Lesson 5-4 Prove ∆ Midsegment Theorem Use ∆ Midsegment Theorem

Prove ∆ Midsegment Theorem A Midsegment of a ∆ is parallel to a side of the ∆ and equals half the length of that side. ∆ Midsegment Thm BC A MN =2●

A B Use ∆ Midsegment Theorem C XY A B C M N EXAMPLE 1

Use ∆ Midsegment Theorem EXAMPLE 2 Find Each measure. C A E BD 26 o F xoxo a)BD b)m<CBD AE 2 = 17 2 = 8.5 = = m<BDF26 o = ∆ Midsegment Thm Alternate interior <s

Question 1 Use the diagram for exercises 1-6 1) Find each measure NM Y ZX ML 29 o 7 10 N = YX 2 = 5 = ) XZ= 2●ML = 2●(7) = 14 3) NZ= ZX 2 = 7 = ) = Alt. int.<s = 29 o m<LMN m<MNZ 29 o 5) = Corresponding.<s = 29 o m<YXZ m<MNZ 6) = m<XLM180 o – m<LMN = 180 o – 29 o Same side.<s = 151 o

Question 2 Use the diagram for exercises ) Find each measure GJ J G H P R 55 o Q = 2●PQ = 38 = 2●19 8) RQ= 9) RJ= GJ 2 = 19 = ) = Alt. int.<s = 55 o m<PQR m<QRJ 55 o 11) = Corresponding.<s = 55 o m<HGJ m<QRJ 12) = m<GPQ180 o – m<PQR = 180 o – 55 o Same side.<s = 125 o HG 2 = 13.5 = 27 2

Use ∆ Midsegment Theorem EXAMPLE 3 Find the value of n C A E BD 5n 8n + 10=2●5n ∆ Midsegment Thm 8n + 10 AE2●BD = 8n + 10=10n 10=2n n= 5

Use ∆ Midsegment Theorem Critical Thinking Draw a scalene ∆DEF. Label X, Y and Z as the midpoints of DE, EF and DF respectively. Connect the three midpoints. List all pairs of congruent <s in your drawing E D F Y Z m<EXY m<XYZ  X m<YZF  m<DXZ m<XZY  m<ZYF  m<ZXY m<XZD  m<XYE  m<D  m<E  m<F 

Question 3 Use the diagrams for exercises ) Find the value of n 13) 3n 54 n – ●3n=54 6n=54 n= 9 2●(n – 9)=35 2n – 18=35 2n=53 n= 26.5

Question 4 Use the diagram for exercises ) Find the perimeter of ∆GHJ Find each measure H JG LK 7 12 M 4 16) Find the perimeter of ∆KLM 17) What is the relationship between the two perimeters? Perimeter=JH+HG+GJ = =34 Perimeter=LK+KM+ML =7+4+6 =17 The perimeter of ∆GHJ is twice the perimeter of ∆KLM

Question 5 PQ is a Midsegment of ∆RST. What is the length of RT? A. 9 C. 45 B. 21 D. 63 S R T P Q x + 9 4x – 27 =2●(x + 9) 4x – 27= 2x x – 2x = x= 45 x= 22.5 RT=4(22.5) – 27=63

Question 6 ∆XYZ is the Midsegment triangle of ∆JKL, XY = 8, YK = 14, and m<YKZ = 67o. Which of the following CANNOT be determined? E. KL G. m<XZL F. JY H. m<KZY J KL YX o Z o