What are the most common Pythagorean triples to memorize for the ACT?

The essentials are 3-4-5, 5-12-13, and 8-15-17, plus their multiples such as 6-8-10 and 9-12-15. Recognizing a triple lets you find the third side of a right triangle instantly without squaring and adding.

What are the side ratios of the special right triangles?

A 45-45-90 triangle has sides in the ratio 1 : 1 : sqrt(2), so its hypotenuse is a leg times sqrt(2). A 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2, opposite the 30, 60, and 90 degree angles respectively.

How do similar triangles work on the ACT?

Similar triangles have equal corresponding angles and proportional corresponding sides. Set up a proportion pairing sides opposite equal angles, then cross-multiply to solve for the unknown side. Remember area scales by the square of the side ratio.

When should you subtract instead of add in the Pythagorean theorem?

Subtract when you know the hypotenuse and one leg and need the other leg: b = sqrt(c^2 - a^2). Add only when you know both legs and need the hypotenuse: c = sqrt(a^2 + b^2). The hypotenuse is always the longest side.

CramClub • Triangles and Right Triangles

$\angle A + \angle B + \angle C = 180^\circ$

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = k$

The Most-Tested Shape on the ACT

No figure appears more often on ACT Math than the triangle. A handful of reliable rules — the angle sum, the exterior-angle relationship, similarity and congruence, the Pythagorean theorem with its famous triples, and the two special right triangles — will carry you through the vast majority of geometry questions. This guide organizes those tools so you know which to reach for.

Angle Rules

The interior angles of any triangle sum to $180^\circ$ :

$\angle A + \angle B + \angle C = 180^\circ$

Two consequences show up constantly. First, the exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles — so an exterior angle is always larger than either of them. Second, angle relationships pin down triangle types: an equilateral triangle has three $60^\circ$ angles; an isosceles triangle has two equal angles opposite its two equal sides (the "base angles are equal" rule the ACT tests again and again).

Similar vs. Congruent

Two triangles are similar if they have the same shape — equal corresponding angles — but possibly different sizes. Their corresponding sides are then proportional:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$

where $k$ is the scale factor. Similar triangles are the engine behind shadow problems, nested-triangle problems, and midsegment problems. Two triangles are congruent — the same shape and size — when the scale factor is exactly $1$ . The ACT rarely asks you to name a congruence postulate (SSS, SAS, ASA); it asks you to use the equal parts that congruence guarantees.

A crucial similarity fact: when you scale a triangle by factor $k$ , its perimeter also scales by $k$ , but its area scales by $k^2$ . Doubling the sides quadruples the area.

The Pythagorean Theorem and Triples

For a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, always the longest):

$a^2 + b^2 = c^2$

Memorizing common Pythagorean triples lets you skip the arithmetic:

Triple	Scaled examples
$3, 4, 5$	$6, 8, 10$ · $9, 12, 15$
$5, 12, 13$	$10, 24, 26$
$8, 15, 17$	—

Whenever two sides match a triple (or a multiple), the third comes for free. But confirm which side is the hypotenuse: if you know the hypotenuse and one leg, you subtract to find the other leg, not add.

The Two Special Right Triangles

These two ratios are worth memorizing cold, because they let you find every side from just one.

45°–45°–90° (isosceles right): the legs are equal and the hypotenuse is a leg times $\sqrt{2}$ .

$\text{sides in ratio } 1 : 1 : \sqrt{2}$

30°–60°–90°: the sides opposite the $30^\circ$ , $60^\circ$ , and $90^\circ$ angles are in a fixed ratio.

$\text{sides in ratio } 1 : \sqrt{3} : 2$

So the side opposite $30^\circ$ is the short leg, the side opposite $60^\circ$ is that short leg times $\sqrt{3}$ , and the hypotenuse (opposite $90^\circ$ ) is twice the short leg. Match the given side to its position in the ratio first, then scale.

A Reliable Game Plan

Missing angle? Use the $180^\circ$ sum, or the exterior-angle rule when an angle sticks outside.
Two triangles, same angles? They are similar — set up a proportion of corresponding sides.
Right triangle, two sides known? Check for a triple; otherwise use $a^2 + b^2 = c^2$ (subtract if the hypotenuse is known).
See a $45^\circ$ or $30^\circ$ / $60^\circ$ angle? Deploy the special-triangle ratios.

Common Mistakes to Avoid

Adding when you should subtract in the Pythagorean theorem because the hypotenuse was the known side.
Pairing the wrong sides in a similarity proportion — match sides opposite equal angles.
Scaling area by $k$ instead of $k^2$ when a triangle is enlarged.
Assigning a special-triangle side to the wrong position — always anchor to the side opposite the known angle.

