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

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Practice 62 Triangles — congruence, similarity, Pythagorean theorem questions in the app

What this topic is

Similar triangles have the same shape but not always the same size, so their corresponding sides keep one fixed ratio while their angles stay equal.

That idea, together with congruence and the Pythagorean theorem, makes up Triangles — congruence, similarity, Pythagorean theorem. ACT questions ask for a missing side from a scale ratio, a hypotenuse from two legs, or an angle in an isosceles triangle. Special right triangles save time: in an isosceles right triangle, the hypotenuse is a leg times the square root of two.

Matching the wrong pair of sides is the classic error, because a ratio only works between corresponding sides. Answer choices often contain radicals.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

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

12✗ Not this one13✗ Not this one15✓ Correct20✗ Not this one

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Answer: C — 15

The scale factor from the smaller triangle to the larger is $\frac{3}{2}$ , so the longest side of the larger triangle is $10 (\frac{3}{2}) = 15$ centimeters.

Question 2Easier

A right triangle has two $4 5^{\circ}$ angles at its non-right vertices, and one of its legs measures 9 centimeters. What is the length, in centimeters, of the hypotenuse?

4.5 \sqrt{2}

✗ Not this one9✗ Not this one

9 \sqrt{2}

✓ Correct18✗ Not this one

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Answer: C — $9 \sqrt{2}$

In a 45-45-90 triangle the hypotenuse equals a leg times $\sqrt{2}$ , so the hypotenuse is $9 \sqrt{2}$ centimeters.

Question 3Easier

In $△ D E F$ , $\overline{D E} ≅ \overline{D F}$ and the measure of $∠ D$ is $5 2^{\circ}$ . What is the measure of $∠ E$ ?

5 2^{\circ}

✗ Not this one

6 4^{\circ}

✓ Correct

7 6^{\circ}

✗ Not this one

12 8^{\circ}

✗ Not this one

Show answer

Answer: B — $6 4^{\circ}$

$\overline{D E} ≅ \overline{D F}$ makes $∠ E$ and $∠ F$ congruent, and they total $18 0^{\circ} - 5 2^{\circ} = 12 8^{\circ}$ . Each measures $6 4^{\circ}$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 62 Triangles — congruence, similarity, Pythagorean theorem questions in the app

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

What this topic is

Sample questions

More ACT topics

Mathematics