NeokaalStudy

Triangles: Practice Worksheet

Proportionality, similarity criteria, missing lengths, and proof-based reasoning.

18 problems·35–45 min·★★★★★
Print worksheet (PDF) →
  1. Problem 1
    Basic Proportionality Theorem·★★★★★
    In triangle ABC, D lies on AB and E lies on AC with $DE\parallel BC$. Prove that $AD/DB=AE/EC$.
    Triangle with an internal segment parallel to its base
    ABCDENot to scale
    ▶Answer
    $AD/DB=AE/EC$.
    ▶Step-by-step solution
    1. Triangles ADE and BDE have the same altitude to AB, so their area ratio is $AD/DB$.
    2. Triangles ADE and CDE have the same altitude to AC, so their area ratio is $AE/EC$.
    3. Triangles BDE and CDE lie between the same parallels DE and BC and have equal areas; equating the two ratios gives $AD/DB=AE/EC$.
    Report an issue →
  2. Problem 2
    Similarity Criteria·★★★★★
    One angle of a triangle equals one angle of another triangle, and two side pairs are proportional. Must the triangles be similar? State the missing condition.
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    Not always. The equal angle must be included between the proportional side pairs for SAS similarity.
    ▶Step-by-step solution
    1. SAS similarity uses two proportional side pairs and the angle between those sides.
    2. If the stated equal angle is not included, the data can describe different triangle shapes.
    3. Therefore similarity follows only when the equal angles are the included corresponding angles.
    Report an issue →
  3. Problem 3
    Similar Triangles·★★★★☆
    D lies on AB in triangle ABC and $\angle ACB=\angle CDA$. If $AC=6$ cm and $AD=3$ cm, find DB.
    Triangle divided by a segment from a vertex to its base
    3?6ADBCNot to scale
    ▶Answer
    $DB=9\text{ cm}$
    ▶Step-by-step solution
    1. $\angle CAB=\angle CAD$ and $\angle ACB=\angle CDA$, so $\triangle ACB\sim\triangle ADC$.
    2. Hence $AC/AD=AB/AC$, so $AB=6^2/3=12$ cm.
    3. Therefore $DB=AB-AD=12-3=9$ cm.
    Triangle divided by a segment from a vertex to its base
    396ADBCNot to scale
    Report an issue →
  4. Problem 4
    Correspondence in Similar Triangles·★★☆☆☆
    Given $\triangle ABC\sim\triangle EDF$, which relation does not follow from the stated correspondence?
    1. (A)$AB\cdot DF=AC\cdot ED$
    2. (B)$AB\cdot EF=AC\cdot ED$
    3. (C)$BC\cdot EF=AC\cdot DF$
    4. (D)$AB\cdot DF=BC\cdot ED$
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    Option A: $AB\cdot DF=AC\cdot ED$
    ▶Step-by-step solution
    1. The order gives $A\leftrightarrow E$, $B\leftrightarrow D$, and $C\leftrightarrow F$.
    2. Hence $AB/ED=BC/DF=AC/EF$.
    3. The relation $AB\cdot DF=AC\cdot ED$ mixes non-corresponding ratios, so option A does not follow.
    Report an issue →
  5. Problem 5
    Correspondence in Similar Triangles·★★☆☆☆
    The side correspondences are XY ↔ FD, YZ ↔ DE, and ZX ↔ EF. Write the similarity statement in matching order.
    ▶Answer
    $\triangle XYZ \sim \triangle FDE$
    ▶Step-by-step solution
    1. Match vertex X with F using the two sides that meet there.
    2. Similarly, Y ↔ D and Z ↔ E.
    3. Therefore the correct order is $\triangle XYZ \sim \triangle FDE$.
    Report an issue →
  6. Problem 6
    Correspondence in Similar Triangles·★★☆☆☆
    Given $\triangle ABC\sim\triangle DFE$, a learner says $\angle A=\angle D$ and $\angle C=\angle F$. Is the complete claim correct? Explain.
