Acute / Obtuse Isosceles 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

$\bigtriangleup ABC$ and $\bigtriangleup DEF$ are both isosceles triangles;

$m \angle B = 120 ^{\circ }$

$m \angle F = 30 ^{\circ }$

True or false: from the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

$\bigtriangleup DEF$ is an isosceles triangle, so it must have two congruent angles. $\angle F$ has measure $30^{\circ }$ , so either $\angle D$ has this measure, $\angle E$ has this measure, or $\angle D \cong \angle E$ . If we examine the second case, it immediately follows that $\angle B cong \angle E$ . One condition of the similarity of triangles is that all pairs of corresponding angles be congruent; since there is at least one case that violates this condition, it does not necessarily follow that $\bigtriangleup ABC \sim \bigtriangleup DEF$ . This makes the correct response "false".

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Question

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

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Question

Refer to the above diagram. .

True or false: From the information given, it follows that $\bigtriangleup AVB \sim \bigtriangleup CVD$ .

Answer

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar.

$\angle AVB$ and $\angle CVD$ are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, $\angle AVB \cong \angle CVD$ .

$\angle ABV$ and $\angle CDV$ are alternating interior angles formed by two parallel lines and cut by a transversal . As a consequence, $\angle ABV \cong \angle CDV$ .

The conditions of the Angle-Angle Similarity Postulate are satisfied, and it holds that $\bigtriangleup AVB \sim \bigtriangleup CVD$ .

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Question

Refer to the above diagram. $\angle ABV \cong \angle DCV$ .

True or false: From the information given, it follows that $\bigtriangleup AVB \sim \bigtriangleup CVD$ .

Answer

The given information is actually inconclusive.

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar. Therefore, we seek to prove two of the following three angle congruence statements:

$\angle AVB \cong \angle CVD$

$\angle ABV \cong \angle CDV$

$\angle BAV \cong \angle DCV$

$\angle AVB$ and $\angle CVD$ are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, $\angle AVB \cong \angle CVD$ .

$\angle ABV \cong \angle DCV$ , but this is not one of the statements we need to prove. Also, without further information - for example, whether and are parallel, which is not given to us - we have no way to prove either of the other two necessary statements.

The correct response is "false".

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that

$\angle A \cong \angle B$

$\angle C \cong \angle F$

$\angle D \cong \angle E$

Which statement(s) must be true?

(a) $\bigtriangleup ABC \sim \bigtriangleup DEF$

(b) $\bigtriangleup ABC \cong \bigtriangleup DEF$

Answer

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ , so

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$m \angle A + m \angle B + m \angle C - m \angle C = 180^{\circ } - m \angle C$

$m \angle A + m \angle B = 180^{\circ } - m \angle C$

Also,

$m \angle A = m \angle B$ ,

so

$m \angle A + m \angle A = 180^{\circ } - m \angle C$

$2 \cdot m \angle A = 180^{\circ } - m \angle C$

$\frac{2 \cdot m \angle A }{2}= \frac{180^{\circ } - m \angle C}{2}$

$m \angle A = \frac{180^{\circ } - m \angle C}{2}$

By similar reasoning, it holds that

$m \angle D = \frac{180^{\circ } - m \angle F}{2}$

Since $m \angle F = m \angle C$ , by substitution,

$m \angle D = \frac{180^{\circ } - m \angle C}{2}$

Therefore,

$m \angle A = m \angle D$ ,

or $\angle A \cong \angle D$

This, along with the statement that $\angle C \cong \angle F$ , sets up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem.

Therefore, statement (a) must hold, but not necessarily statement (b).

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