Proposition XXX

129. Construct two triangles having two sides of one equal to two sides of the other, and the angles included between the sides unequal. How do the third sides compare in length? Which triangle has the greater third side?

Theorem. If two sides of one triangle are equal to two sides of another, each to each, and the included angles are unequal, the remaining sides are unequal, and the greater side is in the triangle which has the greater included angle.

Data: Any two triangles, as ABC and DEF, in which AC = DF, BC EF, and angle ACB is greater than angle F.

Proof. Of the two sides DF and EF, suppose that EF is the side which is not greater. Place ▲ DEF in the position DBC so that the equal sides, EF and BC, coincide.

Proposition XXXI

130. Construct two triangles that have two sides of one equal respectively to two sides of the other, but the third sides unequal. How do the angles opposite the third sides compare in size?

Theorem. If two sides of one triangle are equal to two sides of another, each to each, and the third sides are unequal, the angles opposite the third sides are unequal, and the greater angle is in the triangle which has the greater third side. (Converse of Prop. XXX.)

Data: Any two triangles, as ABC and DEF, in which AC DF, BCEF, and AB is greater than DE.

Therefore, both hypotheses, namely, that ≤C = ≤ F, and that <c is less than ▲ F, are untenable.

Consequently,
Therefore, etc.

ZC is greater than F.

Q.E.D.

Ex. 80. If one angle of a triangle is equal to the sum of the other two, what is the value of that angle? What kind of a triangle is the triangle? Ex. 81. AD is perpendicular to BC, one of the equal sides of the isosceles triangle ABC whose vertical angle is 30°. How many degrees are there in each of the angles CAD, DAB, and ABC?

Proposition XXXII

131. Choose some point within any triangle and from it draw lines to the extremities of one side. How does the sum of these lines compare with the sum of the other two sides of the triangle?

Theorem.

The sum of two lines drawn from a point within a triangle to the extremities of one side is less than the sum of the other two sides.

Data: Any triangle, as ABC; any point
within it, as D; and the two lines, AD and
BD, drawn from D to the extremities of
AB.

To prove AD + BD less than AC + BC.
Proof.

Produce AD to meet BC in E.

In A AEC, § 124,

AE <AC+ CE.

Adding BE to both members of this inequality,

D

E

B

Ax. 4,

or

AE + BE < AC + BC.

Adding AD to both members of this inequality,

AD + BD <AD +DE+ BE,

AD + BD <AE + BE.

BD <DE +BE.

BEAC + CE + BE,

Why?

132. 1. Draw a straight line and a perpendicular to it; select a point in the perpendicular, and from that point draw two oblique lines meeting the given line at equal distances from the foot of the perpendicular. How do the oblique lines compare in length?

2. Draw oblique lines from that point to points unequally distant from the foot of the perpendicular. How do they compare in length? Which is the greater?

3. Draw two unequal lines from that point to the given line. Which one meets the line at the greater distance from the foot of the perpendicular?

4. How many equal straight lines can be drawn from a point to a straight line?

Theorem. If from a point in a perpendicular to a given straight line, oblique lines are drawn to the given line,

1. The oblique lines which meet the given line at equal distances from the foot of the perpendicular are equal.

2. Of oblique lines which meet the given line at unequal distances from the foot of the perpendicular the more remote is the greater.

Data: Any straight line, as AB; any perpendicular to AB, as PD; and any point in PD, as C, from which oblique lines, as CE, CF, and CG, are drawn meeting AB so that DE = DF, and DG is greater than DE.

To prove 1. CE = CF.

2. CG greater than CE.

Proof. 1. Data, CD EF at its middle point.

D

F

B

2. Produce CD to H, making DH = CD; draw EH and GH. Then, data and const., AB ICH at its middle point.

133. Cor. Only two equal straight lines can be drawn from a point to a straight line; and of two unequal lines the greater cuts off the greater distance from the foot of a perpendicular drawn to the line from the given point.

Proposition XXXIV

134. Bisect any angle; from any point in the bisector draw lines per pendicular to the sides of the angle. How do the perpendiculars com pare in length? How do the distances of the point from the sides or the angle compare?

Theorem. Every point in the bisector of an angle is equidistant from the sides of the angle,

Data: Any angle, as ABC, and any point in its bisector BD, as F.

To prove F equidistant from AB and CB.

Proof. Draw the perpendiculars FE and FG representing the distances of the point F from AB and CB respectively.

Ex. 82. The perpendicular let fall from the vertex to the base of a triangle divides the vertical angle into two angles. How does the difference of these angles compare with the difference of the base angles of the triangle?

Ex. 83. ABC is a triangle. Angle A = 60°, angle B = 40°. The bisector of angle A is produced until it cuts the side BC. How many degrees are there in each angle thus formed?

Ex. 84. A perpendicular is let fall from one end of the base of an isosceles triangle upon the opposite side. How does the angle formed by the perpendicular and the base compare with the vertical angle?

Ex. 85. If an angle of a triangle is equal to half the sum of the other two, what is the value of that angle?

Ex. 86. How does the sum of the lines from a point within a triangle to the vertices of the triangle compare with the sum of the sides of the triangle? With half that sum ?