Isosceles Triangles - High School Math

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Question

Points A and B lie on a circle centered at Z, where central angle <AZB measures 140°. What is the measure of angle <ZAB?

Answer

Because line segments ZA and ZB are radii of the circle, they must have the same length. That makes triangle ABZ an isosceles triangle, with <ZAB and <ZBA having the same measure. Because the three angles of a triangle must sum to 180°, you can express this in the equation:

140 + 2x = 180 --> 2x = 40 --> x = 20

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Question

In triangle ABC, Angle A = x degrees, Angle B = 2x degrees, and Angle C = 3x+30 degrees. How many degrees is Angle B?

Answer

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

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Question

Triangle FGH has equal lengths for FG and GH; what is the measure of ∠F, if ∠G measures 40 degrees?

Answer

It's good to draw a diagram for this; we know that it's an isosceles triangle; remember that the angles of a triangle total 180 degrees.

Angle G for this triangle is the one angle that doesn't correspond to an equal side of the isosceles triangle (opposite side to the angle), so that means ∠F = ∠H, and that ∠F + ∠H + 40 = 180,

By substitution we find that ∠F * 2 = 140 and angle F = 70 degrees.

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Question

The vertex angle of an isosceles triangle is $54^{\circ}$ . What is the base angle?

Answer

An isosceles triangle has two congruent base angles and one vertex angle. Each triangle contains $180^{\circ}$ . Let = base angle, so the equation becomes . Solving for gives

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Question

In an isosceles triangle the base angle is five less than twice the vertex angle. What is the sum of the vertex angle and the base angle?

Answer

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle

and = base angle

So the equation to solve becomes

or

Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.

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Question

An isosceles triangle has a base angle that is six more than three times the vertex angle. What is the base angle?

Answer

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle.

Then the equation to solve becomes

or

.

Solving for gives a vertex angle of 24 degrees and a base angle of 78 degrees.

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Question

The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?

Answer

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let be the vertex angle and be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

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Question

Sides and in this triangle are equal. What is the measure of $\angle A$ ?

Answer

This triangle has an angle of $65^{\circ}$ . We also know it has another angle of $65^{\circ}$ at $\angle ABC$ because the two sides are equal. Adding those two angles together gives us $130^{\circ}$ total. Since a triangle has $180^{\circ}$ total, we subtract 130 from 180 and get 50.

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Question

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

Answer

A triangle has 180 degrees. An isoceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes

or

So the vertex angle is $34^{\circ}$ and the base angle is $73^{\circ}$ so the difference is $39^{\circ}$

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Question

An isosceles triangle has a vertex angle that is twenty degrees more than twice the base angle. What is the sum of the vertex and base angles?

Answer

All triangles contain degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let and .

So the equation to solve becomes .

We get and , so the sum of the base and vertex angles is .

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Question

If an isosceles triangle has an angle measuring greater than 100 degrees, and another angle with a measuring degrees, which of the following is true?

Answer

In order for a triangle to be an isosceles triangle, it must contain two equivalent angles and one angle that is different. Given that one angle is greater than 100 degrees: Thus, the sum of the other two angles must be less than 80 degrees. If an angle is represented by :

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Question

An isoceles triangle has a base angle that is twice the vertex angle. What is the sum of the base and vertex angles?

Answer

All triangles have degrees. An isoceles triangle has one vertex angle and two congruent base angles.

Let vertex angle and base angle.

So the equation to solve becomes:

or

Thus for the vertex angle and for the base angle.

The sum of the vertex and one base angle is .

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Question

An isoceles triangle has a vertex angle that is degrees more than twice the base angle. What is the vertex angle?

Answer

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

Let base angle and vertex angle.

So the equation to solve becomes .

Thus the base angles are and the vertex angle is .

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Question

An isoceles triangle has a base angle that is degrees less than three times the vertex angle. What is the product of the vertex angle and the base angle?

Answer

Every triangle has 180 degrees. An isoceles triangle has one vertex angle and two congruent base angles.

Let vertex angle and base angle.

Then the equation to solve becomes:

, or .

Then the vertex angle is , the base angle is , and the product is .

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Question

An isoceles triangle has a base angle that is twice the vertex angle. What is the sum of one base angle and the vertex angle?

Answer

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

Let the vertex angle and the base angle

So the equation to solve becomes or and dividing by gives for the vertex angle and for the base angle, so the sum is

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Question

An isoceles triangle has a base angle that is five less than twice the vertex angle. What is the sum of the base and vertex angles?

Answer

Each triangle has degrees.

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

Let vertex angle and base angle.

Then the equation to solve becomes or .

Add to both sides to get .

Divide both sides by to get vertex angle and base angles, so the sum of the angles is .

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Question

The base of a right isosceles triangle is 8 inches. The hypotenuse is not the base. What is the area of the triangle in inches?

Answer

To find the area of a triangle, multiply the base by the height, then divide by 2. Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other. Since, a short side serves as the base of the triangle, the other short side tells us the height.

$(8\cdot 8)/2=32$

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Question

An isosceles right triangle has a hypotenuse of $h=18\sqrt{2} in$ . Find its area.

Answer

In order to calculate the triangle's area, we need to find the lengths of its legs. An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as $45^{\circ}-45^{\circ}-90^{\circ}$ triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

$x=\frac{h}{\sqrt{2}}$

$x=\frac{18\sqrt{2}in}{\sqrt{2}}$

Now we can calculate the area using the formula

$A=\frac{1}{2}bh$

$A=\frac{1}{2}18in18in$

$A=162in^{2}$

Now, convert to feet.

$\rightarrow \frac{162 in^{2}}{1}\frac{1ft}{12in}\frac{1ft}{12in}=1.125ft^{2}$

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Question

An isosceles triangle has a base of $12\ cm$ and an area of $42\ cm^{2}$ . What must be the height of this triangle?

Answer

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

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Question

In an isosceles right triangle, two sides equal . Find the length of side .

Answer

This problem represents the definition of the side lengths of an isosceles right triangle. By definition the sides equal , , and $x\sqrt{2}$ . However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem $(a^{2}+b^{2}=c^{2})$ .

$h^{2}=x^{2}+x^{2}$

$h^{2}=2x^{2}$

$h=\sqrt{2x^{2}}$

$\rightarrow x\sqrt{2}$

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