If a triangle has an angle of 30 ° and one side is twice as big as the other, is it an acute triangle By tomorrow night

If a triangle has an angle of 30 ° and one side is twice as big as the other, is it an acute triangle By tomorrow night

There are three cases
1. As shown in Figure 1, when the angle of 30 ° is the included angle of the two sides, the angle ACD = ADC = 75 °,
Because BD = ad & gt; CD, angle DCB & gt; DBC, angle DCB & gt; 37.5 ° and angle ACB & gt; 112.5 °, triangle ABC is an obtuse triangle
2. As shown in Figure 2, when the 30 ° angle is the opposite angle of the shorter side of the two sides, the triangle ABC must be a right triangle
If B is not perpendicular to A.C
Because angle ADB = 90 °, angle a = 30 ° and ab = 2bd, we can get BD = BC, angle BDC = angle c, then angle c = 90 degrees, which contradicts that angle c is an acute angle, so angle c is not an acute angle
If angle c is not a right angle, set it as an obtuse angle, and make BD through B perpendicular to AC and D, then D is on the extension line of AC, and angle c = 90 ° can also be obtained, which contradicts angle c as an obtuse angle, so angle c is not an obtuse angle
So angle c is a right angle and triangle ABC is a right triangle
3. When the angle of 30 ° is the diagonal of the larger side of the two sides, the bigger side is opposite to the bigger side, the diagonal of the smaller side is less than 30 ° and the third angle is more than 120 ° and the triangle is an obtuse triangle
To sum up, triangle ABC cannot be an acute triangle

The sum of the internal angles of a triangle is equal to 180 degrees. It is known that the first internal angle of a triangle is equal to three times of the second internal angle, and the third internal angle is 15 degrees larger than the second internal angle. What is the degree of each angle?

Let the second internal angle be x, the first internal angle be 3x, the third internal angle be x + 15 °, the sum of internal angles of ∵ triangle be 180 °, and the solution be x = 33 °, the first internal angle be 3x = 99 °, the second internal angle be 33 ° and the third internal angle be 48 °

In the figure below, the sum of all acute angles is equal to 180 ° and ∠ 1 = 2 = 3. How many degrees is ∠ AOB?

Let ∠ 1 = 2 = 3 = X,
Then: 3x + 2x + 2x + 3x = 180 degree
10x=180°
x=18°
So: ∠ AOB = 3x = 54 degree