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Intermediate Geometry : How to find an angle in an acute / obtuse triangle

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

In ΔABC, A = 75°, a = 13, and b = 6.

Find B (to the nearest tenth).

Possible Answers:

27.8°

26.5°

34.9°

30.4°

28.1°

Correct answer:

26.5°

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:
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ $\displaystyle \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

To solve for b, we can use the first two terms which gives us:
$\frac{\sin 75°}{13}=\frac{\sin B°}{6}$ $\displaystyle \frac{\sin 75°}{13}=\frac{\sin B°}{6}$
$\sin B =6*\frac{\sin 75}{13}$ $\displaystyle \sin B =6*\frac{\sin 75}{13}$
$\displaystyle B=26.5°$

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

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

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

Possible Answers:

$45^{\circ}$ $\displaystyle 45^{\circ}$

$30^{\circ}$ $\displaystyle 30^{\circ}$

$60^{\circ}$ $\displaystyle 60^{\circ}$

$90^{\circ}$ $\displaystyle 90^{\circ}$

$15^{\circ}$ $\displaystyle 15^{\circ}$

Correct answer:

$30^{\circ}$ $\displaystyle 30^{\circ}$

Explanation:

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

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

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

The largest angle in an obtuse scalene triangle is 120 $\displaystyle 120$ degrees. The second largest angle in the triangle is $\frac{1}{3}$ $\displaystyle \frac{1}{3}$ the measurement of the largest angle. What is the measurement of the smallest angle in the obtuse scalene triangle?

Possible Answers:

$15^\circ$ $\displaystyle 15^\circ$

$40^\circ$ $\displaystyle 40^\circ$

$25^\circ$ $\displaystyle 25^\circ$

$20^\circ$ $\displaystyle 20^\circ$

$30^\circ$ $\displaystyle 30^\circ$

Correct answer:

$20^\circ$ $\displaystyle 20^\circ$

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 180 $\displaystyle 180$ degrees.

The largest angle is equal to 120 $\displaystyle 120$ degrees and second interior angle must equal:

$120\div3=40$ $\displaystyle 120\div3=40$

$\displaystyle 120+40=160$

Therefore, the final angle must equal:

$\displaystyle 180-160=20$

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

In an obtuse isosceles triangle the largest angle is 133 $\displaystyle 133$ degrees. Find the measurement of one of the two equivalent interior angles.

Possible Answers:

$31^\circ$ $\displaystyle 31^\circ$

$47^\circ$ $\displaystyle 47^\circ$

$23\frac{1}{2}^\circ$ $\displaystyle 23\frac{1}{2}^\circ$

$24^\circ$ $\displaystyle 24^\circ$

$47\frac{1}{2}^\circ$ $\displaystyle 47\frac{1}{2}^\circ$

Correct answer:

$23\frac{1}{2}^\circ$ $\displaystyle 23\frac{1}{2}^\circ$

Explanation:

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 180 $\displaystyle 180$ degrees, the solution is:

$\displaystyle 180-133=47$

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

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Example Question #3 : How To Find An Angle In An Acute / Obtuse Triangle

In an acute scalene triangle the measurement of the interior angles range from $\displaystyle 79$ degrees to $\displaystyle 34$ degrees. Find the measurement of the median interior angle.

Possible Answers:

$69^\circ$ $\displaystyle 69^\circ$

$67^\circ$ $\displaystyle 67^\circ$

$72^\circ$ $\displaystyle 72^\circ$

$12^\circ$ $\displaystyle 12^\circ$

$47^\circ$ $\displaystyle 47^\circ$

Correct answer:

$67^\circ$ $\displaystyle 67^\circ$

Explanation:

Acute scalene triangles must have three different acute interior angles--which always have a sum of 180 $\displaystyle 180$ degrees.

Thus, the solution is:

$\displaystyle 79+34=113$

$\displaystyle 180-113=67$

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Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

The largest angle in an obtuse scalene triangle is 115 $\displaystyle 115$ degrees. The smallest interior angle is $\frac{1}{5}$ $\displaystyle \frac{1}{5}$ the measurement of the largest interior angle. Find the measurement of the third interior angle.

