For the triangle to be acute, all three angles must measure less than \(\displaystyle 90 ^{\circ }\). We can eliminate \(\displaystyle 32 ^{\circ }, 32 ^{\circ }, 116^{\circ }\) and \(\displaystyle 45 ^{\circ }, 45 ^{\circ }, 90^{\circ }\) for this reason. 

In an isosceles triangle, at least two angles are congruent, so we can eliminate \(\displaystyle 50 ^{\circ }, 60 ^{\circ }, 70 ^{\circ }\).

The degree measures of the three angles of a triangle must total 180, so, since \(\displaystyle 80 ^{\circ }+ 80 ^{\circ }+ 40 ^{\circ } = 200^{\circ }\), we can eliminate \(\displaystyle 80 ^{\circ }, 80 ^{\circ }, 40 ^{\circ }\).

\(\displaystyle 36^{\circ }, 72 ^{\circ }, 72 ^{\circ }\) is correct.

The degree measures of a triangle total \(\displaystyle 180^{\circ }\), so

\(\displaystyle x + (x+24) + 2x = 180\)

\(\displaystyle 4x+24 = 180\)

\(\displaystyle 4x+24 -24 = 180 -24\)

\(\displaystyle 4x = 156\)

\(\displaystyle 4x \div 4 = 156 \div 4\)

\(\displaystyle x = 39\)

The degree measures of the interior angles of a triangle total \(\displaystyle 180 ^{\circ }\), so, if we let \(\displaystyle x^{\circ }\) be the measure of the unmarked angle, then

\(\displaystyle x + 41 + 72 = 180\)

\(\displaystyle x + 113 = 180\)

\(\displaystyle x = 67\)

Three angles with measures \(\displaystyle 67^{\circ}, y^{\circ}, y^{\circ}\) together form a straight angle, so

\(\displaystyle y+ y + 67= 180\)

\(\displaystyle 2y+ 67= 180\)

\(\displaystyle 2y+ 67- 67 = 180 - 67\)

\(\displaystyle 2y = 113\)

\(\displaystyle 2y \div 2 = 113 \div 2\)

\(\displaystyle y = 56 \frac{1}{2}\)

One of the angles of \(\displaystyle \Delta EFC\) - namely, \(\displaystyle \angle EFC\) - can be seen to be an obtuse angle, as it is wider than a right angle. This makes \(\displaystyle \Delta EFC\), by definition, an obtuse triangle.

One of the angles of \(\displaystyle \Delta CED\) - namely, \(\displaystyle \angle CDE\) - is marked as a right angle. This makes \(\displaystyle \Delta CED\), by definition, a right triangle.

A congruency statement about two triangles implies nothing about the relationship between two angles of one of the triangles, so \(\displaystyle \angle M \cong \angle N\) is not correct.

Also, letters in the same position between the two triangles refer to corresponding - and subsequently, congruent - angles. Therefore, \(\displaystyle \Delta MNO \cong \Delta PQR\) implies that:

\(\displaystyle \angle M \cong \angle P\)

\(\displaystyle \angle N \cong \angle Q\)

\(\displaystyle \angle O \cong \angle R\)

Of the given choices, only \(\displaystyle \angle M \cong \angle P\) is a consequence. It is the correct response.

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.