Math by First Principles
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Congruent Triangles: Chapter Test

20 multiple-choice questions and 10 core practice problems, drawn from across the chapter and mixed together.

Multiple choice

20 questions, 100 points in total, 5 points each. Answer in any order and change your mind as often as you like. When you submit, your answers lock and every question shows its worked solution.

Multiple choice 0 / 20 answered
Question 1 of 20
  1. 1

    Given

    △UVW≅△ZXY\triangle UVW\cong\triangle ZXY

    which side corresponds to VWVW?

  2. 2

    Triangle ABCABC has the following measurements:

    AB=ACm∠A=136∘\begin{aligned} AB&=AC \\ m\angle A&=136^\circ \end{aligned}

    What is m∠Bm\angle B?

    Stated measurements in triangle ABCSchematic triangle ABC has one tick on AB and AC and its angle at A labeled 136 degrees. Angles at B and C are unmarked. The drawing is not a measurement scale.ABC136°Schematic
  3. 3

    Triangles ABCABC and GHIGHI have these measurements:

    AB=GHAB=GH

    GH=17 cmGH=17\text{ cm}

    AC=GIAC=GI

    GI=26 cmGI=26\text{ cm}

    m∠BAC=m∠HGIm\angle BAC=m\angle HGI

    m∠HGI=58∘m\angle HGI=58^\circ

    Which ordered match follows?

    Two triangles with the stated givensSchematic triangles ABC and GHI. AB and GH are labeled 17 centimeters with one tick. AC and GI are labeled 26 centimeters with two ticks. Only the included angles at A and G are marked and labeled 58 degrees. No ordered match is printed.ABCGHI17 cm26 cm17 cm26 cm58°58°Schematic
  4. 4

    Two equilateral triangular cutouts have unrecorded side lengths. Which measurements would settle whether they are congruent without measuring any angles?

  5. 5

    To construct △EFG\triangle EFG, you have drawn

    EF=17 cmEF=17\text{ cm}

    and need

    EG=14 cmFG=24 cm\begin{aligned} EG&=14\text{ cm} \\ FG&=24\text{ cm} \end{aligned}

    Which compass settings locate G?

    The constructed baselineOnly the already drawn baseline EF is shown, labeled 17 centimeters. No third vertex, compass arcs, or compass centers have been added.EF17 cm
  6. 6

    Triangle MNP is right at N; triangle RST is right at S. Their hypotenuses satisfy

    MP=RTRT=31 cm\begin{aligned} MP&=RT \\ RT&=31\text{ cm} \end{aligned}

    and their legs satisfy

    NP=STST=11 cm\begin{aligned} NP&=ST \\ ST&=11\text{ cm} \end{aligned}

    Which ordered congruence follows?

    Two triangles with the stated givensSchematic right triangles MNP and RST have squares at N and S. MP and RT have one tick and labels 31 centimeters. NP and ST have two ticks and labels 11 centimeters. No congruence statement is printed.MNPRST31 cm11 cm31 cm11 cmSchematic
  7. 7

    A triangle has three equal interior angles and a perimeter of 117 cm. What are one side length and one interior angle measure?

  8. 8

    A four-sided figure ABCD has every corner pointing outward and

    AB=ADBC=CD\begin{aligned} AB&=AD \\ BC&=CD \end{aligned}

    Which plan can establish

    ∠ABC≅∠ADC\angle ABC\cong\angle ADC

    from these givens?

    Four-sided figure ABCDAB and AD have one tick. BC and CD have two ticks. The interior contains no added segment, and the target angles at B and D have no marks.ABCD
  9. 9

    In an angle-copy procedure, the recorded lengths are

    OA=OBOB=PDPD=PEPE=15 cmAB=18 cm\begin{aligned} OA&=OB \\ OB&=PD \\ PD&=PE \\ PE&=15\text{ cm} \\ AB&=18\text{ cm} \end{aligned}

    Here ABAB is the source chord. Which value of DE supplies the remaining corresponding side match?

