ACT Math Flashcards: Triangles

What this deck covers

This deck focuses on Triangles, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the measure of each angle in an equilateral triangle?

Answer: 60 degrees. Since all sides are equal, all angles are equal: $180° ÷ 3 = 60°$.

Flashcard 2: What is the definition of a right triangle in terms of angles?

Answer: One angle is $90^\circ$. A right angle defines a right triangle.

Flashcard 3: What is the condition for a triangle to be acute using side lengths with largest side $c$?

Answer: $a^2+b^2>c^2$. When legs squared exceed hypotenuse squared, all angles are acute.

Flashcard 4: What is the side opposite the right angle in a right triangle called?

Answer: Hypotenuse. The longest side in a right triangle, opposite the right angle.

Flashcard 5: What is the definition of a congruent triangle?

Answer: Triangles with equal corresponding angles and sides. Triangles that are identical in shape and size.

Flashcard 6: What is the midpoint of the segment with endpoints $(2,4)$ and $(8,10)$?

Answer: $(5,7)$. $\left(\frac{2+8}{2}, \frac{4+10}{2}\right) = (5,7)$

Flashcard 7: What is the slope of the line through points $(1,2)$ and $(5,10)$?

Answer: $2$. $\frac{10-2}{5-1} = \frac{8}{4} = 2$

Flashcard 8: What is the sum of interior angles in any triangle?

Answer: 180 degrees. This is a fundamental property that applies to all triangles.

Flashcard 9: What is the relationship between an exterior angle and its remote interior angles?

Answer: Exterior angle = sum of remote interior angles. Exterior angle equals sum of the two non-adjacent interior angles.

Flashcard 10: What is the distance formula for the hypotenuse when legs are $a$ and $b$ in a right triangle?

Answer: $c=\sqrt{a^2+b^2}$. Take the square root of the sum of squared legs.

Flashcard 11: Determine the longest side in a triangle with sides 7, 10, 5.

Answer:

1. The longest side is opposite the largest angle.

Flashcard 12: What is the side opposite the right angle in a right triangle called?

Answer: Hypotenuse. The longest side in a right triangle, opposite the right angle.

Flashcard 13: What is the altitude in an equilateral triangle with side $s$?

Answer: $\frac{\sqrt{3}}{2} \times s$. Height of equilateral triangle using special right triangle ratios.

Flashcard 14: What is the condition for a triangle to be obtuse using side lengths with largest side $c$?

Answer: $a^2+b^2<c^2$. When legs squared are less than hypotenuse squared, one angle is obtuse.

Flashcard 15: What is the formula for the area of an equilateral triangle with side $s$?

Answer: $\frac{\sqrt{3}}{4} \times s^2$. Derived from the general triangle area formula for equilateral triangles.

Flashcard 16: What is the Triangle Inequality Theorem stated in terms of side lengths $a$, $b$, and $c$?

Answer: $a+b>c$, $a+c>b$, and $b+c>a$. Each side must be less than the sum of the other two sides.

Flashcard 17: What is the definition of an isosceles triangle in terms of side lengths?

Answer: At least two sides are congruent. Two or more equal sides define an isosceles triangle.

Flashcard 18: What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Pythagorean theorem to coordinate differences.

Flashcard 19: Which triangle inequality must hold for any triangle?

Answer: The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.

Flashcard 20: What is the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

Answer: $1:1:\sqrt{2}$. Isosceles right triangle has hypotenuse $\sqrt{2}$ times leg.

Flashcard 21: Find the length of the longer leg in a 30-60-90 triangle with short leg 5.

Answer: $5 \sqrt{3}$. In 30-60-90 triangle, long leg is short leg times $\sqrt{3}$.

Flashcard 22: What is the incenter of a triangle?

Answer: The point where the angle bisectors meet. Center of the circle inscribed within the triangle.

Flashcard 23: Given a triangle with sides 7, 24, 25, classify it.

Answer: Right triangle. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so right triangle.

Flashcard 24: What is the sum of interior angles in a triangle?

Answer: 180°. Sum of all angles in any triangle is always constant.

Flashcard 25: What is each interior angle of an equilateral triangle?

