ADJACENT ANGLES; VERTICAL ANGLES

120. Adjacent angles. Two angles ABC and ABD with a common vertex B and a common side AB, such that the other two sides BC and BD (called exterior sides) are on opposite sides of the common side AB, are said to be adjacent.

121. If two adjacent angles DBA and ABC have their exterior sides in a straight line, their sum is equal to two right angles (2R).

For, draw BE perpendicular to CD. C Then 2 DBA + ABC= 2 DBE + LEBC=2R.

B

-A

D

FIG. 51.

E

A

B

FIG. 52.

D

122. From this we infer immediately the following: When one straight line meets another straight line and makes two adjacent angles equal, each is a right angle, and the lines are perpendicular.

123. Also: If two or more straight lines in

a plane pass through a common point,

I. the sum of the consecutive adjacent angles formed on one side of any line drawn through D the point is two right angles (2R);

II. the sum of all the consecutive adjacent angles formed is four right angles (4R).

E

F

FIG. 53.

α

B

124. Complementary angles. Two angles a and ẞ whose sum is a right angle are said to be complementary, and each is called the complement of the other. Hence, if α, ß are complementary,

a+B=R.

125. Complements of the same angle, or of equal angles, are equal.

For, let a and a' be complements of the same angle or of equal angles. Then R a and Ra' are equal. But from

follows

126. Supplementary angles. Two angles a and B whose sum is two right angles are said to be supplementary, and each is called the supplement of the other. Hence, if a, ẞ are supplementary,

a+B=2 R.

127. Supplements of the same angle or of equal angles are equal. For, let a and a' be supplements of the same angle or of equal angles. Then 2 R a and 2 Ra' are equal. But from

follows

128. If two adjacent angles ABC and CBD are supplementary, their exterior sides BA and BD are

and BA coincides with BE, that is, with DB produced.

129. Vertical angles. Two angles AOC and BOD having the

same vertex O, and such that the sides of the one are the prolongations of the sides of the other, are called vertical angles.

130. Vertical angles are equal. Why? 130 a. Acute and obtuse angles. An angle (less than two right angles) is said to be acute or obtuse according as it is less than or greater than a right angle.

C

FIG. 55.

B

1. If one of the angles formed by two intersecting lines is 45°, what are the other angles ?

2. If one of the angles formed by two intersecting lines is denoted by α, express the other angles.

3. What is the complement of 30° ? of an acute angle a?

4. What is the supplement of 30° ? of an angle a?

5. What angle is equal to its supplement ?

6. Find the angle that is five times its supplement.

7. Find the angle whose supplement is three times its complement.

8. If two adjacent angles are supplementary, prove that their bisectors are perpendicular.

9. Prove that the straight line which bisects one of two vertical angles bisects the other also.

HINT: Turn the figure on the bisector as an axis.

10. To draw a line through the vertex of a given angle and forming equal angles with the sides.

11. A ray of light having its origin at a point A and striking a plane mirror at P is reflected along a straight line PB such that AP and PB make equal angles in the same plane with a perpendicular PC to a certain line MN on the mirror. This is expressed in physics by saying that the angle of reflexion CPB is equal to the angle of incidence APC.

To determine the path of a ray of light having its origin at a given point

A, and which is reflected from the given straight line MN to the given point

CENTRAL SYMMETRY

132. Symmetry with respect to a point. Imagine Fig. 56 to rotate in the plane of the figure about O, the mid-point of MN, until M and N change places. Let Fig. 57 be a duplicate of Fig. 56 together with the position of the latter after the rotation. Then Fig. 57 is symmetric with respect to O as a center of symmetry.

In general, a figure is said to be symmetric with respect to a point as a center of symmetry if, after the figure is rotated in its plane about that point through two right angles, the figure coincides point for point with its original position.

133. Corresponding parts. Any two parts of the figure (points, lines, angles, etc.) which simply change places when the figure is thus rotated are called corresponding parts.

As immediate consequences of the preceding paragraphs we have: If a figure is symmetric with respect to a point,

134. I. Corresponding parts of the figure are congruent.

135. II. A point which corresponds to itself is the center of symmetry.

136. III. A straight line which corresponds to itself passes through the center of symmetry.

137. IV. Segments joining corresponding points pass through the center of symmetry and are bisected by it.

138. A straight line is symmetric with respect to any one of its points.

139. A circle is symmetric with respect to its center.

140. The figure formed by two straight lines which cut each other is symmetric with respect to their common point.

1. Which letters of the alphabet have a center of symmetry ?

2. Given three points A, B, and C. Find points A' and B' in a symmetry with respect to C as a center.

3. Arrange two points and a straight line in a symmetry with respect to a center.

4. Show how to bisect a line segment by the use of a draftsman's triangle or set square alone.

5. Given two equal circles tangent at a point P. Prove that any line segment having its extremities on the circles and passing through P is bisected by the latter.

6. Given a straight line AB and a point O not on AB. Construct a line A'B' corresponding to AB in a symmetry with respect to O as a center.

7. Discover a method for measuring the distance between two points on level ground, when the surface between the points is obstructed by some object, as a house.