77. At a given point in a given line to construct an angle equal to a given angle. Given the angle O and the point P on the line MN, to construct at P an angle equal to the angle O.

AA

M P

FIG. 48.

R

N

CONSTRUCTION. With O as center and any convenient radius draw an arc cutting the sides of ZO at points C and D. With P as center and radius OC draw an arc RX cutting the line PN at R.

With R as center and radius CD draw an arc cutting the arc RX at S. Draw PS.

Then the angle RPS is the desired angle.

Exercise 78

1. Construct a triangle with sides 11⁄2 in., 2 in., and 21⁄2 in. 2. Construct an equilateral triangle with sides 1 in.

3. Construct an isosceles triangle with base 14 in., and the equal sides 17 in. each.

4. Make an angle. Construct an angle equal to it. Test their equality by measuring the angles with the protractor.

5. Make an obtuse angle. Construct an angle equal to it.

6. A triangular field has sides 10 rods, 14 rods, and 17 rods. Make a drawing of this field on the scale of 8 rods to the inch.

7. Construct a triangle with sides twice as long as those of Figure 47.

10. Make a triangle. Construct an angle equal to the sum of the three angles of this triangle.

11. Draw two angles. Construct an angle equal to their difference.

12. Measure the angles of the triangle in exercise 2. How do the angles compare in size? Do the same for the triangle in exercise 3 and compare the angles.

78. To construct a triangle when given two sides and their included angle. In AABC, Figure 50, the A is said to be included by the sides AB and AC, and the side AB is said to be included by the angles A and B.

We wish to construct a triangle having the sides c and b, and the 40 included between them.

CONSTRUCTION. Draw a line AX. At A construct

Z XAY equal to ≤0.

On AX mark off AB equal to line c, and on AY mark off AC equal to line b. Join B and C.

Then A ABC is the triangle required.

79. To construct a triangle when given two angles and their included side. Given 40 and 4Q and their included side a, we wish to construct a triangle having these parts. CONSTRUCTION. Draw a line CB equal to a.

At C construct an angle equal to 20, and at B construct an angle equal to Q.

Prolong the sides of these angles until they meet at a point A.

Then A ABC is the desired triangle.

Exercise 79

1. Construct a triangle given two sides and the included angle. Let the given angle be obtuse.

2. Construct a triangle given two angles and the included side.

=

3. A railroad cuts off from a farm a triangular field with side a 20 rods, side b= 15 rods, and C-90°. Draw such a triangle on the scale of 10 rods to 1 inch. Use the protractor or T-square in making the angle of 90°.

4. Construct an equilateral triangle. Measure the angles. How many degrees in each angle? How do the angles compare in size?

5. Construct an isosceles triangle. Measure its angles. What is true of the angles opposite the equal sides?

6. Construct a triangle with sides 1 in., 11⁄2 in., and 2 in. Measure the angles. Which angle is greatest? Which side is it opposite? Which angle is smallest? Which side is it opposite? Measure the angles of the triangle of example 4 and answer the same questions. Which of the principles stated below is illustrated here?

7. Find the sum of the angles of each of the triangles of examples 4, 5, and 6. If you could measure the angles of a triangle exactly, by how much do you think their sum would differ from 2 right angles?

8. Draw a triangle. Construct with ruler and compasses an angle equal to the sum of the angles of this triangle. Measure with the protractor this angle which you have constructed. How many degrees does it contain?

9. Can you make a triangle with the sides 2 in., 1 in., and 3 in.? Try it.

10. Can you make a triangle with sides 1 in., in., and 2 in.? Try it.

11. Make the Gothic Arch, Figure 52.

PRINCIPLES CONCERNING TRIANGLES

80. Facts to be learned. The following important principles concerning triangles have been illustrated in the preceding exercises. They should be memorized. In what exercise is each illustrated?

I. The sum of the angles of a triangle equals 180°.

II. The angles of an equilateral triangle are equal.

III. In an isosceles triangle the angles opposite the equal sides are equal.

IV. In any triangle if two sides are unequal, the angles opposite those sides are unequal and the angle opposite the greater side is the greater.

V. The sum of two sides of a triangle is greater than the third side.

Exercise 80

1. Can a triangle have two right angles? Why? Two obtuse angles? Why? Two acute angles? What is the least number of acute angles that a triangle can have? Why? 2. In a right triangle how large may one of the acute angles be?

3. In a triangle ABC, a=16 ft., b=20 ft., and c=10 ft. Which is the largest angle? Which is the smallest?

4. In a triangle ABC, ZA=48°, and B=65°. How many degrees in C? Which side is the longest? Which side is the shortest?

5. One angle of a right triangle contains 72°. How many degrees in the smallest angle?

6. One of the equal angles of an isosceles triangle is 50°. How many degrees in each of the other angles?

7. Can the angles of a triangle be 101°, 38°, and 35° respectively?