A triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).
Every triangle has angles that add to 180°. Tick marks show equal sides, and a small square marks a right angle.
Formal definition and notation
A triangle is the figure formed by three noncollinear points A, B, C and the three line segmentsAB, BC and CA joining them. Its interior angles satisfy ∠ A + ∠ B + ∠ C = 180°, and its sidelengths satisfy the triangle inequality: each side is shorter than the sum of the other two.
Symbol
Read it as
Meaning
△ ABC
triangle A B C
the triangle with vertices A, B and C
∠ A
angle A
the angle at vertex A
AB
segment A B
the side joining A and B
tick marks
tick marks
sides with the same number of ticks are equal in length
Key facts
The angle measures in any triangle add to 180 degrees, whatever its size or shape.
A triangle can have at most one right angle or one obtuse angle; the other two angles are always acute.
By sides: equilateral (3 equal sides), isosceles (at least 2 equal sides), scalene (no equal sides).
By angles: acute (all angles under 90°), right (one 90° angle), obtuse (one angle over 90°).
An angle is formed by two rays from one point. A triangle has three angles, one at each vertex, and three sides.
Quick check: 3 questions
Question 1 of 3
Can side lengths of 5 cm, 9 cm and 15 cm form a triangle?
Show answer and explanation
Answer: B. The two shorter sides must add to more than the longest: 5 + 9 = 14, which is less than 15, so they cannot meet. Choices C and D give reasons that do not decide anything; A ignores the triangle inequality.
Question 2 of 3
A triangle has angles of 90°, 45° and 45°. How is it classified?
Show answer and explanation
Answer: D. One 90° angle makes it right. Two equal 45° angles mean the sides opposite them are equal, so it is isosceles. An equilateral triangle needs three 60° angles.
Question 3 of 3
Two interior angles of a triangle are 65° and 45°. What is the exterior angle at the third vertex?
Show answer and explanation
Answer: C. The exterior angle equals the sum of the two remote interior angles: 65 + 45 = 110°. Choice A is the third interior angle (180 − 110), which is next to the exterior angle, not equal to it.
Triangle: frequently asked questions
Why do the angles of a triangle add up to 180 degrees?
Draw a line through one vertex parallel to the opposite side. The three angles at that vertex lie along the straight line, so they add to 180°. Two of them are equal to the triangle’s other two angles (alternate interior angles), and the middle one is the triangle’s own angle. So the triangle’s three angles also add to 180°.
What are the types of triangles?
By sides: equilateral (all three sides equal), isosceles (at least two equal) and scalene (no equal sides). By angles: acute (all angles less than 90°), right (one angle exactly 90°) and obtuse (one angle greater than 90°). Every triangle gets one name from each list, such as right scalene or acute isosceles.
How do you find the area of a triangle?
Use A = 12bh: half the base times the height, where the height is perpendicular to the base. A triangle with base 10 cm and height 6 cm has area 12 × 10 × 6 = 30 square centimeters. It is half of the rectangle that has the same base and height.
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