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    No. $\angle A=\angle D$, but $\angle C=\angle E$.
    ▶Step-by-step solution
    1. Read corresponding vertices in the stated order.
    2. The mapping is $A\leftrightarrow D$, $B\leftrightarrow F$, and $C\leftrightarrow E$.
    3. Therefore the first equality is correct, while the second should be $\angle C=\angle E$.
    Report an issue →
  7. Problem 7
    Congruence and Similarity·★★☆☆☆
    Two triangles have two matching angles, and every side of the second is 2 times the corresponding side of the first. Are they similar, congruent, both, or neither?
    ▶Answer
    They are similar but not congruent.
    ▶Step-by-step solution
    1. Two pairs of equal angles establish AA similarity.
    2. The scale factor is 2, not 1.
    3. Hence corresponding sides are not equal, so the triangles are not congruent.
    Report an issue →
  8. Problem 8
    Correspondence in Similar Triangles·★★★☆☆
    In triangles ABC and DEF, $\angle A=\angle E$ and $\angle B=\angle F$. Which ratio equality is not guaranteed?
    1. (A)$AB/FD=BC/EF$
    2. (B)$BC/FD=AC/ED$
    3. (C)$AB/EF=AC/ED$
    4. (D)$AB/EF=BC/FD$
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    Option A: $AB/FD=BC/EF$
    ▶Step-by-step solution
    1. The third angles are also equal, so $C\leftrightarrow D$.
    2. Thus $AB/EF=BC/FD=AC/ED$.
    3. Option A compares sides outside that correspondence and is not guaranteed.
    Report an issue →
  9. Problem 9
    Correspondence in Similar Triangles·★★★☆☆
    In triangle ABC, the angles at A, B and C are 60°, 61° and 59°. Triangle DEF has the same respective angles at D, E and F. Is $\triangle BCA\sim\triangle EFD$? Explain.
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    Yes, by AA.
    ▶Step-by-step solution
    1. The equal-angle correspondence is $A\leftrightarrow D$, $B\leftrightarrow E$, and $C\leftrightarrow F$.
    2. Starting with BCA therefore requires the order EFD, not any other ordering.
    3. The proposed order is EFD, so AA similarity is correctly stated.
    Report an issue →
  10. Problem 10
    Similarity Criteria·★★★☆☆
    Two right triangles each have an acute angle of $64^\circ$. Must they be similar? Give a reason.
    ▶Answer
    Yes, they are similar by AA.
    ▶Step-by-step solution
    1. Both triangles contain a $90^\circ$ angle.
    2. They also contain matching $64^\circ$ angles.
    3. Two corresponding angles are equal, so AA similarity applies.
    Report an issue →
  11. Problem 11
    Similarity Criteria·★★★☆☆
    In triangles ABC and DEF, $AB/DE=BC/EF$. Which angle equality, together with these ratios, proves similarity by SAS?
    Two triangles in corresponding order
    ABCDEFNot to scale
    ▶Answer
    $\angle B=\angle E$
    ▶Step-by-step solution
    1. The proportional sides in the first triangle meet at B.
    2. The corresponding proportional sides in the second triangle meet at E.
    3. SAS similarity therefore requires the included angles $\angle B$ and $\angle E$ to be equal.
    Report an issue →
  12. Problem 12
    Similar Triangles·★★★☆☆
    Triangles ABC and DEF are similar in that order. If $\angle A=54^\circ$, $\angle B=35^\circ$, $AB=3$ cm, and $DE=9$ cm, find $\angle F$ and the scale factor from ABC to DEF.
    Two triangles in corresponding order
    3 cm9 cmABCDEFNot to scale
    ▶Answer
    $\angle F=91^\circ$ and the scale factor is $3$.