Answer: $60^\circ$. $180^\circ \div 3 = 60^\circ$ for each equal angle.

Flashcard 26: What is each interior angle of an equilateral triangle?

Answer: $60^\circ$. $180^\circ \div 3 = 60^\circ$ for each equal angle.

Flashcard 27: State the Pythagorean Theorem.

Answer: $a^2 + b^2 = c^2$. Relates the squares of sides in a right triangle.

Flashcard 28: Find the hypotenuse in a 45-45-90 triangle with legs of length 7.

Answer: $7\sqrt{2}$. In 45-45-90 triangle, hypotenuse equals leg times $\sqrt{2}$.

Flashcard 29: What is the definition of an equilateral triangle in terms of side lengths?

Answer: All three sides are congruent. Equal sides create an equilateral triangle.

Flashcard 30: Identify whether sides $5$, $6$, and $7$ form an acute, right, or obtuse triangle.

Answer: Acute. $5^2 + 6^2 = 61 > 49 = 7^2$, so acute.

Flashcard 31: What is the altitude in a triangle?

Answer: A perpendicular segment from a vertex to the opposite side. Height drops perpendicularly from vertex to opposite side.

Flashcard 32: What is the area of a right triangle in terms of its legs $a$ and $b$?

Answer: $A=\frac{1}{2}ab$. Right triangles use legs as base and height.

Flashcard 33: What is the orthocenter of a triangle?

Answer: The point where the altitudes meet. Intersection point of all three altitude lines.

Flashcard 34: Identify the type of triangle with angles 90°, 45°, 45°.

Answer: Isosceles right triangle. Right triangle with two equal legs and equal base angles.

Flashcard 35: What type of triangle has no sides of equal length?

Answer: Scalene triangle. All three sides have different lengths.

Flashcard 36: What is the perimeter of a triangle with side lengths $7$, $8$, and $9$?

Answer: $24$. $7 + 8 + 9 = 24$

Flashcard 37: What is the Heron's formula for the area of a triangle?

Answer: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$. Calculates area using only the three side lengths.

Flashcard 38: State the Cosine Rule for triangle sides $a$, $b$, $c$ and angle $C$.

Answer: $c^2 = a^2 + b^2 - 2ab \text{cos} C$. General formula connecting all sides and one angle.

Flashcard 39: What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$ of a segment?

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

Flashcard 40: Given a triangle with sides 7, 24, 25, classify it.

Answer: Right triangle. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so right triangle.

Flashcard 41: Identify the type of triangle with angles 90°, 45°, 45°.

Answer: Isosceles right triangle. Right triangle with two equal legs and equal base angles.

Flashcard 42: What is the missing leg of a right triangle with hypotenuse $13$ and one leg $5$?

Answer: $12$. $\sqrt{13^2 - 5^2} = \sqrt{144} = 12$

Flashcard 43: What is the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

Answer: $1:1:\sqrt{2}$. Isosceles right triangle has hypotenuse $\sqrt{2}$ times leg.

Flashcard 44: Identify the missing angle: a triangle has angles $50^\circ$ and $60^\circ$; what is the third angle?

Answer: $70^\circ$. $180^\circ - 50^\circ - 60^\circ = 70^\circ$

Flashcard 45: Determine the longest side in a triangle with sides 7, 10, 5.

Answer:

1. The longest side is opposite the largest angle.

Flashcard 46: Identify the triangle inequality theorem.

Answer: The sum of any two sides must be greater than the third side. Ensures a valid triangle can be formed with three given lengths.

Flashcard 47: What is the sum of interior angles in any triangle?

Answer: 180 degrees. This is a fundamental property that applies to all triangles.

Flashcard 48: What is the definition of a right triangle?

Answer: A triangle with one 90-degree angle. Contains exactly one angle measuring $90°$.

Flashcard 49: What is a median in a triangle?

Answer: A segment from a vertex to the midpoint of the opposite side. Connects vertex to midpoint of opposite side.

Flashcard 50: What is the definition of a right triangle?

Answer: A triangle with one 90-degree angle. Contains exactly one angle measuring $90°$.

