486. Triangles that have their corresponding sides proportional are similar.

Compare (1) and (2), remembering that BC=EN, and show that AC MN.

Prove ABC and MEN equal in all respects.

▲ DEF and MEN have been proved similar, and since ▲ ABC and MEN are equal in all respects, ▲ DEF and ABC are similar.

Q.E.D

487. EXERCISE. The sides of a triangle are 6 in., 8 in., and 12 in. respectively. The sides of a second triangle are 6 in., 3 in., and 4 in. respectively. Are they similar?

488. SCHOLIUM. Polygons must fulfill two conditions in order to be similar, i.e. they must be mutually equiangular, and must have their corresponding sides proportional. Propositions XV. and XVI. show that in the case of triangles, either of these conditions involves the other. Hence to prove triangles similar, it will be sufficient to show either that they are mutually equiangular, or that their corresponding sides are proportional.

489. EXERCISE.

A piece of cardboard 8 in. square is cut into 4 pieces, A, B, C, and D, as shown in the first figure. These pieces, as placed in the second figure, apparently, form a rectangle whose area is 65 sq. in.

Explain the fallacy by means of similar triangles.

5 3

490. EXERCISE. The sides of a triangle are 12, 16, spectively. A similar triangle has one side 8 ft. in length. length of the other two sides? (Three solutions.)

5

and 24 ft. re

What is the

491. EXERCISE. On a given line as a side construct a triangle similar to a given triangle. [Construct in two ways. Use § 478 and also § 486.]

492. EXERCISE. Construct a triangle that shall have a given perimeter, and shall be similar to a given triangle.

493. EXERCISE. If the sides of one triangle are inversely proportional to the sides of a second triangle, the triangles are not necessarily similar. [Let the sides of the first triangle be in the ratio of 2, 3, and 4. Then the sides of the second triangle are in the ratio of 1, 3, and 4, or f1⁄2, 12, and; and these fractions are in the ratio of the integers 6, 4, and 3. Therefore the triangles are not similar.]

494. EXERCISE. Any two altitudes of a triangle are inversely proportional to the sides to which they are respectively perpendicular.

PROPOSITION XVII. THEOREM.

495. Triangles that have an angle in each equal, and the including sides proportional, are similar.

496. EXERCISE. If a line is drawn parallel to the base of a triangle, and lines are drawn from the vertex to different points of the base, these lines divide the base and the parallel proportionally.

PROPOSITION XVIII. THEOREM.

497. Triangles that have their sides parallel, each to each, or perpendicular, each to each, are similar.

A

E

ал

F

Let AABC and DEF have AB | to DE, BC | to EF, and AC to DF.

To Prove ABC and DEF similar.

(§ 131 and

Proof. The angles of the ▲ ABC are either equal to the angles of ADEF, or are their supplements. § 132.)

There are four possible cases:

1. The three angles of AABC may be supplements of the angles of A Def.

2. Two angles of ▲ ABC may be supplements of two angles of A DEF, and the third angle of AABC equal the third angle of A DEF.

3. One angle of ▲ ABC may be the supplement of an angle of ▲ DEF, and the two remaining angles of ▲ ABC be equal to the two remaining angles of ▲ def.

4. The three angles of ▲ ABC may equal the three angles of A DEF.

Show that in the first case the sum of the angles of the two would be six right angles.

Show that in the second case the sum of the angles of the two would be greater than four right angles.

Show, by means of § 140, that the third case is impossible

unless the angles that are supplementary are right angles, in which case they would also be equal, and the triangles would have three angles of the one equal to three angles of the other.

Therefore if two triangles have their sides parallel, each to each, the triangles are mutually equiangular, and consequently similar.

F

Let AABC and DEF have AB DE, BC 1 EF, and AC LDF.

B

E

To Prove AABC and DEF similar.

A

C

Proof. The angles of ▲ ABC are either equal to the angles of A DEF, or are their supple

ments.

[Show, as was done in the first part of this proposition, that the angles of ▲ ABC are equal to those of A DEF, and consequently AABC and DEF are similar.]

Q.E.D.

NOTE. The equal angles are those that are included between sides that are respectively parallel or perpendicular to each other.

498. EXERCISE.

The bases of a trapezoid are 8 in. and 12 in., and the altitude is 6 in. Find the altitudes of the two triangles formed by producing the non-parallel sides until they meet.

D

499. EXERCISE. The angles ABC, DAE, and DBE are right angles.

Prove that two triangles in the diagram arc similar.

500. EXERCISE. The lines joining the middle points of the sides of a given triangle form a second triangle that is similar to the given triangle.

B

E

501. EXERCISE. The bisectors of the exterior angles of an equilateral triangle form by their intersection a triangle that is also equilateral.