Triangles - Intermediate Geometry

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Question

An isoceles triangle has a vertex angle that is twenty more than twice the base angle. What is the difference between the vertex and base angles?

Answer

A triangle has degrees. An isoceles triangle has one vertex angle and two congruent base angles.

Let = the base angle and $2x\ +\ 20$ = vertex angle

So the equation to solve becomes $x\ +\ x\ +\ 2x\ +\ 20 = 180$

or

$4x\ +\ 20=180$

so the base angle is and the vertex angle is and the difference is .

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Question

An ssosceles triangle has interior angles of degrees and degrees. Find the missing angle.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees.

Thus, the solution is:

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Question

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

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.

Thus, the solution is:

$82\div2=41$

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Question

The two equivalent interior angles of an obtuse isosceles triangle each have a measurement of degrees. Find the measurement of the obtuse angle.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees.

Thus, the solution is:

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Question

In an obtuse isosceles triangle the angle measurements are, $x^\circ$ , $x^\circ$ , and $(10x-2)=128^\circ$ . Find the measurement of one of the acute angles.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.

The solution is:

However, degrees is the measurement of both of the acute angles combined.

Each individual angle is $26\div2=13$ .

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Question

In an acute isosceles triangle the two equivalent interior angles each have a measurement of degrees. Find the missing angle.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an acute isosceles triangle, all of the interior angles must be acute angles.

The solution is:

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Question

In an acute isosceles triangle the two equivalent interior angles are each degrees. Find the missing angle.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an acute isosceles triangle, all of the interior angles must be acute angles.

The solution is:

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Question

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

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an obtuse Isosceles triangle, the two missing angles must be acute angles.

Thus, the solution is:

$42\div2=21$

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Question

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

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.

The solution is:

$62\div2=31$

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Question

In an acute isosceles triangle the measurement of the non-equivalent interior angle is degrees. Find the measurement of one of the equivalent interior angles.

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an acute isosceles triangle, all of the interior angles must be acute angles.

The solution is:

$94\div2=47$

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Question

In an obtuse isosceles triangle, the largest interior angle is degrees. What is the measurement of one of the equivalent interior angles?

Answer

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.

The solution is:

$45\div2=22.5$

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Question

You are flying a kite at an altitude of 40 feet after having let out 75 feet of string. What is the kite's angle of elevation from where you are holding the spool of string at a height of 4 feet off the ground? Round answer to one decimal place.

Answer

First, we must draw a picture to include all important parts given in the problem.

Once this is determined we can use trigonometry to find the angle of elevation.

$sin(x)=\frac{36}{75}$

Use the inverse sin on a calculator to solve.

$sin^{-1}\left (sin(x) \right )=sin^{-1}\left (\frac{36}{75} \right )$

$x=28.7^{0}$

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Question

Two angles in an isosceles triangle are 50 and 80 degrees respectively. What is the measure of the third angle?

Answer

If a triangle is isosceles, two of the angles must be congruent. So the angle must be either 50 degrees or 80 degrees.

We know that the three angles in all triangles must sum to equal 180 degrees. The only answer choice that is both the same as one of the given angles and results in a sum equal to 180 degrees is the 50 degree angle.

50+50+80=180.

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Question

An isosceles triangle has one angle measuring . Which of the following are possible values for angles in this triangle?

Answer

An isosceles triangle has 2 congruent angles and then a third angle. These angles, as in any triangle, must add to 180.

One possibility is that the 25-degree angle is the "different" one, and the other two are congruent. This could be expressed using the algebraic expression . To find the other two angles, solve for x. First combine like terms:

subtract 25 from both sides

divide both sides by 2

The other possibility is that there are 2 25-degree angles and then some different angle measure. This could be expressed using the algebraic expression . Again, solve for x. First add the 2 25's to get 50:

subtract 50 from both sides

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Question

Refer to the above triangle. By what statement does it follow that $\angle B \cong \angle C$ ?

Answer

We are given that, in $\bigtriangleup ABC$ , two sides are congruent; specifically, $\overline{AB} \cong\overline{AC}$ . It is a consequence of the Isosceles Triangle Theorem that the angles opposite the sides are also congruent - that is, $\angle B \cong \angle C$ .

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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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