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# Angles Practice 1

• To convert degrees to radians, multiply the degree measure by  $\frac{\pi&space;radians}{180^{0}}$.

Example:

Solution

$85^{^{0}}=85(\frac{\\\pi&space;radians}{180^{0}})=\frac{17\pi&space;}{36}$

• To convert from radians to degrees, multiply the radian measure by $\frac{180^{0}}{\pi&space;radians}$.

Example

Convert $\frac{\pi&space;}{6}$ radians to degrees.

Solution

$\frac{\pi&space;}{6}=\left&space;(&space;\frac{\pi&space;radians}{6}&space;\right&space;)\left&space;(\frac{180^{0}}{\pi&space;radians}&space;\right&space;)=30^{^{0}}$

1. In the figure below, angles 3m and 4n are vertical angles. What is the value of m in terms of n?

(A) $n$

(B) $12n$

(C) $\frac{n}{12}$

(D) $\frac{3n}{4}$

(E) $\frac{4n}{3}$

Solution

The two angles are vertical angles, so their measures are equal:

3m = 4n

To solve for m, divide both sides by 3:

$m=\frac{4n}{3}$

Thus, the right answer is (E)

2. In the figure below, v = 3u. What is the value of w?

(A) 15

(B) 30

(C) 45

(D) 60

(E) 75

Solution

Angles u and v are supplementary angles because they complete a line, so u + v = 180.

Substitute 3u for v into this equation and solve for u:

u + 3u = 180

4u = 180

u = 45

The two angles u and w are vertical angles, so they’re equal. Thus, w = 45, and the right answer is (C) .

3. In the following diagram, angle 1 is equal to 40°, and angle 2 is equal to 150°. What is the number of degrees in angle 3?

(A) 70°

(B) 90°

(C) 110°

(D) 190°

(E) The answer cannot be determined from the given information.

Solution

Choice C is correct. In the problem it is given that ∠1 = 40° and ∠2 = 150°. The diagram below makes it apparent that:

(1) ∠1 = ∠4 and ∠3 = ∠5 (vertical angles);

(2) ∠6 + ∠2 = 180° (straight angle);

(3) ∠4 + ∠5 + ∠6 = 180° (sum of angles in a triangle).

To solve the problem, ∠3 must be related through the above information to the known quantities in ∠1 and ∠2.

Proceed as follows:

∠3 = ∠5, but ∠5 = 180° – ∠4 – ∠6.

∠4 = ∠1 = 40° and ∠6 = 180° – ∠2 = 180° – 150° = 30°.

Therefore, ∠3 = 180° – 40° – 30° = 110°.

4. In the figure below, angles p and q are complementary, with 2p = 3q. What is the measure of angle p?

Solution

The problem states that angles p and q are complementary, so you have two equations to work with:

p + q = 90

2p = 3q

Solve the top equation for q:

q = 90 – p

Now substitute 90 – p for q in the second equation and solve for p:

2p = 3(90 – p)

2p = 270 – 3p

5p = 270

p = 54