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Triangles — congruence, similarity, Pythagorean theorem — ACT practice questions

On the ACT Mathematics test, Triangles — congruence, similarity, Pythagorean theorem covers corresponding sides and angles, similar-figure scale factors, and right-triangle lengths. Students must match parts, apply a given side ratio, and use the Pythagorean theorem or a forty-five degree isosceles right triangle to find a missing side or angle. Questions often give similar triangles and ask for a corresponding length, give a known leg and ask for the hypotenuse, or show a figure with an angle of fifty-two degrees and ask for another angle. The five answer choices are numeric lengths or degree measures; traps include pairing the wrong sides or swapping the hypotenuse with a leg.

  • Section: Mathematics 45 questions on the paper
  • 59 practice questions in the app

Sample questions

Question 1

The lengths of corresponding sides of 2 similar triangles are in the ratio 2:32{:}3. The longest side of the smaller triangle is 10 centimeters. How many centimeters long is the longest side of the larger triangle?

  1. A12
  2. B13
  3. C15
  4. D20
Show answer

Answer: C — 15

The scale factor from the smaller triangle to the larger is 32\dfrac{3}{2}, so the longest side of the larger triangle is 10(32)=1510\left(\dfrac{3}{2}\right)=15 centimeters.

Question 2

A right triangle has two 45∘45^\circ angles at its non-right vertices, and one of its legs measures 9 centimeters. What is the length, in centimeters, of the hypotenuse?

  1. A4.524.5\sqrt{2}
  2. B9
  3. C929\sqrt{2}
  4. D18
Show answer

Answer: C — 929\sqrt{2}

In a 45-45-90 triangle the hypotenuse equals a leg times 2\sqrt{2}, so the hypotenuse is 929\sqrt{2} centimeters.

Question 3

In △DEF\triangle DEF, DE‾≅DF‾\overline{DE}\cong\overline{DF} and the measure of ∠D\angle D is 52∘52^\circ. What is the measure of ∠E\angle E?

  1. A52∘52^\circ
  2. B64∘64^\circ
  3. C76∘76^\circ
  4. D128∘128^\circ
Show answer

Answer: B — 64∘64^\circ

DE‾≅DF‾\overline{DE}\cong\overline{DF} makes ∠E\angle E and ∠F\angle F congruent, and they total 180∘−52∘=128∘180^\circ-52^\circ=128^\circ. Each measures 64∘64^\circ.

Practice 59 Triangles — congruence, similarity, Pythagorean theorem questions in the app

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