Homework 3 Isosceles And Equilateral Triangles Answers

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How to Solve Homework 3 on Isosceles and Equilateral Triangles

Isosceles and equilateral triangles are two special types of triangles that have some properties in common. In this article, we will review the definitions and theorems related to these triangles, and show you how to solve some problems from Homework 3.

Definitions and Theorems

An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides are called the legs, and the angle between them is called the vertex angle. The other two angles are called the base angles, and they are also congruent.

An equilateral triangle is a triangle that has all three sides congruent. It is also an isosceles triangle, since any two sides are congruent. All three angles of an equilateral triangle are also congruent, and each measure 60 degrees.

The following theorems can be used to prove that a triangle is isosceles or equilateral:

If two sides of a triangle are congruent, then the angles opposite them are congruent. (Converse of Isosceles Triangle Theorem)

If two angles of a triangle are congruent, then the sides opposite them are congruent. (Isosceles Triangle Theorem)

If all three sides of a triangle are congruent, then the triangle is equilateral. (Definition of Equilateral Triangle)

If all three angles of a triangle are congruent, then the triangle is equilateral. (Converse of Equilateral Triangle Theorem)

Solving Problems

Now let's look at some examples of how to apply these definitions and theorems to solve problems from Homework 3.

In ÎABC, AB=CB, mâ ABC = 4x-3, and mâ CAB = x-3. Find mâ ACB.

Solution: Since AB=CB, we know that ÎABC is an isosceles triangle, and mâ ABC=mâ ACB by the Isosceles Triangle Theorem. So we can set up an equation:

4x-3=x-3

Solving for x, we get x=0. Therefore, mâ ACB=4x-3=4(0)-3=-3 degrees.

Note: This answer does not make sense in terms of geometry, since an angle cannot have a negative measure. This means that there is no such triangle that satisfies the given conditions.

The measures of two of the sides of an equilateral triangle are 3x+15 in. and 7x-5 in. What is the measure of the third side in inches

Solution: Since the triangle is equilateral, we know that all three sides are congruent. So we can set up an equation:

3x+15=7x-5

Solving for x, we get x=5. Therefore, the measure of the third side is 3x+15=3(5)+15=30 inches.

In ÎDEF, FDâ ED, mâ FDE = 49Â, and mâ DFE = 118Â. Find mâ DEF.

Solution: Since FDâ ED, we know that ÎDEF is an isosceles triangle, and mâ FDE=mâ EDF by the Isosceles Triangle Theorem. So we can use the Triangle Angle Sum Theorem to find mâ DEF:

mâ DEF+mâ FDE+mâ DFE=180Â

mâ DEF+49Â+118Â=180Â

mâ DEF+167Â=180Â

mâ DEF=180Â-167Â

mâ DEF=13Â

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