    Source and target distance recordsSchematic source OAB has OA and OB labeled 15 centimeters and chord AB labeled 18 centimeters. Target PDE has PD and PE labeled 15 centimeters. The target chord DE is not drawn or assigned a length. No angle-equality marks are shown.OABPDE1515181515Schematic
  10. 10

    In an exact survey model, triangle ABC containing an inaccessible point A is congruent to accessible triangle DCB in that order. The measurements are

    BC=27 mDB=55 mAC=(4r−9) m\begin{aligned} BC&=27\text{ m} \\ DB&=55\text{ m} \\ AC&=(4r-9)\text{ m} \end{aligned}

    What is r?

    Opposite-side survey trianglesSchematic triangle ABC lies above baseline BC and triangle DCB below it. BC is labeled 27 meters, DB 55 meters, and AC with the expression 4r minus 9 meters. A is inaccessible and D accessible. No value of r or evaluated AC length is shown.ABCD27 m55 m4r − 9 mSchematic
  11. 11

    Triangles ABC and DEF satisfy

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

    A rigid motion aligns A with D and B with E; reflection if needed puts the image of C on F’s side of line DE. Why must those third vertices coincide?

    Two triangles with the stated givensBefore any alignment, triangle ABC and triangle DEF are shown separately. AB and DE have one tick, AC and DF two ticks, and only the given included angles at A and D have equal arcs. There is no motion overlay or third-vertex coincidence mark.ABCDEF
  12. 12

    Triangle ABC has

    AB=ACAB=AC

    Point D lies inside BC, and AD bisects the angle at A. The measurements are

    m∠BAD=(2x+9)∘m\angle BAD=(2x+9)^\circ

    m∠DAC=(5x−21)∘m\angle DAC=(5x-21)^\circ

    What is m∠Bm\angle B?

    The stated bisector and equal sidesSchematic triangle ABC has AB and AC marked equal. D lies inside BC, and AD is drawn. Equal arcs at A mark only the given bisector. The two angle expressions are stated in the question text. No equal base segments, right angles, or base-angle values are marked.ABCDSchematic
  13. 13

    Triangles ABC and DEF satisfy

    m∠A=m∠Em∠E=30∘m∠B=m∠Fm∠F=60∘m∠C=m∠Dm∠D=90∘\begin{aligned} m\angle A&=m\angle E \\ m\angle E&=30^\circ \\ m\angle B&=m\angle F \\ m\angle F&=60^\circ \\ m\angle C&=m\angle D \\ m\angle D&=90^\circ \end{aligned}

    Also,

    AB=DFDF=24 cmEF=48 cm\begin{aligned} AB&=DF \\ DF&=24\text{ cm} \\ EF&=48\text{ cm} \end{aligned}

    Which conclusion and reason are justified?

    The recorded angles and lengthsTwo schematic right-triangle outlines show only the recorded measurements. A and E are labeled 30 degrees, B and F 60 degrees, and right-angle squares mark C and D. AB and DF have the stated 24-centimeter labels and matching ticks, while EF is labeled 48 centimeters. No corresponding side pair is identified. The outlines are not drawn to a common measurement scale.ABCDEF30°60°30°60°24 cm24 cm48 cmSchematic
  14. 14

    A first proof establishes

    △KLM≅△PQR\triangle KLM\cong\triangle PQR

    A second comparison has

    AK=BPAL=BQ\begin{aligned} AK&=BP \\ AL&=BQ \end{aligned}

    with target

    ∠KAL≅∠PBQ\angle KAL\cong\angle PBQ

    Which consequence of the first match supplies the missing side pair for the triangles containing the target angles?

    First triangles and the second comparisonKLM and PQR are separate upper triangles. A joins K and L below the first, while B joins P and Q below the second. One tick marks AK and BP, and two ticks mark AL and BQ. No first-match side consequence or target angle equality is marked.KLMAPQRB
  15. 15

    Triangles JKL and PQR have

    m∠J=m∠Pm∠P=44∘m∠K=m∠Rm∠R=57∘JL=PQPQ=16 cm\begin{aligned} m\angle J&=m\angle P \\ m\angle P&=44^\circ \\ m\angle K&=m\angle R \\ m\angle R&=57^\circ \\ JL&=PQ \\ PQ&=16\text{ cm} \end{aligned}

    Which ordered congruence is guaranteed?

