If angles of a triangle are in the ration of 2 : 3 : 4, then the measure of the smallest angle is:
40°
Solving for Triangle Angles Given Ratio
The question provides the ratio of the angles of a triangle as \(2 : 3 : 4\). We need to find the measure of the smallest angle.
Understanding Triangle Angle Properties
A fundamental property of any triangle is that the sum of its interior angles is always \(180^\circ\). This property is key to solving this problem.
Representing the Angles
Since the angles are in the ratio \(2 : 3 : 4\), we can represent the angles as \(2x\), \(3x\), and \(4x\), where \(x\) is a common multiplier.
Setting Up the Equation
Using the property that the sum of angles in a triangle is \(180^\circ\), we can write the equation:
\[2x + 3x + 4x = 180^\circ\]
Solving for x
Combine the terms on the left side of the equation:
\[(2 + 3 + 4)x = 180^\circ\]
\[9x = 180^\circ\]
Now, divide both sides by 9 to find the value of \(x\):
\[x = \frac{180^\circ}{9}\]
\[x = 20^\circ\]
Calculating the Measure of Each Angle
Now that we have the value of \(x\), we can find the measure of each angle:
- First angle = \(2x = 2 \times 20^\circ = 40^\circ\)
- Second angle = \(3x = 3 \times 20^\circ = 60^\circ\)
- Third angle = \(4x = 4 \times 20^\circ = 80^\circ\)
Identifying the Smallest Angle
Comparing the three angle measures \(40^\circ\), \(60^\circ\), and \(80^\circ\), the smallest angle is \(40^\circ\).
Summary of Angles
| Ratio Part | Expression | Angle Measure |
|---|---|---|
| 2 | \(2x\) | \(40^\circ\) |
| 3 | \(3x\) | \(60^\circ\) |
| 4 | \(4x\) | \(80^\circ\) |
The sum of these angles is \(40^\circ + 60^\circ + 80^\circ = 180^\circ\), which confirms our calculation is correct.
Conclusion
The measure of the smallest angle in the triangle is \(40^\circ\).
Revision Table: Triangle Angles Ratio
| Concept | Key Idea | Application in Problem |
|---|---|---|
| Sum of angles in a triangle | Always \(180^\circ\) | Used to form the equation \(2x + 3x + 4x = 180^\circ\) |
| Angles in ratio | Represent angles as multiples of a variable (e.g., \(2x, 3x, 4x\)) | Used to set up terms for the equation |
| Solving linear equation | Find the value of the unknown variable \(x\) | Calculated \(x = 20^\circ\) |
| Finding angle measures | Substitute the value of \(x\) back into angle expressions | Calculated angles as \(40^\circ, 60^\circ, 80^\circ\) |
Additional Information: Types of Triangles by Angles
Triangles can be classified based on their angle measures:
- Acute Triangle: All three angles are less than \(90^\circ\). The triangle in this problem (angles \(40^\circ, 60^\circ, 80^\circ\)) is an acute triangle.
- Right Triangle: One angle is exactly \(90^\circ\). The other two angles must be acute.
- Obtuse Triangle: One angle is greater than \(90^\circ\). The other two angles must be acute.
Understanding the sum of angles and how to work with ratios is fundamental in solving various geometry problems related to triangles.
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