Proportions and Similar Triangles Section 7.5

Objectives Use the Triangle Proportionality Theorem and its converse.

Key Vocabulary Midsegment of a Triangle

Theorems 7.4 Triangle Proportionality Theorem 7.5 Converse of Triangle Proportionality Theorem 7.6 Triangle Midsegment Theorem

Review Parallel Lines & Angle Pairs Corresponding ∠ ’s ≅ –Example: ∠ 11 ≅ ∠ 15 Alternate Interior ∠’ s ≅ –Example: ∠ 12 ≅ ∠ 15 Alternate Exterior ∠’ s ≅ –Example: ∠ 10 ≅ ∠ 17 Consecutive Interior ∠’ s supplementary –Example: ∠ 12 & ∠ 14 Vertical ∠’ s ≅ –Example: ∠ 14 ≅ ∠ 17 Linear ∠’ s supplementary –Example: ∠ 16 & ∠ 17

Proportional Parts Within Triangles When a triangle contains a line that is parallel to one of its sides, the two triangles can be proved similar using the AA shortcut. Example: Since the triangles are similar, their sides are proportional. Which leads us to the Triangle Proportionality Theorem. ∠ D ≅∠ EGH & ∠ F ≅∠ EHGCorresponding ∠ s ∆ DEF ∼∆ GEH AA

Theorem 7.4 Triangle Proportionality Theorem Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths.

Example 1

Substitute the known measures. Cross Products Property Multiply. Divide each side by 8. Simplify.

Your Turn: A.2.29 B C.12 D.15.75

Example 2 Find the length of YZ.

Example 3 Find Segment Lengths Find the value of x. 4 8 x 12 = Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA. 4 · 12 = 8 · x Cross product property 48 = 8x Multiply = 8x8x 8 Divide each side by 8. SOLUTION CD DB = CE EA Triangle Proportionality Theorem 6 = x Simplify.

Example 4 Find Segment Lengths Find the value of y. 3 9 y 20 – y = Substitute 3 for PQ, 9 for QR, y for PT, and (20 – y) for TS. 3(20 – y) = 9 · y Cross product property 60 – 3y = 9y Distributive property PQ QR = PT TS Triangle Proportionality Theorem SOLUTION You know that PS = 20 and PT = y. By the Segment Addition Postulate, TS = 20 – y.

Example 4 Find Segment Lengths = 12y 12 Divide each side by – 3y + 3y = 9y + 3y Add 3y to each side. 60 = 12y Simplify. 5 = y Simplify.

Your Turn Find Segment Lengths and Determine Parallels Find the value of the variable ANSWER 8 10

Proportional Parts Within Triangles The converse of Theorem 7.4, Triangle Proportionality Theorem, is also true. Which is the next theorem.

Theorem 7.5 Converse of Triangle Proportionality Theorem Converse of the Triangle Proportionality Theorem If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.

Example 5 In order to show that we must show that

Example 5 Since the sides are proportional. Answer: Since the segments have proportional lengths, GH || FE.

Your Turn: A.yes B.no C.cannot be determined

Example 6 Determine Parallels Given the diagram, determine whether MN is parallel to GH. SOLUTION Find and simplify the ratios of the two sides divided by MN. LM MG = = 8 3 LN NH = = 3 1 ANSWER Because ≠ , MN is not parallel to GH.

Your Turn Find Segment Lengths and Determine Parallels Given the diagram, determine whether QR is parallel to ST. Explain. ANSWER Converse of the Triangle Proportionality Theorem. = Yes; || so QR ST by the ≠ no; ANSWER

Triangle Midsegment A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. A special case of the triangle Proportionality Theorem is the Triangle Midsegment Theorem.

Theorem 7.6 Triangle Midsegment Theorem A midsegment is a segment whose endpoints are the midpoints of two sides of a Δ.  Triangle Midsegment Theorem: A midsegment of a triangle is parallel to one side of the triangle, and its length is ½ the length of the side its parallel to. If D and E are midpoints of AB and AC respectively, then DE || BC and DE = ½ BC.

Example 7 Use the Midsegment Theorem Find the length of QS. 1 2 QS = PT = (10) = ANSWER The length of QS is 5. SOLUTION From the marks on the diagram, you know S is the midpoint of RT, and Q is the midpoint of RP. Therefore, QS is a midsegment of  PRT. Use the Midsegment Theorem to write the following equation.

Your Turn Use the Midsegment Theorem Find the value of the variable. ANSWER ANSWER Use the Midsegment Theorem to find the perimeter of  ABC.

Example 8a A. In the figure, DE and EF are midsegments of ΔABC. Find AB.

Example 8a Answer: AB = 10 ED = ABTriangle Midsegment Theorem __ 1 2 5= ABSubstitution __ = ABMultiply each side by 2.

Example 8b B. In the figure, DE and EF are midsegments of ΔABC. Find FE.

Example 8b Answer: FE = 9 FE = (18)Substitution __ FE = BCTriangle Midsegment Theorem FE = 9Simplify.

Example 8c C. In the figure, DE and EF are midsegments of ΔABC. Find m  AFE.

Example 8c By the Triangle Midsegment Theorem, AB || ED. Answer:  AFE = 87°  AFE   FEDAlternate Interior Angles Theorem m  AFE = m  FEDDefinition of congruence  AFE = 87Substitution

Your Turn: A.8 B.15 C.16 D.30 A. In the figure, DE and DF are midsegments of ΔABC. Find BC.

Your Turn: B. In the figure, DE and DF are midsegments of ΔABC. Find DE. A.7.5 B.8 C.15 D.16

Your Turn: C. In the figure, DE and DF are midsegments of ΔABC. Find m  AFD. A.48 B.58 C.110 D.122

Proportional Parts with Parallel Lines Another special case of the Triangle Proportionality Theorem involves three or more parallel lines cut by two transversals. Notice that if transversals a and b are extended, they form triangles with the parallel lines.

Example 9 MAPS In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find x.

Example 9 Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem. Answer: x = 32 Triangle Proportionality Theorem Cross Products Property Multiply. Divide each side by 13.

Your Turn: A.4 B.5 C.6 D.7 In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in between city blocks. Find x.

Proportional Parts with Parallel Lines If the scale factor of the proportional segments is 1, then the parallel lines separate the transversals into congruent parts.

Congruent Parts of Parallel Lines If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. B A C D E F

Example 10 ALGEBRA Find x and y. To find x: 3x – 7= x + 5Given 2x – 7= 5Subtract x from each side. 2x= 12Add 7 to each side. x= 6Divide each side by 2.

Example 10 To find y: The segments with lengths 9y – 2 and 6y + 4 are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal.

Example 10 Answer: x = 6; y = 2 9y – 2= 6y + 4Definition of congruence 3y – 2= 4Subtract 6y from each side. 3y= 6Add 2 to each side. y= 2Divide each side by 3.

Your Turn: Find a and b. A. ; B.1; 2 C.11; D.7; 3 __

Assignment Pg #1 – 29 odd

Joke Time How would you describe a frog with a broken leg? Unhoppy What did the horse say when he got to the bottom of his feed bag? That’s the last straw!