The triangle has an exterior angle of \(\displaystyle 130 ^{\circ }\), so it has an interior angle of \(\displaystyle (180-130)^{\circ } = 50^{\circ }\). By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

Case 1: Two angles have measure \(\displaystyle 50^{\circ }\). The third angle will have measure

\(\displaystyle \left [180 - (50 + 50 ) \right ] ^{\circ } = 80 ^{\circ }\).

\(\displaystyle 80 ^{\circ }\) will be the greatest of the angle measures.

Case 2: One angle has measure \(\displaystyle 50^{\circ }\) and the others are congruent. Their common measure will be

\(\displaystyle \frac{1}{2} (180^{\circ }- 50^{\circ } ) = 65 ^{\circ }\).

\(\displaystyle 65 ^{\circ }\) will be the greatest of the angle measures.

The given information is therefore inconclusive.

The triangle has an exterior angle of \(\displaystyle 130 ^{\circ }\), so it has an interior angle of \(\displaystyle (180-130)^{\circ } = 50^{\circ }\). By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

Case 1: Two angles have measure \(\displaystyle 50^{\circ }\). The third angle will have measure

\(\displaystyle \left [180 - (50 + 50 ) \right ] ^{\circ } = 80 ^{\circ }\).

\(\displaystyle 50^{\circ }\) will be the least of the angle measures.

Case 2: One angle has measure \(\displaystyle 50^{\circ }\) and the others are congruent. Their common measure will be

\(\displaystyle \frac{1}{2} (180^{\circ }- 50^{\circ } ) = 65 ^{\circ }\).

\(\displaystyle 50^{\circ }\) will be the least of the angle measures.

In both cases, the least of the degree measures of the angles will be \(\displaystyle 50^{\circ }\).

The triangle has an exterior angle of \(\displaystyle 70 ^{\circ }\), so it has an interior angle of \(\displaystyle (180-70)^{\circ } = 110 ^{\circ }\).

Since this is an obtuse angle, its other two angles must be acute. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles - the acute angles are those. Since the sum of their measures is the same as their remote exterior angle - \(\displaystyle 70 ^{\circ }\) - each has measure \(\displaystyle 35 ^{\circ }\).This is the correct response.

The triangle has an exterior angle of \(\displaystyle 80 ^{\circ }\), so it has an interior angle of \(\displaystyle (180-80)^{\circ } = 100 ^{\circ }\).

Since this is an obtuse angle, its other two angles must be acute. Therefore, this angle is the one of greatest measure.

The degree measures of a triangle must total \(\displaystyle 180 ^{\circ }\), so we find the sum of the degree measures in each choice.

\(\displaystyle 75^{\circ }, 75^{\circ }, 65^{\circ }\): 

\(\displaystyle 75 + 75 + 65 = 215\)

\(\displaystyle 25^{\circ }, 65^{\circ }, 100^{\circ }\):

\(\displaystyle 25 + 65 + 100 = 190\)

\(\displaystyle 35^{\circ },40^{\circ }, 95^{\circ }\):

\(\displaystyle 35 + 40 + 95 = 170\)

\(\displaystyle 50 ^{\circ }, 55^{\circ }, 75^{\circ }\):

\(\displaystyle 50 + 55 + 75 = 180\)

This is the correct choice.

Since corresponding parts of congruent triangles are congruent, it follows that \(\displaystyle \angle BAX \cong \angle CAX\). Therefore, \(\displaystyle \overline{AX}\), by definition, bisects \(\displaystyle \angle BAC\), and the first statement is true. A bisector of an angle of an equilateral triangle is also the perpendicular bisector of the opposite side, so the other two statements are immediate consequences. 

Every equilateral triangle has three congruent angles - all of which have measure \(\displaystyle 60 ^{\circ }\) - as a result of the Isosceles Triangle Theorem. Therefore, the statements:

 \(\displaystyle m \angle C = 60 ^{\circ }\)

\(\displaystyle m \angle B = m \angle C\)

\(\displaystyle \Delta ABC\) is an acute triangle

immediately follow.

However, if \(\displaystyle \angle A\) and \(\displaystyle \angle B\) are complementary, then the sum of their measures is \(\displaystyle 90 ^{\circ }\); since each angle measures \(\displaystyle 60 ^{\circ }\), then their measures add up to \(\displaystyle 120^{\circ }\). This is the correct choice.

By the Isosceles Triangle Theorem, angles opposite congruent sides of a triangle are congruent. Therefore, a scalene triangle, having three noncongrent sides, must have three noncongruent angles.

\(\displaystyle 50 ^{\circ } , 50 ^{\circ }\) can be eliminated immediately.

For the other three choices, we need to find the measure of the third angle using the fact that the degree measures of the angles of a triangle must total \(\displaystyle 180 ^{\circ }\).

\(\displaystyle 140^{\circ } , 20^{\circ }\): 

The third angle measure is \(\displaystyle 180^{\circ } - \left (140^{\circ } + 20^{\circ } \right ) = 180^{\circ } - 160^{\circ } = 20^{\circ }\).

This triangle has two \(\displaystyle 20^{\circ }\) angles and can be eliminated.

\(\displaystyle 70 ^{\circ } , 40 ^{\circ }\):

The third angle measure is \(\displaystyle 180^{\circ } - \left (70^{\circ } + 40^{\circ } \right ) = 180^{\circ } - 110^{\circ } = 70^{\circ }\).

This triangle has two \(\displaystyle 70^{\circ }\) angles and can be eliminated.

\(\displaystyle 80^{\circ } , 40^{\circ }\):

The third angle measure is \(\displaystyle 180^{\circ } - \left (80^{\circ } + 40^{\circ } \right ) = 180^{\circ } - 120^{\circ } = 60^{\circ }\).

This triangle has three angles of different measure, making it scalene. This is the correct choice.

One yard is equal to three feet, and one foot is equal to twelve inches. Therefore, 9 feet is equal to \(\displaystyle 9 \times 12 = 108\) inches, and 3 yards is equal to \(\displaystyle 3 \times 36 = 108\) inches. The triangle has sides of measure 90 inches, 108 inches, and 108 inches. Exactly two sides are of equal measure, so it is isosceles but not equilateral.

The sum of the three interior angles of a triangle is 180 degrees.

Set up an equation such that all three angles are equal to 180 degrees.

\(\displaystyle 40+3x-3+3x-3 = 180\)

Combine like-terms.

\(\displaystyle 6x+34 = 180\)

Subtract 34 on both sides.

\(\displaystyle 6x+34-34 = 180-34\)

\(\displaystyle 6x=146\)

Divide by 6 on both sides.

\(\displaystyle \frac{146}{6} =\frac{73}{3}\)

The answer is:  \(\displaystyle \frac{73}{3}\)