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250. THEOREM. The bisector of an angle of a triangle divides the opposite side into segments whose ratio is the

same as that of the adjacent sides.

Given CD bisecting / C in ABC.

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Now show that ABCE is isosceles, and hence that BC

may be substituted for CE in the above proportion. Complete the proof.

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1. Fill in the blank spaces in the table, if in the figure of $ 250 CD is the bisector of ∠ACB.

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24.5

18.3

32.6

252. Definition. A segment is said to be divided externally by any point which lies on the line of the segment but not on the segment itself.

E.g. point C divides the segment AB externally, the parts being AC and CB, while point B divides the segment AC internally, the parts being AB and BC.

A

B

C

253. THEOREM. A line which bisects an exterior angle of a triangle divides the opposite side externally into two segments whose ratio is the same as that of the adjacent sides of the triangle.

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Given CD, the bisector of the exterior angle at C of the triangle

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1. Draw a triangle with an acute exterior angle bisected. Using different lettering from that in § 253, prove the theorem again.

2. Compare the proofs in §§ 250 and 253. Give the proof in § 253 for a figure in which AC <BC.

3. Fill in the blank spaces in the table below if CD is the bisector of the exterior angle BCK.

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-K

4. To measure indirectly the distance from an accessible point A to an inaccessible point B by means of § 253.

SUGGESTION. Through C, a point where A and B are both visible, draw CK making 21 = 22. Produce KC to a point Don the line BA extended. D.

C

K

2

A

B

What lines must now be measured in order to compute AB?

5. What methods have been used so far for the indirect measurement of the distance from an accessible to an inaccessible point? Compare these

(a) As to the simplicity of the theory involved.

(b) As to the simplicity and ease of the direct measurements required.

6. Divide a given line-segment in a given ratio without constructing a line parallel to another.

7. Similarly divide a given line-segment externally in a given ratio.

8. Solve Ex. 3, § 249, if x is on Cy extended.

9. Solve Ex. 4, § 249, if x is on Oy extended.

SIMILAR POLYGONS.

255. Two polygons, in which the angles of the one are equal respectively to the angles of the other, taken in order, are said to be mutually equiangular.

The angles of the two polygons are thus arranged in pairs of equal angles, which are called corresponding angles.

Two sides, one of each polygon, included between corresponding angles, are called corresponding sides.

256. Two polygons are similar if (1) they are mutually equiangular and if (2) their pairs of corresponding sides are proportional.

Two polygons may have property (1) but not (2). For example, a rectangle and a square. Or they may have property (2) and not (1). For example, a square and a rhombus.

Hence any proof that two polygons are similar must show that both (1) and (2) hold concerning them.

In the case of triangles it will be proved that either property specified in the definition of similar polygons is sufficient to make them similar.

257. THEOREM. If two triangles are mutually equiangular, they are similar.

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ABC and A'B'C', in which ∠ A = A', ∠B = B',

Given

and ∠C=∠C'.

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To prove that the other property of similarity holds,

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Proof: Place

A'B'C' on ABC with A' upon its

AB

AC

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equal ∠A, and B'C' taking the position DE.

Now show that DE || BC and hence

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In like manner, placing ∠B' upon ∠ B,

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1. To measure indirectly the distance from an accessible point A

to an inaccessible point B.

SUGGESTION. Construct AD 1 the line of sight

A

B

CL

D

from A to B, and ED 1 AD. Let C be the point E---

on AD which lies in line with E and B.

Now show that & EDC and BAC are mutually equiangular and hence similar. What segments need to be measured in order to compute AB? Give full details of proof.

2. Prove that two right triangles are similar if they have an acute angle of one equal to an acute angle of the other.

3. Two isosceles triangles are similar if they have the vertical angle of one equal to the vertical angle of the other.

4. Two triangles which have the sides of one respectively parallel or perpendicular to the sides of the other are similar.

5. Show by similar triangles that the segment joining the midpoints of two sides of a triangle is equal to one half the third side.

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