Answer:
16.:
x = 5
m ∠ABC = 60°
17.:
m ∠RQS is 102°.
m ∠TQS is 78°
18.:
m ∠EFG = 75°
m ∠GFH = 105°
Explanation:
16.:
- An angle bisector splits one angle into two equal angles.
- Thus, ∠ABD + ∠CBD = ∠ABC and ∠ABD = ∠CBD.
Solving for x:
We can now find x by setting angles ABD and CBD equal to each other:
ABD = CBD
(7x - 5 = 5x + 5) + 5
(7x = 5x + 10) - 5x
(2x + 10) / 2
x = 5
Thus, x = 5.
Finding the measure of ∠ABC:
Now we can find the measure of ∠ABC by plugging in 5 for x in either 7x - 5 or 5x + 5 and multiplying by 2. Let's do 7x - 5:
m∠ABC = 2 * (7(5) - 5)
m ∠ABC = 2 * (35 - 5)
m ∠ABC = 2 * (30)
m ∠ABC = 60
Thus, the measure of ∠ABC is 60°.
17.:
- A straight angle is 180°.
- Thus, the measure of ∠RQT is 180.
Solving for x:
Since RQS + TQS = 180°, we can first find x by setting the sum of the two angles equal to 180:
14x + 4 + 11x + 1 = 180
(14x + 11x) + (4 + 1) = 180
(25x + 5 = 180) - 5
(25x = 175) / 25
x = 7
Thus, x = 7
Finding m∠RQS:
Now we plug in 7 for x in 14x + 4 to find m ∠RQS:
m ∠RQS = 14(7) + 4
m ∠RQS = 98 + 4
m ∠RQS = 102
Thus, m ∠RQS is 102°.
Finding m ∠TQS:
Since m ∠RQS + m ∠TQS = 180°, we can find m ∠TQS by subtracting 102 from 180:
m ∠RQS + m ∠TQS = 180
m ∠TQS = 180 - m ∠RQS
m ∠TQS = 180 - 102
m ∠TQS = 78
Thus, m ∠TQS is 78°
18.:
- A linear pair is defined as two adjacent angles that form a straight angle.
- This means that the sum of a linear pair is 180°.
Finding n:
We can first find n by setting the sum of m ∠EFG and m ∠GFH equal to 180°:
m ∠EFG + m ∠GFH = 180°
3n + 24 + 4n + 37 = 180
(3n + 4n) + (24 + 37) = 180
(7n + 61 = 180) - 61
(7n = 119) / 7
n = 17
Thus, n = 17.
Finding m ∠EFG:
Now we can find m ∠EFG by plugging in 17 for n in 3n + 24:
m ∠EFG = 3(17) + 24
m ∠EFG = 51 + 24
m ∠EFG = 75
Thus, m ∠EFG = 75°.
Finding m ∠GFH:
Now we can find m ∠GFH by plugging in 17 for n in 4n + 37:
Now we can find m ∠GFH by plugging in 17 for n in 4n + 37:
m ∠GFH = 4(17) + 37
m ∠GFH = 68 + 37
m ∠GFH = 105
Thus ,m ∠GFH = 105°.