How to Solve Triangles Problems? (+FREE Worksheet!)

How to solve triangles problems, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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

1. Count the sides to confirm it is a triangle (three sides).
2. Check the sides to classify: equal (color{blue}{text{ sides } = text{ equilateral }}) or isosceles; all (color{blue}{text{ different } = text{ scalene }}).
3. Check the angles: all <90° = acute; (color{blue}{text{ one } = 90})° = right; one >90° = obtuse.
4. Use (color{blue}{A + B + C = 180^{circ}}) to find a missing angle.
5. 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

1. A triangle has angles 40° and 65°. Find the third angle.
2. Classify a triangle with all three sides equal to 6 cm.
3. A triangle has angles 90°, 45°, and 45°. What type is it?
4. Find the perimeter of a triangle with sides 8, 10, and 12.
5. One angle of a triangle is 110°. What type of triangle is it?
6. A triangle has sides 7, 7, and 10. Classify it by sides and find the perimeter.

Answers

1. (color{blue}{180 – 40 – 65}) = 75°
2. Equilateral (and equiangular, all angles 60°)
3. Right isosceles triangle
4. (color{blue}{8 + 10 + 12}) = 30 units
5. Obtuse triangle (one angle > 90°)
6. Isosceles; (color{blue}{P = 7 + 7 + 10}) = 24 units

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Frequently Asked Questions

What are the types of triangles?

Triangles are classified by sides and by angles. By sides, an equilateral triangle has three equal sides, an isosceles triangle has at least two, and a scalene triangle has none. By angles, an acute triangle has three angles under 90°, a right triangle has one 90° angle, and an obtuse triangle has one angle over 90°. A triangle gets one name from each list, such as right isosceles.

Do the angles of a triangle add up to 180?

Yes. The three interior angles of any triangle always add up to 180°. That makes missing angles quick to find: if two angles are 45° and 70°, the third is 180° − 115° = 65°. It also explains why a triangle can have at most one right angle or one obtuse angle.

How do you find the area of a triangle?

Multiply the base by the height and divide by 2: A = ½ × b × h. The height must be perpendicular to the base, so it may fall outside an obtuse triangle. For a triangle with base 10 cm and height 6 cm, A = ½ × 10 × 6 = 30 cm². In a right triangle, the two legs work as base and height.

How do you know if triangles are congruent?

Two triangles are congruent if they have the same size and shape. You can prove it without checking all six parts by using SSS (three pairs of equal sides), SAS, ASA, AAS, or HL for right triangles. Watch out: AAA and SSA do not prove congruence, because two triangles can match those parts and still be different.

How do you know if triangles are similar?

Similar triangles have the same shape but not necessarily the same size. They are similar if two pairs of angles are equal (AA), if all three side ratios match (SSS similarity), or if two side ratios match with an equal included angle (SAS similarity). A 3-4-5 triangle and a 6-8-10 triangle are similar because every side doubles.

Why are triangles the strongest shape?

Triangles are rigid: once the three side lengths are fixed, the shape cannot change without bending or breaking a side. A square frame can be pushed into a slanted rhombus, but a triangle cannot be squashed that way. That is why bridges, roof trusses, and cranes are built from triangles. It is the same idea as SSS: three sides determine one triangle.

Related Topics

• Triangle Inequality
• Relationship Between Sides and Angles in a Triangle
• Area of a Triangle
• Polygons
• Area of a Parallelogram