Possible Answers:

$42^\circ$ $\displaystyle 42^\circ$

$23^\circ$ $\displaystyle 23^\circ$

$44^\circ$ $\displaystyle 44^\circ$

$38^\circ$ $\displaystyle 38^\circ$

$46^\circ$ $\displaystyle 46^\circ$

Correct answer:

$42^\circ$ $\displaystyle 42^\circ$

Explanation:

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

Therefore the solution is:

$115\div5=23$ $\displaystyle 115\div5=23$

$\displaystyle 115+23=138$

All triangles have three interior angles with a sum total of 180 $\displaystyle 180$ degrees.

Thus,

$\displaystyle 180-138=42$

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Example Question #7 : How To Find An Angle In An Acute / Obtuse Triangle

The largest angle in an obtuse isosceles triangle is $\displaystyle 94$ degrees. Find the measurement for one of the equivalent acute interior angles.

Possible Answers:

$86^\circ$ $\displaystyle 86^\circ$

$49^\circ$ $\displaystyle 49^\circ$

$43^\circ$ $\displaystyle 43^\circ$

$84^\circ$ $\displaystyle 84^\circ$

$37^\circ$ $\displaystyle 37^\circ$

Correct answer:

$43^\circ$ $\displaystyle 43^\circ$

Explanation:

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Example Question #8 : How To Find An Angle In An Acute / Obtuse Triangle

In an acute isosceles triangle the largest interior angle is $\displaystyle 88$ degrees. Find the measure for one of the two equivalent acute interior angles.

Possible Answers:

$39^\circ$ $\displaystyle 39^\circ$

$46^\circ$ $\displaystyle 46^\circ$

$49^\circ$ $\displaystyle 49^\circ$

$44^\circ$ $\displaystyle 44^\circ$

$48^\circ$ $\displaystyle 48^\circ$

Correct answer:

$46^\circ$ $\displaystyle 46^\circ$

Explanation:

The sum of the three interior angles in any triangle must equal 180 $\displaystyle 180$ degrees. Therefore, the solution is:

$\displaystyle 180-88=92$

$92\div 2= 46$ $\displaystyle 92\div 2= 46$

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Example Question #9 : How To Find An Angle In An Acute / Obtuse Triangle

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Find the value of $\displaystyle x$.

Possible Answers:

$28^\circ$ $\displaystyle 28^\circ$

$34^\circ$ $\displaystyle 34^\circ$

$38^\circ$ $\displaystyle 38^\circ$

$37^\circ$ $\displaystyle 37^\circ$

$27^\circ$ $\displaystyle 27^\circ$

Correct answer:

$38^\circ$ $\displaystyle 38^\circ$

Explanation:

To find the value of $\displaystyle x$, consider the fundamental notion that the sum of the three interior angles of any triangle must equal 180 $\displaystyle 180$ degrees.

Thus, the solution is:

$\displaystyle 90+52=142$

$\displaystyle 180-142=38$

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Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

An obtuse isosceles triangle has an interior angle with a measure of 140 $\displaystyle 140$ degrees.

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

Possible Answers:

$10^\circ$ $\displaystyle 10^\circ$, $15^\circ$ $\displaystyle 15^\circ$

$15^\circ$ $\displaystyle 15^\circ$, $25^\circ$ $\displaystyle 25^\circ$

$30^\circ$ $\displaystyle 30^\circ$, $10^\circ$ $\displaystyle 10^\circ$

$20^\circ$ $\displaystyle 20^\circ$, $20^\circ$ $\displaystyle 20^\circ$

$20^\circ$ $\displaystyle 20^\circ$, $40^\circ$ $\displaystyle 40^\circ$

Correct answer:

$20^\circ$ $\displaystyle 20^\circ$, $20^\circ$ $\displaystyle 20^\circ$

Explanation:

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 180 $\displaystyle 180$ degrees, the solution is:

$\displaystyle 180-140=40$

$40\div2=20$ $\displaystyle 40\div2=20$

Therefore, each of the two equivalent interior angles must have a measurement of $\displaystyle 20$ degrees each.

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Intermediate Geometry : How to find an angle in an acute / obtuse triangle

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #3 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #7 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #8 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #9 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

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