Let us call the third side x. According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be larger than the other two sides. Thus, all of the following must be true:

x + 7 > 9

x + 9 > 7

7 + 9 > x

We can solve these three inequalities to determine the possible values of x.

x + 7 > 9

Subtract 7 from both sides.

x > 2

Now, we can look at x + 9 > 7. Subtracting 9 from both sides, we obtain

x > –2

Finally, 7 + 9 > x, which means that 16 > x.

Therefore, x must be greater than 2, greater than –2, but also less than 16. The only number that satisfies all of these requirements is 12.

The answer is 12. 

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater (not greater than or equal) than the remaining side. Thus, the following inequalities must all be true:

x + 8 > 12

x + 12 > 8

8 + 12 > x

Let's solve each inequality.

x + 8 > 12

Subtract 8 from both sides.

x > 4

Next, let's look at the inequality x + 12 > 8

x + 12 > 8

Subtract 12 from both sides.

x > –4

Lastly, 8 + 12 > x, which means that x < 20.

This means that x must be less than twenty, but greater than 4 and greater than –4. Since any number greater than 4 is also greater than –4, we can exclude the inequality x > –4.

To summarize, x must be greater than 4 and less than 20. We can write this as 4 < x < 20. 

The answer is 4 < x < 20.

The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.

Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.

Given lengths of two of the sides of the are 15 and 5. The length of the third side must be greater than 15-5 or 10 and less than 15+5 or 20.

The question asks what is the least possible integer length of BC, which would be 11

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

The triangles are similar because of the side-angle-side postulate.

Side:

Angle:

Side:

The third side must also have a 2:3 ratio.

Cross-multiply, and solve for .

The triangles are not similar, as proven by the side-side-side postulate.

The third side does not follow the same ratio of the other two, thus the triangles are not similar.

The perimeter is equal to the sum of the three sides.  In similar triangles, each side is in proportion to its correlating side.  The perimeters are also in equal proportion.

Perimeter A = 45” and perimeter B = 135”

The proportion of Perimeter A to Perimeter B is . 

This applies to the sides of the triangle.  Therefore to get the any side of Triangle B, just multiply the correlating side by 3.

15” x 3 = 45”

10” x 3 = 30“

Angle-Angle Postulate: if two angles of one triangle are equal to two corresponding angles of another triangle, the triangles must be similar.

In this example, the triangles share one angle, which must be equal. Additionally, the triangles contain segments that are parallel to each other. When two parallel lines are crossed by another line, the corresponding angles must be equal. Each angle in one triangle is congruent with its corresponding angle in the other triangle, indicating that they are similar.