Question

explain 4 using the angle addition postulate
teacher voice:
postulate 1 - 4 angle addition postulate
words: if p is the interior of ∠rst, then the measure of ∠rst is equal to the sum of the measures of ∠rsp and ∠pst.
symbols: if p is in the interior of ∠rst, then m∠rst = m∠rsp + m∠pst.
example 4 - finding angle measures
given that ∠klm is a straight angle, find m∠kln and m∠nlm.
solution
step 1: write and solve an equation to find the value of x.
m∠kln + m∠nlm = m∠klm (write the angle addition postulate)
(10x - 5)° + (4x + 3)° = 180° (substitute angle measures)
14x - 2 = 180 (combine like terms)
14x = 182 (add 2 to both sides)
x = 13 (divide each side by 14)
step 2: evaluate the given expressions when x = 13.
m∠kln = (10x - 5)° = 10(13) - 5 = 125
m∠nlm = (4x + 3)° = 4(13) + 3 = 55
so, m∠kln = 125°, and m∠nlm = 55°.
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try this video for extra example 4 - finding angle measures

1. given that ∠pqr = 102°, find m∠sqr and m∠pqs.

Explanation:

Step1: Apply Angle Addition Postulate

$m\angle PQS + m\angle SQR = m\angle PQR$

Step2: Substitute angle measures

$(7x - 5)^\circ + (9x + 11)^\circ = 102^\circ$

Step3: Combine like terms

$16x + 6 = 102$

Step4: Isolate term with $x$

$16x = 102 - 6$
$16x = 96$

Step5: Solve for $x$

$x = \frac{96}{16} = 6$

Step6: Calculate $m\angle PQS$

$m\angle PQS = (7x - 5)^\circ = 7(6) - 5 = 37^\circ$

Step7: Calculate $m\angle SQR$

$m\angle SQR = (9x + 11)^\circ = 9(6) + 11 = 65^\circ$

Answer:

$m\angle PQS = 37^\circ$, $m\angle SQR = 65^\circ$