If  and  have the same perimeter, , and , it follows that . The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate. 

If , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that . Since  and  are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate. 

In both of the above cases, it follows that .

However, similarly to the previous situation, if , then it follows that , meaning that we have congruent sides and congruent nonincluded angles. However, this is not sufficient to prove congruence.

"Statement III" is the correct response.

Consider the perimeter of a triangle:

  P = a + b + c

Since we know a and b, we can find c. 

In I:

  11 = 7 + 4 + c

  11 = 11 + c 

  c = 0

Note that if c = 0, the shape is no longer a trial. Thus, we can eliminate I.

In II:

  15 = 7 + 4 + c

  15 = 11 + c

   c = 4.

This is plausible given that the other sides are 7 and 4. 

In III:

  25 = 7 + 4 + c

  25 = 11 + c

  c = 14.

It is not possible for one side of a triangle to be greater than the sum of both of the other sides, so eliminate III. 

Thus we are left with only II.

The sum of the angles in a triangle is 180^o:  a + b + c = 180

In this case, the average of a and b is 75:

(a + b)/2 = 75, then multiply both sides by 2

(a + b) = 150, then substitute into first equation

150 + c = 180

c = 30

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

The measures of the three angles are x, y, and z. Because the sum of the measures of the angles in any triangle must be 180 degrees, we know that x + y + z = 180. We can use this equation, along with the other two equations given, to form this system of equations:

x + y + z = 180

y = 2z

z = 0.5x - 30

If we can solve for both y and x in terms of z, then we can substitute these values into the first equation and create an equation with only one variable.

Because we are told already that y = 2z, we alreay have the value of y in terms of z.

We must solve the equation z = 0.5x - 30 for x in terms of z.

Add thirty to both sides.

z + 30 = 0.5x

Mutliply both sides by 2

2(z + 30) = 2z + 60 = x

x = 2z + 60

Now we have the values of x and y in terms of z. Let's substitute these values for x and y into the equation x + y + z = 180.

(2z + 60) + 2z + z = 180

5z + 60 = 180

5z = 120

z = 24

Because y = 2z, we know that y = 2(24) = 48. We also determined earlier that x = 2z + 60, so x = 2(24) + 60 = 108.

Thus, the measures of the three angles of the triangle are 24, 48, and 108. The question asks for the largest of these measures, which is 108.

The answer is 108. 

The answer is 18

We know that the sum of all the angles is 180. Using the rest of the information given we can write the other two equations:

x + y + z = 180      

x = 3y      

2x = z

We can solve for y and z in the second and third equations and then plug into the first to solve.

x + (1/3)x + 2x = 180

3[x + (1/3)x + 2x = 180]

3x + x + 6x = 540

10x = 540

x = 54

y = 18

z = 108

A straight line segment has 180 degrees. Therefore, the angle that is not labelled must have:

We know that the sum of the angles in a triangle is 180 degrees. As a result, we can set up the following algebraic equation:

Subtract 70 from both sides:

Divide by 2:

The triangle has an exterior angle of , so it has an interior angle of . By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

I: One of the other angles also has measure .

In this case, since the angles' measures must total  the third angle has measure 

In this case, the least measure is .

II: The other two angles are the congruent angles.

In this case, the other two angles have measures totaling 

.

They have the same measure, so each has measure half this, or

.

In this case, the least measure is 

Therefore, insufficient information is given to answer this question.

The triangle has an exterior angle of , so it has an interior angle of . This is an obtuse angle; the other two angles must be acute, and therefore, they will have measure less than  - and, subsequently, the  angle will be the one of greatest measure.