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

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.

$28.7^{o}$

$25.6^{o}$

$65.8^{o}$

$45.3^{o}$

Explanation

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

Kite1

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.

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

$x=28.7^{0}$

Isosceles

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

The Isosceles Triangle Theorem

The Converse of the Isosceles Triangle Theorem

The Side-Side-Side Postulate

The Side-Angle-Side Postulate

The Pythagorean Theorem

Explanation

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, $\angle B \cong \angle C$ .

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.

$13^\circ$

$26^\circ$

$32^\circ$

$4^\circ$

$10^\circ$

Explanation

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 $26\div2=13$ .

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?

Explanation

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 $x\ +\ x\ +\ 2x\ +\ 20 = 180$

$4x\ +\ 20=180$

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

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

$74^\circ$

$32^\circ$

$37^\circ$

$90^\circ$

$15^\circ$

Explanation

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:

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

$41^\circ$

$82^\circ$

$42^\circ$

$90^\circ$

$2^\circ$

Explanation

Thus, the solution is:

$82\div2=41$

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

$124^\circ$

$134^\circ$

$56^\circ$

$112^\circ$

$99^\circ$

Explanation

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:

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

$74^\circ$

$75^\circ$

$62^\circ$

$53^\circ$

$42^\circ$

Explanation

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:

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

$70^\circ$

$78^\circ$

$140^\circ$

$62^\circ$

$13^\circ$

Explanation

The solution is:

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

$21^\circ$

$42^\circ$

$44^\circ$

$23^\circ$

$15^\circ$

Explanation

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:

$42\div2=21$

How to find an angle in an acute / obtuse isosceles triangle

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