begin{aligned} AB&=DE \\ \angle A&\cong\angle D \end{aligned}

    Which pair of plans correctly describes two different ways to finish a congruence argument?

    The recorded triangle partsTriangle ABC is above triangle DEF. Sides AB and DE have single ticks. Single arcs mark angles A and D. No other sides or angles are marked.ABCDEF
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