    Two angle pairs and the stated sideSchematic triangles JKL and PQR show 44-degree labels at J and P, 57-degree labels at K and R, and the 16-centimeter sides JL and PQ with matching ticks. The third angles and the ordered match are not marked.JKLPRQ44°44°57°57°16 cm16 cmSchematic
  16. 16

    Two perpendicular rays start at A. Point C lies on one ray with

    AC=19 cmAC=19\text{ cm}

    A circle centered at C with radius 29 cm meets the other ray at B. How do any two exact successful drawings of triangle ABC compare?

    A circle and perpendicular raysA circle centered at C is drawn behind triangle ABC. C lies on the upward ray from A, and B lies on the rightward ray. AC is labeled 19, and CB is labeled 29. No length is supplied for AB.ABC1929
  17. 17

    Triangle XYZ has congruent angles at X and Z. Its side lengths are

    XY=(6h+1) cmXY=(6h+1)\text{ cm}

    YZ=(3h+25) cmYZ=(3h+25)\text{ cm}

    XZ=30 cmXZ=30\text{ cm}

    What is its perimeter?

    Equal angles and recorded side expressionsSchematic triangle XYZ has equal-angle arcs at X and Z. XY is labeled 6h plus 1 centimeters, YZ 3h plus 25 centimeters, and XZ 30 centimeters. No side-equality ticks or evaluated lengths are shown.XYZ6h + 13h + 2530 cmSchematic
  18. 18

    Two students copy the same angle. One places the second target mark above the starting line, and the other below their own starting line. Either side was allowed, and all three transferred side pairs are correct in both copies. What is guaranteed about the angle measures?

    The source and opposite-side target placementsThe source OAB and upper target PDE open above their starting lines; lower target XYZ opens below its starting line. One tick marks OA, PD, and XY. Two ticks mark OB, PE, and XZ. Three ticks mark AB, DE, and YZ. Only transferred side pairs are marked; no copied angle conclusions are shown.OABPDEXYZ
  19. 19

    An accessible triangle and an inaccessible triangle share the same positive baseline. One matching endpoint angle is recorded. Which additional record guarantees the intended triangle match without measuring another side?

    One recorded endpoint angle pairTwo schematic triangles share baseline BC, with inaccessible A above it and accessible D below it. Only the given matching endpoint angles at B have arcs. No matching angle at C, side-length pair, or additional record is marked.ABCDSchematic
  20. 20

    Triangle ABC has congruent angles at B and C. Point D lies strictly inside BC with

    BD=DCBD=DC

    Points E and F lie strictly between A,B and A,C, respectively, with

    AE=AFAE=AF

    Which chain of reasoning establishes

    DE=DF?DE=DF?
    The given nested configurationTriangle ABC has equal-angle arcs at B and C. D is inside BC, with one tick on BD and DC. E is between A and B and F between A and C, with two ticks on AE and AF. DE and DF are drawn without equality marks. AD is not drawn, and no outer-side equality or vertex-angle conclusion is marked.ABCDEF

Core practice

10 problems from across the chapter. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

  1. Problem 1 A stated match

    The ordered congruence is:

    △RST≅△WYV\begin{gathered}\triangle RST\cong\triangle WYV\end{gathered}

    The corresponding side lengths are:

    RT=(7a−3) cmWV=(4a+24) cm\begin{gathered}RT=(7a-3)\text{ cm}\\WV=(4a+24)\text{ cm}\end{gathered}

    Find a and their common length. Also name the angle corresponding to ∠RST\angle RST.

  2. Problem 2 Measurements in a triangle

    Triangle ABC has AB=ACAB=AC. Point D lies strictly inside BC, and AD bisects the interior angle at A. The given measurements are:

    BD=15 cmm∠B=(3u+5)∘m∠BAD=(2u+10)∘\begin{gathered}BD=15\text{ cm}\\m\angle B=(3u+5)^\circ\\m\angle BAD=(2u+10)^\circ\end{gathered}

    Find u, the whole length BC, and the whole angle A. Justify the geometric relationships used.

