Triangles
45 flashcards covering Triangles for the ACT Math section.
Triangles are one of the most fundamental shapes in geometry, consisting of three straight sides that connect to form three angles. They come in various types, such as equilateral, isosceles, and scalene, based on their side lengths and angles. Understanding triangles helps you grasp concepts like area, perimeter, and the Pythagorean theorem, which are essential for solving real-world problems in fields like engineering and architecture. On the ACT, triangles matter because they frequently test your ability to apply geometric principles to practical scenarios.
On the ACT Math section, triangle questions often involve calculating areas, identifying similar or congruent triangles, or using the Pythagorean theorem in word problems. Common traps include overlooking the triangle inequality theorem or confusing acute and obtuse angles, which can lead to incorrect assumptions. Focus on mastering key formulas, visualizing diagrams accurately, and practicing multi-step problems to build speed and accuracy.
Remember to double-check your angle sums—every triangle adds up to 180 degrees.
Terms (45)
- 01
Equilateral triangle
A triangle with all three sides equal in length and all three angles equal to 60 degrees.
- 02
Isosceles triangle
A triangle with at least two sides equal in length, which also means it has at least two equal angles.
- 03
Scalene triangle
A triangle with all three sides of different lengths and all three angles of different measures.
- 04
Acute triangle
A triangle where all three interior angles are less than 90 degrees.
- 05
Obtuse triangle
A triangle with one interior angle greater than 90 degrees and the other two less than 90 degrees.
- 06
Right triangle
A triangle with one interior angle exactly equal to 90 degrees.
- 07
Area of a triangle
The measure of the space inside a triangle, calculated using the formula (1/2) times base times height.
- 08
Perimeter of a triangle
The total length around a triangle, found by adding the lengths of all three sides.
- 09
Pythagorean theorem
In a right triangle, the relationship where the square of the hypotenuse equals the sum of the squares of the other two sides.
- 10
Triangle inequality theorem
A rule stating that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- 11
Congruent triangles
Two triangles that are identical in shape and size, meaning all corresponding sides and angles are equal.
- 12
Similar triangles
Two triangles that have the same shape but not necessarily the same size, with equal corresponding angles and proportional sides.
- 13
Side-Side-Side (SSS) congruence
A criterion for triangle congruence where if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
- 14
Side-Angle-Side (SAS) congruence
A criterion for triangle congruence where two sides and the included angle of one triangle are equal to two sides and the included angle of another.
- 15
Angle-Side-Angle (ASA) congruence
A criterion for triangle congruence where two angles and the included side of one triangle are equal to two angles and the included side of another.
- 16
Angle-Angle-Side (AAS) congruence
A criterion for triangle congruence where two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another.
- 17
Hypotenuse-Leg (HL) congruence
A criterion for right triangle congruence where the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another.
- 18
Angle-Angle (AA) similarity
A criterion for triangle similarity where two angles of one triangle are equal to two angles of another triangle.
- 19
Side-Side-Side (SSS) similarity
A criterion for triangle similarity where the corresponding sides of two triangles are proportional.
- 20
Side-Angle-Side (SAS) similarity
A criterion for triangle similarity where two sides are proportional and the included angles are equal.
- 21
triangle
A special right triangle with angles of 30 degrees, 60 degrees, and 90 degrees, where the sides are in the ratio 1 : √3 : 2.
- 22
Sine ratio
In a right triangle, the ratio of the length of the opposite side to the hypotenuse for a given acute angle.
- 23
Cosine ratio
In a right triangle, the ratio of the length of the adjacent side to the hypotenuse for a given acute angle.
- 24
Tangent ratio
In a right triangle, the ratio of the length of the opposite side to the adjacent side for a given acute angle.
- 25
Opposite side
In a right triangle, the side directly across from a given acute angle.
- 26
Adjacent side
In a right triangle, the side next to a given acute angle, excluding the hypotenuse.
- 27
Hypotenuse
In a right triangle, the longest side, which is opposite the right angle.
- 28
Sum of angles in a triangle
The total measure of the three interior angles of any triangle, which always equals 180 degrees.
- 29
Exterior angle theorem
The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.
- 30
Median of a triangle
A line segment joining a vertex to the midpoint of the opposite side.
- 31
Altitude of a triangle
A perpendicular line segment from a vertex to the line containing the opposite side.
- 32
Angle bisector
A line that divides an angle of a triangle into two equal angles.
- 33
Perpendicular bisector
A line that cuts a side of a triangle into two equal parts at a right angle.
- 34
Centroid of a triangle
The point where the three medians intersect, also the center of mass of the triangle.
- 35
Orthocenter of a triangle
The point where the three altitudes intersect.
- 36
Circumcenter of a triangle
The center of the circle that passes through all three vertices of the triangle.
- 37
Incenter of a triangle
The center of the circle that is tangent to all three sides of the triangle.
- 38
Heron's formula
A method to calculate the area of a triangle when all three side lengths are known, using the square root of [s(s-a)(s-b)(s-c)], where s is the semi-perimeter.
- 39
Strategy for solving similar triangles
Identify corresponding angles and set up proportions between corresponding sides to find unknown lengths.
- 40
Common mistake in Pythagorean theorem
Assuming it applies to non-right triangles, which it does not, as it only works for right triangles.
- 41
Example of area calculation
To find the area of a triangle with base 5 and height 10, multiply 5 by 10 and divide by 2, resulting in 25 square units.
For a triangle with base 5 units and height 10 units, area is (1/2)510 = 25 square units.
- 42
Example of Pythagorean theorem
In a right triangle with legs 3 and 4, the hypotenuse is found by calculating the square root of (3 squared plus 4 squared), which is 5.
For legs of 3 and 4, hypotenuse = √(9 + 16) = √25 = 5.
- 43
Example of triangle inequality
For sides 3, 4, and 7, check if 3 + 4 > 7 (7 > 7 is false), so these lengths cannot form a triangle.
Sides 3, 4, and 7 do not satisfy the inequality because 3 + 4 is not greater than 7.
- 44
Example of congruent triangles
Two triangles with sides 5, 5, 6 and angles 70, 70, 40 are congruent by SAS if the equal sides include the 70-degree angle.
Triangle ABC with sides AB=5, BC=5, angle B=70° is congruent to another with the same.
- 45
Example of similar triangles
Two triangles with angles 30, 60, 90 and sides in ratio 1:2 are similar, as their angles match.
A triangle with sides 3, 4√3, 7.5 is similar to one with sides 6, 8√3, 15.