8. Construct a triangle having a side 3 in. and the angles at its ends 40° and 120°; a side 4 in. and the angles at its ends 100° and 45°.

9. Find the third angle of a triangle if two of its angles are 40° and 56°; 63° 4′ 17′′ and 107° 47' 32".

10. Draw any triangle. Find its perimeter.

11. Find the sum of the angles of the triangle used in Ex. 10. If the sum of the angles of a triangle is 180°, what was your error in measuring the angles of this triangle? What was the per cent of error?

12. Construct an isosceles triangle whose equal sides are each 4.5 in. and the third side 3.25 in.

13. Measure any one angle in the triangle constructed in Ex. 12 and from that compute the other angles.

14. Construct a triangle whose sides are 4.5 in., 3.2 in., and 5.6 in. Measure two angles and compute the third angle. 15. A watch keeping accurate time can be used on a sunny day to tell directions. Hold the watch so that the reflection of the sun seen on the face of the watch appears midway between 12 and the place to which the hour hand points. The 12 will then be south. Try this experiment. 16. When the 12 of a watch is held due south, where will the reflection of the sun be seen at 9 A.M.? 2 P.M.?

noon?

132. Quadrilaterals.-A quadrilateral having no sides parallel is called a trapezium (I); having only two sides parallel is called a trapezoid (II); having two sets of parallel sides (top page 115) is called a parallelogram. A parallelogram with unequal sides and angles not right angles is called a rhomboid; with equal sides but angles not right angles is called a rhom

bus; with unequal sides but angles right angles is called a rectangle; with sides equal and angles right angles is called a square.

In any parallelogram, opposite sides are equal; opposite angles are equal; a diagonal divides it into two equal triangles; each diagonal bisects the other..

EXERCISES

1. Draw free hand and label each kind of quadrilateral mentioned in Art. 132. Note differences closely.

2. Draw any quadrilateral and one diagonal. Note that the sums of the angles of the quadrilateral equals the sum of the angles of the triangles into which the diagonal divides the quadrilateral. What is the sum of the angles of a triangle? Hence what is the sum of the angles of any quadrilateral ?

3. Find the fourth angle in a quadrilateral which has three angles, 46° 14' 35", 135° 34' 17", 97° 45' 16"

4. Find the fourth angle in a quadrilateral which has the three angles: 46° 38′ 47′′, 78° 19′ 17′′, 109° 35′ 29′′.

5. One angle of a parallelogram is 78° 45′ 16′′. What are the other 3 angles?

6. One side of a rhombus is 6.75 in. What is its perimeter?

7. Play a number game on finding angles, complements of angles, and supplements of angles.

133. Construction of Parallel Lines.-A draftsman would draw a line through a point T, parallel to a line AB, by first placing a triangle in the position ABC. He would then place a straight edge along one side of the triangle, as BH, and slide the triangle along BH until the side AB falls upon the point T. Why will a line then drawn along the side of the triangle through T be parallel to AB?

Note that the parallel lines AB, DE, and GH cut HB at the same angle. This is used in drawing a line through a point, Z, parallel to a

angle SZR equal to the angle ZWU. Line ZR is parallel to UV. Why?

EXERCISES

1. Draw any line and mark a point outside. Through the point construct a line parallel to the first line.

2. Draw two intersecting lines. Place a point upon each line. Through these points construct lines parallel to the opposite lines. What quadrilateral is this?

134. Drawing Perpendiculars.-When are lines perpendicular? Suppose that triangle ABC is pushed along until AC falls upon point T. A line now drawn along AC will pass through T and be perpendicular to HB. Why? How can a line be

Join D to C. 4 JCD=? Why is CJ LCD?

first

Here is a plan much used by architects and draftsmen. To construct a line through Q perpendicular to XY, drav a semicircle about Q, with any radius. With a radius longer thar BQ draw arcs using B and C

as ntres. Join their point of intersection, D, to Q, hence DQ 1 BC.

Joining B and C to D forms two

X

=

B

equal triangles. Why? Then 4 DQC 4 DQB. Why? How many degrees are there in each of these two angles? Why is BC 1 DQ?

EXERCISES

Draw any straight line. Through any point on the line construct a perpendicular by use of—

1. The right triangle.

2. The protractor.

3. The compass.