    The recorded triangle measurementsSchematic triangle ABC has AB and AC marked equal. D lies inside BC, and AD is the stated internal bisector with matching arcs at A. BD is labeled 15 centimeters. The two given angle expressions remain in the statement. DC, the whole base, and the whole angle A are not evaluated or marked.ABCD15 cmSchematic
    Text description of this figure

    Schematic triangle ABC has AB and AC marked equal. D lies inside BC, and AD is the stated internal bisector with matching arcs at A. BD is labeled 15 centimeters. The two given angle expressions remain in the statement. DC, the whole base, and the whole angle A are not evaluated or marked.

  3. Problem 3 Three independent records

    Treat each record separately and use only its stated information. Classify each pair as guaranteed congruent, proven noncongruent, or undecided, and justify the classification. Give one ordered triangle match whenever congruence is guaranteed. All triangles are nondegenerate.

    Record A. Triangles ABC and DEF have:

    ∠A≅∠E∠B≅∠FAC=ED\begin{aligned}\angle A&\cong\angle E \\ \angle B&\cong\angle F \\ AC&=ED\end{aligned}

    Record B. Triangles GHJ and KLM have:

    GH=KLGJ=KM∠GHJ≅∠KLMHJ≠LM\begin{aligned}GH&=KL \\ GJ&=KM \\ \angle GHJ&\cong\angle KLM \\ HJ&\ne LM\end{aligned}

    Record C. Triangles PQR and STU have:

    PQ=STPR=SU∠PQR≅∠STU\begin{aligned}PQ&=ST \\ PR&=SU \\ \angle PQR&\cong\angle STU\end{aligned}

    No other relationships or measurements are supplied.

  4. Problem 4 Two sets of measurements

    For each case, construct the triangle on paper, describe the essential steps, and explain why every exact successful drawing has the same shape and size. A ruler and protractor may be used for the numerical drawings. A common stated scale is permitted; apply the same scale to every length within each case.

    Case A. Construct PQR with these side lengths:

    PQ=18 cmPR=23 cmQR=29 cm\begin{aligned}PQ&=18\text{ cm} \\ PR&=23\text{ cm} \\ QR&=29\text{ cm}\end{aligned}

    Use PQ as the base and keep the locating arcs.

    Case B. Construct UVW with these measurements:

    UV=12 cmm∠U=57∘m∠V=64∘\begin{aligned}UV&=12\text{ cm} \\ m\angle U&=57^\circ \\ m\angle V&=64^\circ\end{aligned}

    Also find angle W.

    For each case, state how a successful drawing on the other side of the same base compares with the first.

  5. Problem 5 Marked interior points

    Triangles ABC and DEF are nondegenerate, with these measurements:

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

    Point G lies strictly between B and A, and H strictly between B and C. Point J lies strictly between E and D, and K strictly between E and F.

    The additional givens are:

    BG=EJ∠BGH≅∠EJK\begin{gathered}BG=EJ\\\angle BGH\cong\angle EJK\end{gathered}

    Prove the following equality and give ordered triangle comparisons for any congruences used:

    GH=JK\begin{gathered}GH=JK\end{gathered}
    Marked interior pointsIn the upper triangle ABC, G is between B and A and H between B and C. In the lower triangle DEF, J is between E and D and K between E and F. The inner segments GH and JK are drawn. One tick marks BG and EJ, and one arc marks angles BGH and EJK. The other stated side equalities belong to the text.ABCGHDEFJK
    Text description of this figure

    In the upper triangle ABC, G is between B and A and H between B and C. In the lower triangle DEF, J is between E and D and K between E and F. The inner segments GH and JK are drawn. One tick marks BG and EJ, and one arc marks angles BGH and EJK. The other stated side equalities belong to the text.

  6. Problem 6 An opening at a new point

    On paper, construct triangle OAB with these measurements:

    OA=OBOB=14 cmAB=17 cm\begin{aligned}OA&=OB \\ OB&=14\text{ cm} \\ AB&=17\text{ cm}\end{aligned}

    Extend OA and OB as rays. At a new point P with a starting ray, reproduce the opening ∠AOB\angle AOB with a compass and straightedge. Retain the relevant arcs. A common stated scale may be used for both the source and target drawings.

    Describe the distance settings and justify the equality of the openings without treating that angle equality as a starting fact. Label D and E as the two target radius marks.

