To determine which of the statements regarding triangles is true, we will analyze each statement step by step.
### Step 1: Analyze Statement 1
**Statement 1:** If in a triangle, two angles are equal to 60 degrees, then the triangle is equilateral.
1. Let's denote the angles of the triangle as A, B, and C.
2. Given: A = 60°, B = 60°.
3. We can find angle C using the angle sum property of triangles, which states that the sum of the angles in a triangle is 180°.
\[
A + B + C = 180°
\]
Substituting the known values:
\[
60° + 60° + C = 180°
\]
Simplifying:
\[
120° + C = 180°
\]
\[
C = 180° - 120° = 60°
\]
4. Since all angles A, B, and C are equal to 60°, the triangle is equilateral.
**Conclusion for Statement 1:** True.
### Step 2: Analyze Statement 2
**Statement 2:** If the angles of a triangle are in the ratio of 1:1:2, then it is a right angle isosceles triangle.
1. Let the angles be X, X, and 2X.
2. Using the angle sum property:
\[
X + X + 2X = 180°
\]
Simplifying:
\[
4X = 180°
\]
\[
X = \frac{180°}{4} = 45°
\]
3. Therefore, the angles are:
- Angle 1 = 45°
- Angle 2 = 45°
- Angle 3 = 2 * 45° = 90°
4. Since two angles are equal (45°) and one angle is 90°, the triangle is indeed an isosceles right triangle.
**Conclusion for Statement 2:** True.
### Step 3: Analyze Statement 3
**Statement 3:** If the angles of a triangle are in the ratio of 1:2:3, then it is a right angle triangle.
1. Let the angles be X, 2X, and 3X.
2. Using the angle sum property:
\[
X + 2X + 3X = 180°
\]
Simplifying:
\[
6X = 180°
\]
\[
X = \frac{180°}{6} = 30°
\]
3. Therefore, the angles are:
- Angle 1 = 30°
- Angle 2 = 2 * 30° = 60°
- Angle 3 = 3 * 30° = 90°
4. Since one angle is 90°, the triangle is a right angle triangle.
**Conclusion for Statement 3:** True.
### Step 4: Conclusion
All three statements are true. Therefore, the answer is that all statements are correct.
### Final Answer
**All statements are true.**
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