How to find length of an isosceles triangle given one side and angle

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Find the perimeter of the triangle below.

Correct answer:

Explanation:

Use the Pythagorean Theorem to find the base of the right triangle.

Now, because two of the angles in this triangle are the same, this is an isosceles triangle. In an isosceles triangle, the sides that are directly across from the congruent angles are also congruent.

In addition, the height in an isosceles triangle will always cut the 3rd side in half. With this information, fill out the triangle as shown below:

To find the perimeter, add up all the sides.

Find the perimeter of the triangle below.

Correct answer:

Explanation:

Use the Pythagorean Theorem to find the base of the right triangle.

Now, because two of the angles in this triangle are the same, this is an isosceles triangle. In an isosceles triangle, the sides that are directly across from the congruent angles are also congruent.

In addition, the height in an isosceles triangle will always cut the 3rd side in half. With this information, fill out the triangle as shown below:

To find the perimeter, add up all the sides.

A triangle has sides of lengths 14, 18, and 20. Is the triangle scalene or isosceles?

Explanation:

A triangle with three sides of different length is, by definition, scalene.

Explanation:

Below is regular Pentagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

By symmetry, all of the radii of a regular pentagon are congruent - specifically, . This triangle has at least two congruent sides, so it is isosceles.

If a triangle has side lengths of  and , which of the following can be a length of the third side?

Correct answer:

Explanation:

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

If a triangle has side lengths of  and , which of the following could be the length of the third side?

Correct answer:

Explanation:

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

Figure NOT drawn to scale.

Refer to the above diagram.

True or false: .

Explanation:

The sum of the measures of the interior angles of a triangle is , so 

Substitute the given two angle measures and solve for :

Subtract  from both sides:

, so, by the Converse of the Isosceles Triangle Theorem, their opposite sides are also congruent - that is,

Refer to the above triangle. By what statement does it follow that ?

Possible Answers:

The Angle-Side-Angle Postulate

The Angle-Angle Postulate

The Converse of the Isosceles Triangle Theorem

The Hinge Theorem

The Converse of the Pythagorean Theorem

Correct answer:

The Converse of the Isosceles Triangle Theorem

Explanation:

We are given that, in , two angles are congruent; specifically, . It is a consequence of the Converse of the Isosceles Triangle Theorem that the sides opposite the angles are also congruent - that is, ..

Given  such that , , , which of the following statements is true?

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

Having three sides of different lengths, this triangle is scalene. In any scalene triangle, the angle with greatest measure is opposite the longest side, and the angle with least measure is opposite the shortest side. Therefore, since , their opposite angles would be in order from greatest to least measure - that is, .

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How do I find the length of an isosceles triangle?