If the measure of angle F is x, then the sum of the measures of angles D and E is 2x. Since there are 180° in a triangle, x + 2x = 180, 3x = 180, and x = 60. Since the measure of an exterior angle is equal to the sum of the measures of the interior angles to which the exterior angle is not adjacent, the measure of an angle exterior to F is equal to 2(60) = 120°.

Since there are 180° in a triangle, x² + 1 + 9x – 7 + 6x + 2 = 180, x² + 15x – 4 = 180, x² + 15x – 184 = 0, (x – 8)(x + 23) = 0, x = 8 (disregard the negative value of x since an angle cannot have a negative measure). Therefore, the measure of angle 1 is (8)² + 1 = 64 + 1 = 65, the measure of angle 2 is 9(8) – 7 = 72 – 7 = 65, and the measure of angle 3 is 6(8) + 2 = 48 + 2 = 50°. Since exactly two of the angles of triangle ABC are equal, triangle ABC is isosceles

If one exterior angle is taken from each vertex of a triangle, the sum of these exterior angles is 360°; 7x + 2 + 8x + 8x – 10 = 23x – 8 = 360, 23x = 368, x = 16. Therefore, angle 4 measures 7(16) + 2 = 112 + 2 = 114. An exterior angle and its adjacent interior angle are supplementary, so the measure of angle 1 is equal to: 180 – 114 = 66°.

Angles 3 and 6 are supplementary. Therefore, the measure of angle 3 = 180 – 115 = 65. The measure of an exterior angle is equal to the sum of the measures of the interior angles to which the exterior angle is not adjacent. Therefore, the sum of angles 2 and 3 is equal to the measure of angle 4: 75 + 65 = 140°.

The measure of an exterior angle is equal to the sum of the measures of the interior angles to which the exterior angle is not adjacent. Therefore, the measure of angle 5 is equal to the sum of the measures of angles 1 and 3, not angles 1 and 2. It is true that the sum of angles 4 and 1 is equal to the sum of angles 3 and 6, since both pairs of angles form lines. It is also true that the sum of angles 2 and 3 is equal to the measure of angle 4, since angle 4 is an exterior angle that is not adjacent to 2 or 3. Since there are 180° in a triangle, the sum of angles 1, 2, and 3 is equal to 180°. The sum of the measures of one exterior angle from each vertex of a triangle is 360°, so the statement in choice e is also true.

Since an angle exterior to angle F is 120°, the measure of interior angle F is 60°, and the sum of the measures of interior angles D and E is 120°. Angles D and E could each measure 60°, making triangle DEF acute and equilateral, but these angles could also measure 100° and 20° respectively, making triangle DEF obtuse and scalene However, triangle DEF cannot be isosceles. Angle F measures 60°; if either angles D or E measure 60°, the other must also measure 60°, making triangle DEF equilateral. Angles D and E cannot be congruent to each other without also being congruent to angle F. Therefore, triangle DEF can be acute, obtuse, scalene, or equilateral, but not isosceles.

The measures of the angles of a triangle add to 180°. Therefore, 2x + 5 + 2x + 5 + 3x – 5 = 180, 7x + 5 = 180, 7x = 175, and x = 25. The measure of angle A is 2(25) + 5 = 50 + 5 = 55. Since an angle and its exterior angle are supplementary, the measure of an angle exterior to A is 180 – 55 = 125°

An angle and its exterior angle are supplementary. Therefore, 8x + 16x + 12 = 180, 24x + 12 = 180, 24x = 168, x = 7. Since x = 7, the measure of angle F = 8(7) = 56°.

The measures of the angles of a triangle add to 180°. Therefore, 5x + 10 + x + 10 + 2x = 180, 8x + 20 = 180, 8x = 160, and x = 20. The measure of angle A is 5(20) + 10 = 110, the measure of angle B is (20) + 10 = 30, and the measure of angle C is 2(20) = 40. Since the largest angle of triangle ABC is greater than 90° and no two angles of the triangle are equal in measure, triangle ABC is obtuse and scalene.

The measures of the angles of a triangle add to 180°. Therefore, 3x + 4x + 5x = 180, 12x = 180,and x = 15.