PLANE GEOMETRY.

I.

1. DEFINE a Surface, a Plane, a Plane Figure, a Polygon. Mention all the different kinds of quadrilaterals.

2. Prove that if two angles of a triangle are equal, the sides opposite these angles are also equal.

3. How many degrees in each interior angle of a regular decagon? State and prove the proposition which enables you to answer this question.

4. What is the measure of an angle made by two tangents? by two chords which intersect? by two chords which do not intersect? by a tangent and a chord drawn through the point of contact? Draw a figure for each case.

5. What is the length of the longest line that can be drawn through a rectangular block of marble 12 feet long, 4 feet wide, and 3 feet thick?

6. On a given line as chord, to construct an arc of a given number of degrees.

7. Two tangents drawn to a circle make with each other an angle of 60 degrees; how many degrees of arc between the two points of contact?

8. What is meant by the equation 3.1416? Calculate the difference in area between a circle whose diameter is 20, and the square inscribed in it.

9. Construct a triangle, having given the base, an adjacent angle, and the altitude.

II.

1. Define a Point; a Surface; a Plane; an Angle. What is assumed as the measure of angles?

2. Prove that when two oblique lines are drawn at unequal distances from the perpendicular, the more remote is the greater.

3. Prove that when the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

4. Two angles of a triangle being given, to find the third by geometric construction.

5. What is the measure of an inscribed angle? State and prove.

6. Two tangents drawn to a circle make with each other an angle of 20°; how many degrees of arc between the two points of contact?

7. The side of an equilateral triangle is 12; what is its altitude?

8. Construct a triangle, having given the base and adjacent angle, and the altitude.

III.

1. Define a Right Angle, a Perpendicular, Parallel Lines. On what does the magnitude of an angle depend? What arc is assumed as the usual measure of an angle? Why? 2. To inscribe a circle in a given triangle.

3. Prove that two triangles are equal if the three sides of one are equal respectively to the three sides of the other. 4. Define Similar Polygons.

5. To find a mean proportional between two given lines. Prove the theorem on which your solution depends.

6. Prove that every equilateral polygon inscribed in a circle is regular.

7. The ratio of the squares described on the two legs of a right triangle is equal to the ratio of what two lines?

8. To construct a square which shall be to a given square in a given ratio. Take for the given ratio 2 : 3.

9. What are the expressions for the circumference and area of a circle in terms of " and the radius ?

IV.

1. Define a Plane, a Plane Figure, a Parallelogram. 2. