PROPOSITION XII. THEOREM

89. Two right triangles are congruent if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other.

Given the right triangles ABC and A'B'C', with the hypotenuse AC equal to the hypotenuse A'C', and with BC equal to B'C'. To prove that AABC is congruent to ▲ A'B'C'.

Proof. Place ▲ ABC next to ▲ A'B'C', so that BC shall fall along B'C', B shall fall on B', and A and A' shall fall on opposite sides of B'C'.

Post. 5

.. AABC is congruent to ▲ A'B'C'.

(Two are congruent if the three sides of the one are equal respectively

to the three sides of the other.)

90. COROLLARY. Two right triangles are congruent if any two sides of the one are equal respectively to the corresponding two sides of the other.

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Q. E. D.

EXERCISE 9

1. ABCD is a square and M is the mid-point of AB. With M as a center an arc is drawn, cutting BC at P and AD at Q. Prove that ▲ MBP is congruent to ▲ MAQ, and D

write the general statement of this theorem without using letters as is done here.

C

This would read, "If an arc is drawn, with the midpoint of one side of a square as a center, cutting the A M B sides perpendicular to that side, then the triangles cut off by," etc.

2. Draw a figure similar to that of Ex. 1, but take a radius such that the arc cuts BC produced at a point above C, and AD above D. Then prove that ▲ MBP is congruent to ▲ MAQ. 3. Prove that if from the point P the perpendiculars PM, PN to the sides of an angle AOB are equal, the point P lies on the bisector of the angle AOB. Write the general statement of this theorem without using letters as is done here.

N

B

A

M

4. Prove that if the perpendiculars from the mid-point M of

=

the base AB of a triangle ABC to the sides of the triangle are equal, then ZA ZB. What then follows as to the sides AC and BC? Write the general statement of this theorem without referring to a special figure.

5. Prove that if the perpendiculars from the extremities of the base of a triangle to the other two sides are equal, the triangle is isosceles.

B

6. Suppose OYOX. With O as a center an are is drawn cutting OX at A and OY at B. Then with A as a center an arc is drawn cutting OY at P, and with B as a center and the same radius an arc is drawn cutting OX at Q. Prove that OP = OQ.

What triangles are congruent by Prop. XII?

P

X

PROPOSITION XIII. THEOREM

91. Two right triangles are congruent if the hypotenuse and an adjacent angle of the one are equal respectively to the hypotenuse and an adjacent angle of the other.

Given the right triangles ABC, A'B'C', with the hypotenuse AC equal to the hypotenuse A'C', and with angle A equal to angle A'. То prove that AABC is congruent to ▲ A'B'C'.

Proof. Place ▲ ABC upon ▲ A'B'C' so that A shall fall upon A' and AC shall fall along A'C'.

Post. 5

Given

.. CB will coincide with C'B'.

(Only one perpendicular can be drawn to a given line from a

given external point.)

.. AABC is congruent to ▲ A'B'C'.

(If two figures can be made to coincide in all their parts,
they are said to be congruent.)

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Q. E.D.

PROPOSITION XIV. THEOREM

92. Two lines in the same plane perpendicular to the same line cannot meet however far they are produced.

Given the lines AB and CD perpendicular to XY at A and C respectively.

To prove that AB and CD cannot meet however far they are produced.

Proof. If AB and CD can meet if sufficiently produced, we shall have two perpendicular lines from their point of meeting to the same line.

But this is impossible.

.. AB and CD cannot meet.

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Q.E.D.

93. Parallel Lines. Lines that lie in the same plane and cannot meet however far produced are called parallel lines. 94. Postulate of Parallels. Through a given point only one line can be drawn parallel to a given line.

As always in such cases the word line means straight line.

95. COROLLARY 1. Two lines in the same plane perpendicular to the same line are parallel.

96. COROLLARY 2. Two lines in the same plane parallel to a third line are parallel to each other.

For if they could meet, we should have two lines through a point parallel to a line. Why is this impossible?

PROPOSITION XV. THEOREM

97. If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also.

Given AB and CD, two parallel lines, with XY perpendicular to AB and cutting CD at P.

Proof. Suppose MN drawn through PL to XY.

98. Transversal. A line that cuts two or more lines is called

The angles b and ƒ, c and g, e and a, h and d, are called exterior interior angles.

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