How to Solve Triangles Problems? (+FREE Worksheet!)
Triangles are the simplest polygon — a closed figure with three sides, three angles, and three vertices. Understanding triangle types and their properties is a foundational GED geometry skill. This lesson covers classification, the angle-sum rule, perimeter, and several worked examples.
What Is a Triangle?
A triangle is a polygon with exactly three sides and three interior angles. The sum of the three angles always equals 180°. This is true for every triangle, no matter its size or shape.
angle \(\color{blue}{A + \text{ angle }}\) \(\color{blue}{B + \text{ angle }}\) \(\color{blue}{C = 180}\)°
Types of Triangles
By Side Length
- Equilateral triangle: All three sides are equal in length; all three angles are 60°.
- Isosceles triangle: Exactly two sides are equal; the angles opposite the equal sides are also equal.
- Scalene triangle: All three sides have different lengths; all three angles are different.
By Angle
- Acute triangle: All three angles are less than 90°.
- Right triangle: One angle is exactly 90°. The longest side (opposite the right angle) is called the hypotenuse.
- Obtuse triangle: One angle is greater than 90°.
Perimeter of a Triangle
Add all three side lengths: \(\color{blue}{P = a + b + c}\). For sides 5, 7, and 9: \(\color{blue}{P = 5 + 7 + 9 = 21}\).
Step-by-Step Summary
- Count the sides to confirm it is a triangle (three sides).
- Check the sides to classify: equal \(\color{blue}{\text{ sides } = \text{ equilateral }}\) or isosceles; all \(\color{blue}{\text{ different } = \text{ scalene }}\).
- Check the angles: all <90° = acute; \(\color{blue}{\text{ one } = 90}\)° = right; one >90° = obtuse.
- Use \(\color{blue}{A + B + C = 180^{\circ}}\) to find a missing angle.
- Add all sides to find the perimeter.
Watch: Math Antics — Triangles (Video Lesson)
Math Antics walks through every type of triangle with clear diagrams and examples:
Worked Examples
Example 1: A triangle has angles 50° and 70°. Find the third angle.
\(\color{blue}{50 + 70 + x = 180}\) → \(\color{blue}{x = 180 – 120}\) = 60°. The triangle is acute (all angles < 90°).
Example 2: Classify the triangle with sides 5, 5, and 8.
Two equal sides → isosceles. Angles are not all equal.
Example 3: A right triangle has legs of length 3 and 4. What is the hypotenuse?
\(\color{blue}{3^{2} + 4^{2} = 9 + 16 = 25}\) → hypotenuse = √25 = 5. This is the famous 3-4-5 right triangle.
Example 4: Find the perimeter of a triangle with sides 5, 7, and 9.
\(\color{blue}{P = 5 + 7 + 9}\) = 21 units.
More Practice: Triangle Inequality Theorem (Video)
Khan Academy explains the triangle inequality and how side lengths constrain angle sizes:
Exercises
- A triangle has angles 40° and 65°. Find the third angle.
- Classify a triangle with all three sides equal to 6 cm.
- A triangle has angles 90°, 45°, and 45°. What type is it?
- Find the perimeter of a triangle with sides 8, 10, and 12.
- One angle of a triangle is 110°. What type of triangle is it?
- A triangle has sides 7, 7, and 10. Classify it by sides and find the perimeter.
Answers
- \(\color{blue}{180 – 40 – 65}\) = 75°
- Equilateral (and equiangular — all angles 60°)
- Right isosceles triangle
- \(\color{blue}{8 + 10 + 12}\) = 30 units
- Obtuse triangle (one angle > 90°)
- Isosceles; \(\color{blue}{P = 7 + 7 + 10}\) = 24 units
Frequently Asked Questions
Why do the angles of a triangle always add up to 180°?
This comes from the parallel postulate of Euclidean geometry. If you extend one side of a triangle and draw a line parallel to the opposite side through a vertex, the three angles form a straight line (180°).
Can a triangle have two right angles?
No. If two angles were each 90°, their sum would already be 180°, leaving 0° for the third angle — which is impossible.
What is the difference between an isosceles and an equilateral triangle?
An equilateral triangle has all three sides (and all three angles) equal. An isosceles triangle has exactly two sides equal — all equilateral triangles are also isosceles, but not vice versa.
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