Two angles of a triangle measure $47^\circ$ and $68^\circ$ . What is the measure of the third angle?

A right triangle has legs of length $6$ and $8$ . How long is the hypotenuse?

A right triangle has a hypotenuse of $17$ and one leg of $8$ . Find the other leg.

In a 30°–60°–90° triangle, the side opposite the $30^\circ$ angle is $7$ . How long is the hypotenuse?

Triangle $ABC$ is similar to triangle $DEF$ . In $ABC$ , side $AB = 6$ and side $BC = 9$ . In $DEF$ , the corresponding side $DE = 10$ . How long is side $EF$ ?

In triangle $ABC$ , sides $AB$ and $AC$ are equal, and the exterior angle at vertex $C$ measures $110^\circ$ . What is the measure of angle $A$ ?

Adding the two known sides in the Pythagorean theorem when one of them is the hypotenuse.

If the hypotenuse is known, subtract: $b = \sqrt{c^2 - a^2}$ . Adding would make the "leg" longer than the hypotenuse, which is impossible.

Matching non-corresponding sides in a similarity proportion.

Pair sides that lie opposite equal angles, and keep both ratios in the same order (small triangle over large, top and bottom).

Placing a special-triangle side in the wrong slot — e.g. treating a side opposite $60^\circ$ as the short leg.

Anchor to the angle: the side opposite $30^\circ$ is the " $1$ ," opposite $60^\circ$ is the " $\sqrt{3}$ ," opposite $90^\circ$ is the " $2$ ."

Scaling area by the linear factor $k$ when a triangle is enlarged, instead of $k^2$ .

Perimeter scales by $k$ , but area scales by $k^2$ . Doubling every side multiplies the area by $4$ .

Two angles of a triangle measure $52^\circ$ and $73^\circ$ . What is the measure of the third angle?

A right triangle has legs of length $9$ and $12$ . What is the length of the hypotenuse?

In a 45°–45°–90° triangle, each leg has length $5$ . What is the length of the hypotenuse?

A right triangle has a hypotenuse of length $26$ and one leg of length $10$ . What is the length of the other leg?

Triangle $RST$ is similar to triangle $XYZ$ with $\dfrac{RS}{XY} = \dfrac{2}{3}$ . If the area of triangle $RST$ is $12$ square units, what is the area of triangle $XYZ$ ?

In triangle $ABC$ , angle $A = 90^\circ$ and $AB = AC$ . If the hypotenuse $BC$ has length $8$ , what is the length of leg $AB$ ?

$\angle A + \angle B + \angle C = 180^\circ$

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = k$

The Most-Tested Shape on the ACT

Angle Rules

The interior angles of any triangle sum to $180^\circ$ :

$\angle A + \angle B + \angle C = 180^\circ$

Similar vs. Congruent

Two triangles are similar if they have the same shape — equal corresponding angles — but possibly different sizes. Their corresponding sides are then proportional:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$

A crucial similarity fact: when you scale a triangle by factor $k$ , its perimeter also scales by $k$ , but its area scales by $k^2$ . Doubling the sides quadruples the area.

The Pythagorean Theorem and Triples

For a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, always the longest):

$a^2 + b^2 = c^2$

Memorizing common Pythagorean triples lets you skip the arithmetic:

Triple	Scaled examples
$3, 4, 5$	$6, 8, 10$ · $9, 12, 15$
$5, 12, 13$	$10, 24, 26$
$8, 15, 17$	—

The Two Special Right Triangles

These two ratios are worth memorizing cold, because they let you find every side from just one.

45°–45°–90° (isosceles right): the legs are equal and the hypotenuse is a leg times $\sqrt{2}$ .

$\text{sides in ratio } 1 : 1 : \sqrt{2}$

30°–60°–90°: the sides opposite the $30^\circ$ , $60^\circ$ , and $90^\circ$ angles are in a fixed ratio.

$\text{sides in ratio } 1 : \sqrt{3} : 2$

A Reliable Game Plan

Missing angle? Use the $180^\circ$ sum, or the exterior-angle rule when an angle sticks outside.
Two triangles, same angles? They are similar — set up a proportion of corresponding sides.
Right triangle, two sides known? Check for a triple; otherwise use $a^2 + b^2 = c^2$ (subtract if the hypotenuse is known).
See a $45^\circ$ or $30^\circ$ / $60^\circ$ angle? Deploy the special-triangle ratios.

Common Mistakes to Avoid

Adding when you should subtract in the Pythagorean theorem because the hypotenuse was the known side.
Pairing the wrong sides in a similarity proportion — match sides opposite equal angles.
Scaling area by $k$ instead of $k^2$ when a triangle is enlarged.
Assigning a special-triangle side to the wrong position — always anchor to the side opposite the known angle.