80. Polygons classified as to the number of sides. - Polygons are classified according to the number of sides. A polygon of three sides is a triangle. A polygon of four sides is called a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of eight sides, an octagon; one of ten sides, a decagon; etc.

81. Quadrilaterals.

no sides parallel.

A trapezium is a quadrilateral having

A trapezoid is a quadrilateral having one and only one pair of opposite sides parallel. The parallel sides of a trapezoid are called the bases. The perpendicular distance between the bases is called the altitude. An isosceles trapezoid is one of which the non-parallel sides are equal.

A parallelogram is a quadrilateral having both pairs of opposite sides parallel. Any one of the four sides of a parallelogram may be considered the base, and the perpendicular distance between it and the opposite side, the altitude.

PARALLELOGRAM

RECTANGLE SQUARE

RHOMBUS

A rectangle is a parallelogram all of whose angles are

right angles.

A square is a rectangle all of whose sides are equal.

A rhombus is a parallelogram all of whose sides are equal

but whose angles are not right angles.

82. Theorem. - The opposite sides of a parallelogram are equal.

4. Then in △ ABD and △ BCD, DB is a common side.

5. ∠ABD = ∠ BDC and ∠ ADB = ∠ DBC.

6. .·. Δ ABD = △ BCD.

7. ... AB = DC and AD = BC.

§ 29

§ 65

Def. Congruence

83. Corollary 1. - A diagonal divides a parallelogram into

two congruent triangles.

This follows from step 6 in § 82.

84. Corollary 2. - Segments of parallel lines cut off by two parallel lines are equal.

The proof is left to the student.

85. Corollary 3. – Two parallel lines are everywhere equi

distant.

A

B

Suggestion. If AB || CD, they may be

proved everywhere equidistant by prov

ing what equal? If A and B are any

two points of AB, and ACLCD and

BD CD, prove that AC and BD are parallel and hence

equal.

86. Theorem. The opposite angles of a parallelogram are equal, and the consecutive angles supplementary.

The proof is left to the student. Write the proof in full. 87. Theorem. - If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

Conclusion. ABCD is a parallelogram.

Suggestions. Draw diagonal DB. It may be proved that ABCD is a parallelogram by proving what lines parallel? For proving these lines parallel, what angles should be proved equal? How may these angles be proved equal? Hence begin by proving A ABD = △ BCD. Write out the complete proof.

88. Construction. Construct a parallelogram which shall have two adjacent sides and the included angle equal respectively to two given line-segments and a given angle.

Given line-segments m and n and ∠x.

Required to construct a parallelogram which shall have two adjacent sides and the included angle equal respectively to m, n, and ∠ x.

Construction. 1. Construct ∠ BAD = ∠ x.

2. Mark off AB = m and AD

n.

3. With center B and radius equal to n, draw an arc ;

and with center D and radius equal to m, draw an arc, intersecting the first arc at C.

1. If two forces are exerted in different directions upon the same object at A, they have the same effect as a single force called their

resultant. If the directions and magnitudes of

C

D

the two forces are represented by the line

segments AB and AC, the direction and magnitude of the resultant will be represented by the line-segment AD, diagonal of the parallelogram A

B

ABDC.

A force of 100 lb. and another of 200 lb. are exerted upon an object at an angle of 45° with each other. Representing 100 lb. by a line-segment 2 in. long, draw the forces to scale and find the resultant. (Use protractor. Measure diagonal with ruler and compute resultant in pounds.)

2. Two forces are exerted upon an object at right angles with each other. One force is 400 lb. and the other 600 lb. Construct and compute their resultant as in Ex. 1.

3. Two forces, one of 48 lb. and the other of 60 lb., are exerted upon an object at an angle of 120° with each other. Construct and compute their resultant.

4. When a train is approaching a station at a speed of 40 ft. per second, a mail bag is thrown from the car, at right angles to the railroad track, with a speed of 20 ft. per second. Find the horizontal direction and speed of the bag. (Construct as in case of forces.)

89. Theorem. - If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

A

D

B

Hypothesis.

∠ B = ∠ D.

In quadrilateral ABCD, L∠A = ∠C and

Conclusion. ABCD is a parallelogram.

Suggestions. It may be proved that ABEDC by first proving that ∠Band ∠Care supplementary. Draw AC. What is the sum of the angles of △ABC and △ ADC, or of ABCD? Since ∠A = ∠C and ∠ B = ∠ D, show by equations that ∠ B + ∠ C = st. 2. Hence show that AB | DC. By similar proof, AD || BC. Write the proof in full.

90. Theorem. - If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

Hypothesis. AB || DC and AB = DC.

Conclusion. ABCD is a parallelogram.

Suggestions. It may be proved that ABCD is a parallelogram by proving that AD || BC. For proving AD || BC, what angles must first be proved equal? How may these angles be proved equal? Hence begin by drawing DB and proving the triangles congruent. Write out the complete proof.