64. Corollary. — Two right triangles which have the legs of one equal respectively to the legs of the other are congruent. The proof is left to the student.

EXERCISES

1. Prove orally the theorem of § 63 by showing first that A ABC may be placed upon A DEF so that AC coincides with DF, A falling upon D and C upon F.

2. Prove orally the theorem of § 63 by showing first that A ABC may be placed upon A DEF so that ∠ A coincides with ∠ D.

65. Theorem. - If two angles and the included side of one triangle are equal respectively to two angles and the included side of another, the triangles are congruent.

NOTE. - If desirable, before attempting the following proof, triangles may be constructed on paper or cardboard with two angles and the included side of one equal respectively to two angles and the included side of the other, and one of them cut out and applied to the other as in § 61.

Hypothesis. In ABC and △DEF, ∠ A = ∠ D, ∠ B = ∠E, and AB = DE.

Conclusion. △ABC ≃ △ DEF.

Suggestions. Since △ABC may be placed upon A DEF SO that one pair of equal parts coincide, in how many ways may this be done? If AB is made to coincide with its equal DE, A falling upon D and B upon E, where must AC fall? Why? At the same time where must BC fall? Why? Why, then, at the same time must fall upon F? Write out a complete proof.

66. Corollary 1. - If two angles and any side of one triangle are equal respectively to two angles and the similarly placed side of the other, the triangles are congruent.

For, by § 49, the third pair of angles must be equal, and hence the triangles must be congruent by § 65.

67. Corollary 2. — Two right triangles are congruent if a leg and an acute angle of one are equal respectively to a leg and an acute angle of the other.

The proof is left to the student.

68. Corollary 3. - Two right triangles are congruent if the hypotenuse and an acute angle of one are equal respectively to the hypotenuse and an acute angle of the other.

The proof is left to the student.

69. Distances and angles compared by congruence of triangles. - Since the corresponding parts of congruent triangles are equal, angles or distances may be proved equal by proving that triangles of which they are corresponding parts are congruent. This is one of the chief uses of congruent triangles, and is illustrated in the following exercises.

EXERCISES

1. In order to find the distance from A to B across a lake, surveyors measured off a straight line AC and extended it to D so that CD = AC. Then they measured the distance BC and

extended BC to E so that CE = BC.
they measured the distance from E to D. Prove
that AB equals the measured distance DE.

Then

SUGGESTION. - Prove the triangles congruent.

2. The distance AB across a stream may be found as follows: Measure AC at right angles to AB. Locate a point D halfway between A and C. Measure CE at right angles to AC, to a point E in line with B and D. Prove that AB equals CE.

3. The Greeks, as early as the time of Thales (640-546 в.с.), determined the distance AP of a ship from shore by

P

means of the congruence of triangles having two angles and the included side of one equal respectively to two angles and the included side of the A other. They measured 21 and 22. Explain how they were then able to mark off a distance on the

C

shore equal to AP and thus find the distance AP.

4. The distance from A to an inaccessible point B may be found approximately without instruments as follows: Stand at A and look at B, raising or lowering the head until your

Pace the distance from A to C. In the diagram, D represents the position of the eyes. Prove that AB = AC.

5. Thales (See Ex. 3) is said to have made an instrument for determining the distance BP of a ship from shore. It consisted of two rods AC and AD, hinged together at A. Rod

A

D

AD was held vertically over point B, while
rod AC was pointed toward P. Then,
without changing ∠DAC, the instrument
was revolved about AD, and point Q noted
on the ground at which the arm AC was
directed. BQ was then measured. Prove that BP = BQ.

C

P

B

6. An instrument used as late as the sixteenth century for finding the distance from A to an inaccessible point B was called a cross-staff. It consisted of a vertical staff AC to which

C

was attached a horizontal cross-bar DE

D

E

that could be moved up or down on

the staff. Sighting from C to B, DE
was lowered or raised until C, E, and
B were in a straight line. Then the
instrument was revolved about CA and
the point Fat which the line of sight
CE met the ground marked. The F
distance AF was then measured. Prove that AB = AF.

B

NOTE. - Students may easily make and use this instrument.

7. The calipers shown in the drawing may be used for measuring the thickness of objects, such as the diameters of pipes, the diameters of trees in forestry, etc. AD and BC are

distance between A and B equals the distance between C and D. Hence explain how the instrument may be used.

E

8. Every one is familiar with the fact that if an object is placed before a plane mirror, its image appears to be as far behind the mirror as the object is in front. Mis an edge view of a mirror. Light from an object at A strikes the mirror at D and is reflected to the eye at E. The mind projects the line ED through the mirror to B, forming the image at B. It is known from science that ∠ MDE = 2 ADC and that CD ⊥ AB. Prove that CB = AC.

M
D

A

B

C

9. If two oblique line-segments drawn from a point on a perpendicular to a line cut off equal distances on the line from the foot of the perpen- A dicular, prove that the oblique segments are equal.

10. In the adjoining figure, AB bisects ∠A and also ∠B. Prove that ZC = 2 D.

11. Prove that if the bisector of the angle at a vertex of a triangle is perpendicular to the opposite side, the triangle must be isosceles.

12. In the annexed figure, BP is the bisector of ∠ABC, and PC ⊥BC and PA I BA. Prove that PC = PA.

13. Prove that the medians drawn to the equal sides of an isosceles triangle are equal.

C

A

B

D

C

P

B

A

14. Prove that all of the medians of an equilateral triangle are equal.

15. Prove that the altitudes to the equal sides of an isosceles triangle are equal.

16. Prove that all of the altitudes of an equilateral triangle are equal.

70. Theorem. - In any isosceles triangle the angles opposite the equal sides are equal.

Suggestions. It may be shown that ∠ A = ∠ B by showing that they are corresponding angles of congruent triangles. How must CD be drawn in order to prove that AACD = ABCD by means of § 63? Hence begin by drawing the bisector CD of ∠ C. Draw an accurate figure with straightedge and compasses, and write the complete proof.

71. Corollary 1. - An equilateral triangle is also equiangular. The proof is left to the student.

72. Corollary 2. - Each angle of an equilateral triangle is 60°. The proof is left to the student.

73. Corollary 3. - In any isosceles triangle the bisector of the angle at the vertex, the median to the base, the altitude to the base, and the perpendicular bisector of the base all coincide.

Suggestion. In the proof of § 70 it was proved that CD, the bisector of the angle at the vertex, formed congruent triangles, △ ACD and △BCD. Show from this that CD is also the median to the base, altitude to the base, and perpendicular bisector of the base.