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

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

What this topic is

On the ACT Mathematics test, Triangles — congruence, similarity, Pythagorean theorem covers corresponding sides and angles, similar-figure scale factors, and right-triangle lengths.

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?

A12✗ Not this oneB13✗ Not this oneC15✓ CorrectD20✗ Not this one

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Answer: C — 15

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

Question 2Easier

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

A$4.5\sqrt{2}$✗ Not this oneB9✗ Not this oneC$9\sqrt{2}$✓ CorrectD18✗ 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 $\triangle DEF$, $\overline{DE}\cong\overline{DF}$ and the measure of $\angle D$ is $52^\circ$. What is the measure of $\angle E$?

A$52^\circ$✗ Not this oneB$64^\circ$✓ CorrectC$76^\circ$✗ Not this oneD$128^\circ$✗ Not this one

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Answer: B — $64^\circ$

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

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