Geometry Puzzle – Challenge 62
In a triangle whose three angles are in the ratio 1:2:3, what is the smallest angle? The angles must sum to 180 degrees, so if the angles are X, 2X, and 3X, then X + 2X + 3X = 6X = 180, giving X = 30 degrees. The triangle is the classic 30-60-90 right triangle.
Key takeaways:
- The interior angles of any triangle sum to 180 degrees.
- When angles are in the ratio 1:2:3, the angle pieces are X, 2X, 3X.
- Summing: X + 2X + 3X = 6X = 180, so X = 30. The angles are 30, 60, and 90 degrees.
- This is the famous 30-60-90 right triangle, the side ratios are 1: sqrt(3): 2.
- Don’t confuse the angle ratio (1:2:3, possible) with the side ratio (1:2:3, impossible by triangle inequality).
This is a great mathematics puzzle to measure your basic math knowledge. Let’s challenge your brain!
Challenge:
The ratio of three sides of a triangle is 1:2:3. What is the smallest angle?
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The correct answer is 30 degree.
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There is the same ratio for the angels. Therefore, the ratio of three angles of the triangle is also 1:2:3.
Let X be the smallest angle. Then, the next two angels are 2X and 3X.
X + 2X + 3X = 180 → 6X = 180 → X = 30
The smallest angel is 30 degree.The Best Books to Ace Algebra
Frequently Asked Questions
How do you solve math puzzles step by step?
Read the puzzle slowly, write down what is given and what is asked, then pick a rule that links them. For a triangle with angles in the ratio 1:2:3, call the angles x, 2x and 3x. The angle sum rule gives x + 2x + 3x = 180, so 6x = 180 and x = 30. Finally check: 30 + 60 + 90 = 180.
How do you solve math shape puzzles?
Start with the fixed facts about the shape, such as the angles of a triangle adding to 180° or a rectangle’s opposite sides being equal. Then turn the clues into an equation. In this triangle puzzle the ratio 1:2:3 becomes 6 equal parts of 180°, so each part is 30° and the angles are 30°, 60° and 90°.
Are puzzles considered math?
Many puzzles are math, because solving them uses the same reasoning: spotting patterns, setting up equations and checking answers. This geometry puzzle uses ratios and the triangle angle sum. It also has a lesson in reading carefully: angles in a 1:2:3 ratio form a 30-60-90 triangle, but sides in a 1:2:3 ratio cannot form a triangle, since 1 + 2 is not greater than 3.
What are good math challenge puzzles for 6th grade?
Good 6th grade puzzles use one key fact and a little algebra. Angle-ratio triangles are a great example: with angles in the ratio 2:3:4, there are 9 parts of 180°, so each part is 20° and the angles are 40°, 60° and 80°. Try 1:1:2 next, which gives 45°, 45° and 90°.
Related Lessons You May Like
- How to classify triangles
- How to find the area of triangles
- How to use the Pythagorean Theorem
- How to find complementary, supplementary, vertical, adjacent angles
- How to find similar figures
If your student enjoys puzzles like this, Pre-Algebra for Beginners covers the algebraic reasoning they tap. For deeper geometry exploration, Geometry for Beginners takes you further.
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