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Ex. 182. In the annexed

diagram, if AB = AC, prove

that BD > DC.

Ex. 183. In the diagram of Ex. 182, prove that BE> EC.

Ex. 184. In the diagram of Ex. 182, prove that AFAB.

Ex. 185. In the diagram of Ex. 182, prove that AB> AH.

* Ex. 186. If the opposite sides of a quadrilateral ABCD are equal, but AB> AD, prove that LAOB>ZAOD.

* Ex. 187. The sum of the lines drawn from any point in a triangle to its vertices is less than the perimeter, but greater than the semiperimeter of the triangle.

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Ex. 188. The sum of the diagonals of any quadrilateral is less than the perimeter, but greater than the semiperimeter of the quadrilateral.

QUADRILATERALS

131. DEF. A trapezoid is a quadrilateral that has one pair of sides parallel. A parallelogram has its opposite sides parallel.

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A rhombus is an equilateral parallelogram, whose angles are oblique. A rectangle is a parallelogram, whose angles are right angles. A square is an equilateral rectangle.

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132. DEF. An isosceles trapezoid is one whose non-parallel sides are equal. The parallel sides of a trapezoid are called its bases, and are distinguished as upper and lower.

133. DEF. A diagonal of a quadrilateral is a straight line joining opposite vertices. The altitude of a parallelogram or trapezoid is the perpendicular distance between the two bases.

PROPOSITION XXXII. THEOREM

134. The opposite sides and angles of a parallelogram are equal.

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Hyp.

D

ABCD is a parallelogram.

To prove AD = BC; AB = CD; LA=LC;

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B=LD. '

- What is the usual means of proving the equality of lines and

135. COR. 1. A diagonal divides a parallelogram into two equal triangles.

136. COR. 2. If one angle of a parallelogram is a right angle, the figure is a rectangle.

137. COR. 3. Parallels included between parallels are equal.

Ex. 189. The perpendiculars to a diagonal of a parallelogram from the opposite vertices are equal.

Ex. 190. The diagonals of a parallelogram bisect each other.

* Ex. 191. A line bisecting one side of a triangle and parallel to another side, bisects the third side also.

PROPOSITION XXXIII. THEOREM

138. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

A

B

Нур.

To prove

D

In quadrilateral ABDC

AB= CD; AD = BC.

AD || BC; AB || CD.

HINT. -Prove the equality of a pair of alternate interior angles by means of equal triangles.

Ex. 192. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

PROPOSITION XXXIV. THEOREM

139. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

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Ex. 193. The bisectors of two opposite angles of a parallelogram are parallel.

Ex. 194. If two opposite sides of a parallelogram are produced by the same length in opposite directions and their ends joined to the nearest vertices, another parallelogram is formed.

140. REMARK.

- Lines may be shown to be parallel by proving

them to be opposite sides of a parallelogram.

Ex. 195. If two opposite sides of a parallelogram are divided into three equal parts, and the respective points of division are joined, the lines are parallel.

Ex. 196. If two opposite sides of a parallelogram are produced by the same length in the same direction, a line joining the ends is parallel to the other sides of the parallelogram.

Ex. 197. If from two opposite vertices of a parallelogram lines be drawn bisecting two opposite sides, respectively, the lines are parallel.

PROPOSITION XXXV. THEOREM

141. The diagonals of a parallelogram bisect each other.

B

Hyp.

To prove

ABDC is a parallelogram.

AO = OD; BO = OC.

[The proof is left to the student.]

Ex. 198. The diagonals of a rhombus are perpendicular to each other. Ex. 199. If the diagonals of a parallelogram are perpendicular to each other, the figure is a rhombus, or a square.

Ex. 200. If each half of the diagonal of a parallelogram is bisected, and the midpoints are joined in order, another parallelogram is formed.

Ex. 201. If a diagonal bisects an angle of a parallelogram, the figure is a rhombus.

Ex. 202. The diagonals of a rectangle are equal.

Ex. 203. State and prove the converse of the preceding exercise.

Ex. 204. If the ends of two diameters of a circle be joined in succession a rectangle is formed.

Ex. 205. The base angles of an isosceles trapezoid are equal.
Ex. 206. State and prove the converse of the preceding exercise.

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142. Two parallelograms are equal if two adjacent sides and the included angle of one are equal, respectively, to two adjacent sides and the included angle of the other.

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Hyp. In ABCD and A'B'C'D', AB = A'B', AD = A'D', ZA = ZA'.

To prove

Proof. Apply

with A'B'.

ABCD =A'B'C'D'.

ABCD to A'B'C'D', so that AB coincides

Then AD takes the direction A'D',

And D coincides with D',

(ZA = ZA').

(for AD = A'D').

BC takes the direction of B'C',

and C must lie in B'C' or in B'C' produced.

(Ax. 11.)

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