Geometry - How to find an angle in an acute / obtuse triangle

The largest angle in an obtuse scalene triangle is degrees. The second largest angle in the triangle is $\frac{1}{3}$ the measurement of the largest angle. What is the measurement of the smallest angle in the obtuse scalene triangle?

$20^\circ$

$40^\circ$

$30^\circ$

$25^\circ$

$15^\circ$

Explanation

Since this is a scalene triangle, all of the interior angles will have different measures. However, it's fundemental to note that in any triangle the sum of the measurements of the three interior angles must equal degrees.

The largest angle is equal to degrees and second interior angle must equal:

$120\div3=40$

Therefore, the final angle must equal:

A triangle has sides of lengths 19.5, 46.8, and 50.7. Is the triangle acute, right, or obtuse?

Right

Acute

Obtuse

Explanation

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

$19.5 ^{2}+ 46.8^{2} = 380.25+2,190.24 =2,570.49$

Calculate the square of the length of the longest side:

$50.7^{2} =2,570.49$

The two quantities are equal, so by the Converse of the Pythagorean Theorem, the triangle is right.

A triangle has sides of lengths 18.4, 18.4, and 26.0. Is the triangle acute, right, or obtuse?

Acute

Right

Obtuse

Explanation

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

$18.4^{2} + 18.4 ^{2} = 338.56 + 338.56 = 677.12$

Calculate the square of the length of the longest side:

$26 .0^{2} = 676$

The former is greater than the latter. This indicates that the triangle is acute.

A triangle has sides of lengths 9, 12, and 18. Is the triangle acute, right, or obtuse?

Obtuse

Right

Acute

Explanation

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

$9^{2} + 12^{2} = 81 + 144 = 225$

Calculate the square of the length of the longest side:

$18^{2} = 324$

The former is less than the latter. This indicates that the triangle is obtuse.

A triangle has sides of lengths 14, 18, and 20. Is the triangle acute, right, or obtuse?

Acute

Right

Obtuse

Explanation

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

$14^{2}+ 18 ^{2} = 196 + 324 = 520$

Calculate the square of the length of the longest side:

$2 0 ^{2} = 400$

The former is greater than the latter. This indicates that the triangle is acute.

The largest angle in an obtuse scalene triangle is degrees. The smallest interior angle is $\frac{1}{5}$ the measurement of the largest interior angle. Find the measurement of the third interior angle.

$42^\circ$

$38^\circ$

$44^\circ$

$23^\circ$

$46^\circ$

Explanation

An obtuse scalene triangle must have one obtuse interior angle and two acute angles.

Therefore the solution is:

$115\div5=23$

All triangles have three interior angles with a sum total of degrees.

Thus,

In ΔABC, A = 75°, a = 13, and b = 6.

Find B (to the nearest tenth).

26.5°

30.4°

28.1°

34.9°

27.8°

Explanation

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 75° a = 13

B = ? b = 6

C = ? c = ?

Now we can easily see that we have a complete pair, A and a. This tells us that we can use the Law of Sines. (We use the Law of Cosines when we do not have a complete pair).

Law of Sines:
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

To solve for b, we can use the first two terms which gives us:
$\frac{\sin 75°}{13}=\frac{\sin B°}{6}$
$\sin B =6*\frac{\sin 75}{13}$

Given: $\bigtriangleup ABC$ with perimeter 40;

True or false: $\angle A \cong \angle B$

True

False

Explanation

The perimeter of $\bigtriangleup ABC$ is the sum of the lengths of its sides - that is,

The perimeter is 40, so set , and solve for :

Subtract 26 from both sides:

$\overline{BC}\cong \overline{AC}$ , so by the Isosceles Triangle Theorem, their opposite angles are congruent - that is,

$\angle A \cong \angle B$ .

$\bigtriangleup ABC$ is an equilateral triangle; is the midpoint of $\overline{BC}$ ; the segment $\overline{AM }$ is constructed.

True or false: $m \angle BAM = 40^{\circ }$ .

False

True

Explanation

The referenced triangle is below:

Equilateral 2

In an equilateral triangle, the median from - the segment from to , the midpoint of the opposite side $\overline{BC}$ - is also the bisector of the angle $\angle BAC$ , so