Flashcard 51: Find the missing angle if two angles are 45° and 60°.

Answer: 75°. Sum of angles is $180°$: $180° - 45° - 60° = 75°$.

Flashcard 52: What is an altitude in a triangle?

Answer: A perpendicular segment from a vertex to the opposite side. Creates the shortest distance from vertex to opposite side.

Flashcard 53: Find the hypotenuse in a 30-60-90 triangle with short leg 5.

Answer:

1. In 30-60-90 triangle, hypotenuse is twice the short leg.

Flashcard 54: What is the semi-perimeter of a triangle with sides $a$, $b$, $c$?

Answer: $\frac{a+b+c}{2}$. Half the total perimeter, used in area formulas.

Flashcard 55: What is the orthocenter of a triangle?

Answer: The intersection of the altitudes. Point where all three altitudes intersect.

Flashcard 56: What is the measure of each angle in an equilateral triangle?

Answer: 60 degrees. Since all sides are equal, all angles are equal: $180° ÷ 3 = 60°$.

Flashcard 57: What is the altitude in a triangle?

Answer: A perpendicular segment from a vertex to the opposite side. Height drops perpendicularly from vertex to opposite side.

Flashcard 58: What is the area formula for a triangle with base $b$ and height $h$?

Answer: $A=\frac{1}{2}bh$. Base times height divided by 2 gives triangle area.

Flashcard 59: What is the hypotenuse of a right triangle with legs $6$ and $8$?

Answer: $10$. $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$

Flashcard 60: What is the name of a triangle with angles 60°, 60°, 60°?

Answer: Equilateral triangle. All angles are equal, so each measures $60°$.

Flashcard 61: Which formula represents the Law of Cosines?

Answer: $c^2 = a^2 + b^2 - 2ab \times \cos C$. Generalizes Pythagorean theorem for any triangle using cosine.

Flashcard 62: Identify whether side lengths $2$, $3$, and $6$ can form a triangle using the triangle inequality.

Answer: No, it cannot form a triangle. $2 + 3 = 5 < 6$, violating triangle inequality.

Flashcard 63: Find the third angle in a triangle with angles 35° and 65°.

Answer: 80°. Sum of angles is $180°$: $180° - 35° - 65° = 80°$.

Flashcard 64: What is the perimeter of a triangle with side lengths $7$, $8$, and $9$?

Answer: $24$. $7 + 8 + 9 = 24$

Flashcard 65: What is the area of a right triangle in terms of its legs $a$ and $b$?

Answer: $A=\frac{1}{2}ab$. Right triangles use legs as base and height.

Flashcard 66: Identify the triangle inequality theorem.

Answer: The sum of any two sides must be greater than the third side. Ensures a valid triangle can be formed with three given lengths.

Flashcard 67: Which triangle has one angle greater than 90°?

Answer: Obtuse triangle. One angle exceeds $90°$, making it wider than a right angle.

Flashcard 68: Find the length of the hypotenuse: legs are 3 and 4.

Answer:

1. Using Pythagorean theorem: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Flashcard 69: What is the definition of an exterior angle in a triangle?

Answer: An angle formed by one side and the extension of another. Formed by extending one side beyond a vertex.

Flashcard 70: What is the definition of an obtuse triangle in terms of angles?

Answer: One angle is greater than $90^\circ$. One wide angle defines an obtuse triangle.

Flashcard 71: What is the sum of the exterior angles of any triangle?

Answer: 360 degrees. Constant sum regardless of triangle shape or size.

Flashcard 72: Calculate the area of a triangle with base 10 and height 5.

Answer: 25 square units. Using $\frac{1}{2} \times 10 \times 5 = 25$.

Flashcard 73: What is the slope formula for points $(x_1, y_1)$ and $(x_2, y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

Flashcard 74: Identify the type of triangle with sides 5, 5, and 8.

Answer: Isosceles triangle. Two sides are equal ($5$ each), third side different.

Flashcard 75: What is the centroid of a triangle?

Answer: The point where the medians meet. Balance point where triangle would rest on a pin.

Flashcard 76: What is the definition of an isosceles triangle in terms of side lengths?