  7. Problem 7 Which sides agree?

    In nondegenerate triangle LMN, angles L and N are congruent. The side lengths are:

    LM=(4b+7) cmMN=(7b−29) cmLN=32 cm\begin{aligned}LM&=(4b+7)\text{ cm} \\ MN&=(7b-29)\text{ cm} \\ LN&=32\text{ cm}\end{aligned}

    Find b and both equal side lengths.

    Triangle PQR has these lengths:

    PQ=PRPR=55 cmQR=32 cm\begin{aligned}PQ&=PR \\ PR&=55\text{ cm} \\ QR&=32\text{ cm}\end{aligned}

    Decide whether it is congruent to LMN and give one correct ordered statement beginning with triangle LMN.

    The first triangle recordSchematic triangle LMN has equal-angle arcs at L and N. LM is labeled 4b plus 7 centimeters, MN 7b minus 29 centimeters, and LN 32 centimeters. No evaluated side length or side-equality tick is shown.LMN4b + 77b − 2932 cmSchematic
    Text description of this figure

    Schematic triangle LMN has equal-angle arcs at L and N. LM is labeled 4b plus 7 centimeters, MN 7b minus 29 centimeters, and LN 32 centimeters. No evaluated side length or side-equality tick is shown.

  8. Problem 8 An inaccessible endpoint

    Accessible points P and Q form a baseline with

    PQ=34 mPQ=34\text{ m}

    Points R and S lie on opposite sides of line PQ; R is inaccessible and S is accessible. All points lie in one plane. In the exact model, the givens are:

    QR=QS∠RQP≅∠SQPSP=73 m\begin{aligned}QR&=QS \\ \angle RQP&\cong\angle SQP \\ SP&=73\text{ m}\end{aligned}

    Find RP and justify why the accessible measurement can be transferred. Now remove the equal-side condition:

    QR=QS\begin{gathered}QR=QS\end{gathered}

    Can remeasuring baseline PQ replace that condition and guarantee the following equality? Explain.

    RP=SP\begin{gathered}RP=SP\end{gathered}
    The exact survey recordSchematic triangles RQP and SQP lie on opposite sides of PQ. PQ is labeled 34 meters, SP 73 meters, QR and QS have matching ticks, and the stated equal angles at Q have matching arcs. R is inaccessible and S accessible. RP has no numerical label. This drawing is not a length scale.RPQS34 m73 mSchematic
    Text description of this figure

    Schematic triangles RQP and SQP lie on opposite sides of PQ. PQ is labeled 34 meters, SP 73 meters, QR and QS have matching ticks, and the stated equal angles at Q have matching arcs. R is inaccessible and S accessible. RP has no numerical label. This drawing is not a length scale.

  9. Problem 9 A measurement card

    Design one measurement card by choosing two positive side lengths and one acute angle that is not between those sides. Name the sides and angle on your card. On paper, construct two nondegenerate, noncongruent triangles satisfying that same card.

    Label the vertices consistently to show the intended correspondence, retain the locating arcs, and use one stated scale for both drawings. Explain why each triangle satisfies the card and why the two triangles are noncongruent. You may use a ruler, protractor, and compass; exact calculations of the unrecorded side lengths are not required.

  10. Problem 10 Two right triangles

    Nondegenerate triangle ABC is right at A and has congruent angles at B and C. Triangle DEF is right at D. The additional givens are:

    DE=ACEF=BCAB=(4t+9) cmDF=(6t−13) cm\begin{aligned}DE&=AC \\ EF&=BC \\ AB&=(4t+9)\text{ cm} \\ DF&=(6t-13)\text{ cm}\end{aligned}

    Find t, AC, and DF.

    Justify any triangle comparison used and the side equalities that support your calculation.

    Two separately placed right trianglesThe upper triangle ABC is solid blue, and the lower triangle DEF is red and dashed. Right-angle squares mark A and D. The given equal angles at B and C have one arc each. AC and DE have one side tick; BC and EF have two. The remaining legs carry the stated expressions.ABCDEF4t+96t−13
    Text description of this figure

    The upper triangle ABC is solid blue, and the lower triangle DEF is red and dashed. Right-angle squares mark A and D. The given equal angles at B and C have one arc each. AC and DE have one side tick; BC and EF have two. The remaining legs carry the stated expressions, measured in centimeters. No derived side equality is marked.