If the measure of angle F in triangle DEF is half the sum of the measures of angles D and E, what is the measure of an angle exterior to F?

Solution
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°.
If the measure of angle 1 is x^{2} + 1, the measure of angle 2 is 9x – 7, and the measure of angle 3 is 6x + 2, which of the following is true?
The diagram is not to scale, and every angle is greater than 0°.

Solution
Since there are 180° in a triangle, x^{2} + 1 + 9x – 7 + 6x + 2 = 180, x^{2} + 15x – 4 = 180, x^{2} + 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 angle 4 measures 7x + 2, angle 5 measures 8x, and angle 6 measures 8x – 10, what is the measure of angle 1?
The diagram is not to scale, and every angle is greater than 0°.

Solution
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°.
If angle 6 measures 115° and angle 2 measures 75°, what is the measure of angle 4?
The diagram is not to scale, and every angle is greater than 0°.

Solution
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°.
Which of the following number sentences is NOT true?
The diagram is not to scale, and every angle is greater than 0°.

Solution
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.
The measure of an angle exterior to angle F of triangle DEF measures 120°. Which of the following must be true?

Solution
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.
If the measures of angles A and B of triangle ABC are each 2x + 5 and the measure of angle C is 3x – 5, what is the measure of angle exterior to angle A?

Solution
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°
The measure of an angle exterior to angle F of triangle DEF measures 16x + 12. If angle F measures 8x, what is the measure of angle F?

Solution
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°.
If the measure of angle A of triangle ABC is 5x + 10, the measure of angle B is x + 10, and the measure of angle C is 2x, which of the following is true of triangle ABC?

Solution
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.
If the measure of angle A of triangle ABC is 3x, the measure of angle B is 5x, and the measure of angle C is 4x, what is the value of x?

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