Recall the Angle-Bisector Theorem.If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.The following question is based on a theorem (not stated) that is the converse of the Angle-Bisector Theorem.iGiven:NP = 12, MN = 18, PQ = 8, and MQ = 12;%3D%3DmzP = 62° and mzM = 36°Find:M QNM in degreesNPHint:PQMNMQPQNPShow thatMNby finding the product of the means and the product of the extremes.MQ%3DMN · PQ%3DNP· MQ%3DThis implies that NQ bisects MNPFind the measures of the following angles in degrees.%3DNeed Help?Read It

1 MN.PQ=188= 144 NP.MQ=1212= 144 ∴ MN.PQ=NP.MQ ⇒NPMN=PQMQ proved2 This implies NQ→ bisects ∠ MNP 3 ∵m∠P =62° , m∠M=36° ∵ m∠P+m∠M+∠PNM = 180° ⇒ m∠PNM = 180°-(62+36) ∴ M∠PNM = 82° 4 m∠QNM =12m∠PNM=1282 ⇒ m∠QNM = 41°

Math

Geometry

Recall the Angle-Bisector Theorem. If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle. The following question is based on a theorem (not stated) that is the converse of the Angle-Bisector Theorem. Given: NP = 12, MN = 18, PQ = 8, and MQ = 12; mzP = 62° and m.M = 36° Find: M QNM in degrees NP Hint: MN PQ MQ PO NP Show that MN by finding the product of the means and the product of the extremes. MQ %3D MN · PQ = NP · MQ = This implies that NQ bisects MNP Find the measures of the following angles in degrees. m.PNM = M.QNM = Need Help? Read It

Recall the Angle-Bisector Theorem. If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle. The following question is based on a theorem (not stated) that is the converse of the Angle-Bisector Theorem. Given: NP = 12, MN = 18, PQ = 8, and MQ = 12; mzP = 62° and m.M = 36° Find: M QNM in degrees NP Hint: MN PQ MQ PO NP Show that MN by finding the product of the means and the product of the extremes. MQ %3D MN · PQ = NP · MQ = This implies that NQ bisects MNP Find the measures of the following angles in degrees. m.PNM = M.QNM = Need Help? Read It

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

Recall the Angle-Bisector Theorem.
If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.
The following question is based on a theorem (not stated) that is the converse of the Angle-Bisector Theorem.
i
Given:
NP = 12, MN = 18, PQ = 8, and MQ = 12;
%3D
%3D
mzP = 62° and mzM = 36°
Find:
M QNM in degrees
NP
Hint:
PQ
MN
MQ
PQ
NP
Show that
MN
by finding the product of the means and the product of the extremes.
MQ
%3D
MN · PQ
%3D
NP· MQ
%3D
This implies that NQ bisects MNP
Find the measures of the following angles in degrees.
%3D
Need Help?
Read It

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