Acute / Obtuse Triangles - Intermediate Geometry

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Question

Points A, B, and C are collinear (they lie along the same line). The measure of angle CAD is $30^{\circ}$ . The measure of angle CBD is $60^{\circ}$ . The length of segment $\overline{AD}$ is 4.

Find the measure of $\dpi{100} \small \angle ADB$ .

Answer

The measure of $\dpi{100} \small \angle ADB$ is $30^{\circ}$ . Since $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear, and the measure of $\dpi{100} \small \angle CBD$ is $60^{\circ}$ , we know that the measure of $\dpi{100} \small \angle ABD$ is $120^{\circ}$ .

Because the measures of the three angles in a triangle must add up to $180^{\circ}$ , and two of the angles in triangle $\dpi{100} \small ABD$ are $30^{\circ}$ and $120^{\circ}$ , the third angle, $\dpi{100} \small \angle ADB$ , is $30^{\circ}$ .

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Question

In ΔABC, A = 75°, a = 13, and b = 6.

Find B (to the nearest tenth).

Answer

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:
$\frac{\sin 75°}{13}=\frac{\sin B°}{6}$
$\sin B =6*\frac{\sin 75}{13}$

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Question

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?

Answer

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:

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Question

In an obtuse isosceles triangle the largest angle is degrees. Find the measurement of one of the two equivalent interior angles.

Answer

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal degrees, the solution is:

$47\div2=23\frac{1}{2}$

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Question

In an acute scalene triangle the measurement of the interior angles range from degrees to degrees. Find the measurement of the median interior angle.

Answer

Acute scalene triangles must have three different acute interior angles--which always have a sum of degrees.

Thus, the solution is:

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Question

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.

Answer

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,

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Question

The largest angle in an obtuse isosceles triangle is degrees. Find the measurement for one of the equivalent acute interior angles.

Answer

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal degrees, the solution is:

$86\div2=43$

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Question

In an acute isosceles triangle the largest interior angle is degrees. Find the measure for one of the two equivalent acute interior angles.

Answer

The sum of the three interior angles in any triangle must equal degrees. Therefore, the solution is:

$92\div 2= 46$

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Question

Find the value of .

Answer

To find the value of , consider the fundamental notion that the sum of the three interior angles of any triangle must equal degrees.

Thus, the solution is:

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Question

An obtuse isosceles triangle has an interior angle with a measure of degrees.

Select the answer choice that displays the correct measurements for the two other interior angles of the triangle.

Answer

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal degrees, the solution is:

$40\div2=20$

Therefore, each of the two equivalent interior angles must have a measurement of degrees each.

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Question

A triangle has two equivalent interior angles, each with a measurement of degrees. Select the most accurate categorization of this triangle.

Answer

The sum of the three interior angles in any triangle must equal degrees. Since this triangle has two equivalent interior angles, it must be an isosceles triangle. Additionally, the sum of the two equivalent interior angles is: .

Since, degrees, the third interior angle must be a right angle--which makes this triangle an isosceles right triangle.

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Question

A particular triangle does not have any equivalent sides, and has one interior angle with a measurement of degrees.

Select the most accurate categorization of this triangle.

Answer

A scalene triangle doesn't have any equivalent sides or equivalent interior angles. Since this triangle has one obtuse interior angle, it must be an obtuse scalene triangle.

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Question

A triangle has interior angles that range from degrees to degrees. What is the most accurate categorization of this triangle?

Answer

Acute scalene triangles must have three different acute interior angles--which always have a sum of degrees.

Thus, the solution is:

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Question

A particular triangle has interior angle measurements that range from degrees to degrees. What is the most accurate categorization of this triangle?

Answer

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal degrees, the solution is:

Therefore, this triangle has one interior angle of degrees and two equivalent interior angles of degrees--making this an obtuse isosceles triangle.

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

Two of the interior angles of a triangle have measure $45^{\circ }$ and $80^{\circ }$ . Is the triangle acute, right, or obtuse?

Answer

The measures of the interior angles of a triangle add up to $180^{\circ }$ . If is the measure of the third angle, then

$t + 45 ^{\circ }+ 80^{\circ } = 180 ^{\circ }$

Solve for :

$t + 125^{\circ } = 180 ^{\circ }$

$t + 125^{\circ } - 125^{\circ } = 180 ^{\circ } - 125^{\circ }$

$t = 55^{\circ }$

Each of the three angles has measure less than $90^{\circ }$ , so each angle is, by definition, acute. This makes the triangle acute.

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Question

Two of the exterior angles of a triangle, taken at different vertices, measure $100^{\circ }$ . Is the triangle acute, right, or obtuse?

Answer

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total $180^{\circ }$ . The measures of two interior angles can be calculated by subtracting the exterior angle measures from $180^{\circ }$ :

$180^{\circ } - 100 ^{\circ } = 80 ^{\circ }$

The triangle has two interior angles of measure $80 ^{\circ }$ .

The measures of the interior angles of a triangle add up to $180^{\circ }$ . If is the measure of the third angle, then

$t + 80^{\circ } + 80^{\circ } = 180^{\circ }$

$t +160^{\circ } = 180^{\circ }$

$t +160^{\circ }- 160 ^{\circ } = 180^{\circ } - 160 ^{\circ }$

$t = 20 ^{\circ }$

All three interior angles measure less than $90^{\circ }$ , making them acute. The triangle is, by definition, acute.

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