Question

angle addition postulate & segment addition postulate date

1. find m∠lmn if m∠lmt = 23° and m∠tmn = 144°.
2. find x if m∠ghi = 173°, m∠ghq = 29 + x, and m∠qhi = x + 162
3. m∠tmn = x + 72, m∠lmn = 166°, and m∠lmt = 96 + x. find m∠lmt.

find the length indicated.

1. p 7? r

q 8
solve for x.

1. t 2x - 4 2x - 7 v

17

Explanation:

Step1: Use angle - addition postulate for question 1

By the angle - addition postulate, \(m\angle LMN=m\angle LMT + m\angle TMN\). Given \(m\angle LMT = 23^{\circ}\) and \(m\angle TMN=144^{\circ}\), then \(m\angle LMN=23^{\circ}+144^{\circ}=167^{\circ}\).

Step2: Use angle - addition postulate for question 2

Since \(m\angle GHI=m\angle GHQ + m\angle QHI\), we have the equation \(173=(29 + x)+(x + 162)\).
Combine like - terms: \(173=2x+191\).
Subtract 191 from both sides: \(2x=173 - 191=-18\).
Divide both sides by 2: \(x=-9\).

Step3: Use angle - addition postulate for question 3

By the angle - addition postulate, \(m\angle LMN=m\angle LMT + m\angle TMN\). Substitute the given expressions: \(166=(96 + x)+(x + 72)\).
Combine like - terms: \(166=2x + 168\).
Subtract 168 from both sides: \(2x=166 - 168=-2\).
Divide both sides by 2: \(x=-1\). Then \(m\angle LMT=96+x=96-1 = 95^{\circ}\).

Step4: Use segment - addition postulate for question 4

By the segment - addition postulate, if \(PR = 8\) and \(PQ = 7\), then \(QR=PR - PQ\). So \(QR=8 - 7 = 1\).

Step5: Use segment - addition postulate for question 5

By the segment - addition postulate, \(TV=TU+UV\). So \(17=(2x - 4)+(2x - 7)\).
Combine like - terms: \(17 = 4x-11\).
Add 11 to both sides: \(4x=17 + 11=28\).
Divide both sides by 4: \(x = 7\).

Answer:

1. \(m\angle LMN = 167^{\circ}\)
2. \(x=-9\)
3. \(m\angle LMT = 95^{\circ}\)
4. The length of \(QR\) is 1
5. \(x = 7\)