    ▶Step-by-step solution
    1. The third angle is $180^\circ-54^\circ-35^\circ=91^\circ$.
    2. Since C corresponds to F, $\angle F=\angle C$.
    3. The scale factor is $DE/AB=9/3=3$.
    Report an issue →
  13. Problem 13
    Similar Triangles·★★★★☆
    Triangles ABC and DEF are similar. If $DE=6$, $EF=8$, $FD=10$ and $AB=3$ cm, find the perimeter of ABC.
    Two triangles in corresponding order
    3??6810ABCDEFNot to scale
    ▶Answer
    $12\text{ cm}$
    ▶Step-by-step solution
    1. The scale factor from ABC to DEF is $DE/AB=6/3=2$.
    2. Therefore $BC=8/2=4$ and $CA=10/2=5$.
    3. The perimeter is $3+4+5=12$ cm.
    Two triangles in corresponding order
    3456810ABCDEFNot to scale
    Report an issue →
  14. Problem 14
    Basic Proportionality Theorem·★★★★☆
    Points D and E divide sides AB and AC of triangle ABC with $AD=2$, $DB=4$, $AE=4$ and $EC=8$. Is DE parallel to BC? Explain.
    Triangle with an internal segment parallel to its base
    2448ABCDENot to scale
    ▶Answer
    Yes.
    ▶Step-by-step solution
    1. Compute $AD/DB=2/4$.
    2. Compute $AE/EC=4/8$.
    3. The ratios are equal, so the converse of the Basic Proportionality Theorem gives $DE\parallel BC$.
    Report an issue →
  15. Problem 15
    Basic Proportionality Theorem·★★★★☆
    In triangle CAB, DE is parallel to AB. If $AD=x+14$, $DC=6$, $BE=20$ and $EC=5$, find $x$.
    ▶Answer
    $x=10$
    ▶Step-by-step solution
    1. A parallel segment divides the two sides proportionally.
    2. Thus $(x+14)/6=20/5$.
    3. Solving the linear equation gives $x=10$.
    Report an issue →
  16. Problem 16
    Similarity Criteria·★★★★☆
    Points D and E lie on AB and AC respectively. If $\angle ADE=\angle ACB$, is $\triangle ADE$ similar to $\triangle ACB$? Explain.
    Triangle with an internal segment
    ABCDENot to scale
    ▶Answer
    Yes. $\triangle ADE\sim\triangle ACB$ by AA.
    ▶Step-by-step solution
    1. $\angle ADE=\angle ACB$ is given.
    2. $\angle DAE=\angle BAC$ because AD lies along AB and AE lies along AC.
    3. Two corresponding angles are equal, so $\triangle ADE\sim\triangle ACB$ by AA.
    Report an issue →
  17. Problem 17
    Similarity Criteria·★★★★☆
    Two corresponding sides and the perimeter of triangle DEF are each 2 times those of triangle ABC. Must the triangles be similar? Explain.
    Two triangles in corresponding order
    512?1024?ABCDEFNot to scale
    ▶Answer
    Yes, the triangles are similar by SSS.
    ▶Step-by-step solution
    1. The two known side pairs have the same scale factor 2.
    2. Subtract those two sides from each perimeter; the remaining side pair has the same scale factor.
    3. All three corresponding sides are proportional, so SSS similarity applies.
    Two triangles in corresponding order
    51213102426ABCDEFNot to scale
    Report an issue →
  18. Problem 18
    Similar Triangles·★★★★★
    Triangles ABC and DEF are similar in order. Given $AB=5$, $AC=13$, $DE=15$ and $DF=39$, find BC and EF.
    Two triangles in corresponding order
    5?1315?39ABCDEFNot to scale
    ▶Answer
    $BC=12$ and $EF=36$.
    ▶Step-by-step solution
    1. The scale factor is $DE/AB=15/5=3$.
    2. Since BC corresponds to EF, $EF=3BC$.
    3. The compatible side lengths are $BC=12$ and $EF=36$.
    Two triangles in corresponding order
    51213153639ABCDEFNot to scale
    Report an issue →