Question

8 multiple choice 2 points
triangle fgh and two of its angle measures are shown below.

what is the measure of angle f?
29°
91°
90°
89°
9 multiple choice 2 points
in △abc, m∠a = x, m∠b = 2x + 2, and m∠c = 3x + 4. what is the value of x?
59
61
29
31
10 multiple choice 2 points
the sides of a triangle measure 2, 3, and 4. if the smallest side of a similar triangle measures 8, then the measure of the largest side is
10
16
14
12
11 multiple choice 2 points
which is a correct statement about vertical angles?
vertical angles are always complementary.
vertical angles are always supplementary.
vertical angles are always congruent.
vertical angles are always acute.

Explanation:

Problem 8 Step1: Sum of triangle angles is $180^\circ$

Let $m\angle F = y$. The formula is $y + 31^\circ + 60^\circ = 180^\circ$

Problem 8 Step2: Calculate $\angle F$

$y = 180^\circ - 31^\circ - 60^\circ = 91^\circ$

Problem 9 Step1: Sum of triangle angles is $180^\circ$

The formula is $x + (2x+2) + (3x+4) = 180$

Problem 9 Step2: Simplify the equation

$6x + 6 = 180$

Problem 9 Step3: Solve for $x$

$6x = 174$, so $x = \frac{174}{6} = 29$

Problem 10 Step1: Find similarity ratio

Ratio: $\frac{8}{2} = 4$

Problem 10 Step2: Calculate largest side

Largest side: $4 \times 4 = 16$

Problem 11 Step1: Recall vertical angles property

Vertical angles are defined as congruent (equal in measure). Complementary/supplementary/acute do not always apply.

Answer:

1. 8. B. $91^\circ$
2. 9. C. 29
3. 10. B. 16
4. 11. C. Vertical angles are always congruent.