Answer: At least two sides are congruent. Two or more equal sides define an isosceles triangle.

Flashcard 77: State the Cosine Rule for triangle sides $a$, $b$, $c$ and angle $C$.

Answer: $c^2 = a^2 + b^2 - 2ab \cos C$. General formula connecting all sides and one angle.

Flashcard 78: What is the definition of an equilateral triangle in terms of side lengths?

Answer: All three sides are congruent. Equal sides create an equilateral triangle.

Flashcard 79: What is an altitude in a triangle?

Answer: A perpendicular segment from a vertex to the opposite side. Creates the shortest distance from vertex to opposite side.

Flashcard 80: What is the definition of an exterior angle in a triangle?

Answer: An angle formed by one side and the extension of another. Formed by extending one side beyond a vertex.

Flashcard 81: Find the length of the hypotenuse: legs are 3 and 4.

Answer:

1. Using Pythagorean theorem: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Flashcard 82: What is the definition of a congruent triangle?

Answer: Triangles with equal corresponding angles and sides. Triangles that are identical in shape and size.

Flashcard 83: State the formula for the area of a triangle.

Answer: $\frac{1}{2} \times \text{base} \times \text{height}$. Standard formula where height is perpendicular distance to the base.

Flashcard 84: What is the angle opposite the largest side in a triangle?

Answer: Largest angle. In any triangle, the largest angle faces the longest side.

Flashcard 85: What type of triangle has no sides of equal length?

Answer: Scalene triangle. All three sides have different lengths.

Flashcard 86: Define an isosceles triangle.

Answer: A triangle with two equal sides. Two sides have equal length, creating two equal angles.

Flashcard 87: What do the exterior angles of a triangle sum to?

Answer: 360°. Each exterior angle plus its interior angle equals $180°$.

Flashcard 88: Find the third angle in a triangle with angles 35° and 65°.

Answer: 80°. Sum of angles is $180°$: $180° - 35° - 65° = 80°$.

Flashcard 89: Identify whether sides $3$, $4$, and $5$ form a right triangle using $a^2+b^2=c^2$.

Answer: Yes, it is a right triangle. $3^2 + 4^2 = 9 + 16 = 25 = 5^2$

Flashcard 90: What is the definition of a scalene triangle in terms of side lengths?

Answer: No sides are congruent. All sides have different lengths in a scalene triangle.

Flashcard 91: What is the distance formula for the hypotenuse when legs are $a$ and $b$ in a right triangle?

Answer: $c=\sqrt{a^2+b^2}$. Take the square root of the sum of squared legs.

Flashcard 92: Identify the angle bisector theorem.

Answer: Divides the opposite side into segments proportional to the other two sides. Angle bisector creates proportional segments on the opposite side.

Flashcard 93: What is the semi-perimeter of a triangle with sides $a$, $b$, $c$?

Answer: $\frac{a+b+c}{2}$. Half the total perimeter, used in area formulas.

Flashcard 94: Which formula represents the Law of Sines?

Answer: $\frac{a}{\text{sin } A} = \frac{b}{\text{sin } B} = \frac{c}{\text{sin } C}$. Relates sides to opposite angles in any triangle.

Flashcard 95: What is the Pythagorean Theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$?

Answer: $a^2+b^2=c^2$. Sum of squares of legs equals square of hypotenuse.

Flashcard 96: State the formula for the area of an equilateral triangle with side $s$.

Answer: $\frac{\sqrt{3}}{4} s^2$. Derived from base-height formula using equilateral properties.

Flashcard 97: Calculate the perimeter of a triangle with sides 3, 4, 5.

Answer:

1. Sum of all three sides: $3 + 4 + 5 = 12$.

Flashcard 98: What is the relationship between an exterior angle and its remote interior angles?

Answer: Exterior angle = sum of remote interior angles. Exterior angle equals sum of the two non-adjacent interior angles.

Flashcard 99: Define a scalene triangle.

Answer: A triangle with all sides of different lengths. No two sides have the same length.

Flashcard 100: What type of triangle has all sides of equal length?

Answer: Equilateral triangle. All three sides are congruent in this regular triangle.