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.
Important Questions from Lines and Angles
In the triangle, if AB = AC and ∠ABC = 72°, then ∠BAC is:
The angles of a triangle are (8x - 15)°,(6x - 11)° and ( 4x – 10)°. What is the value of x ?
In a ΔABC, the bisectors of ∠B and ∠C meet at point O, inside the triangle. If ∠BOC = 122°, then the measure of ∠A is:
In ΔABC, D is a point on side BC such that ∠ADC = 2∠BAD. If ∠A = 80° and ∠C = 38°, then what is the measure of ∠ADB?
In ∆ABC, ∠B = 68° and ∠C = 32°. Sides AB and AC are produced to points D and E respectively. The bisectors of ∠DBC and ∠BCE meet at F. what is the measure of ∠BFC?
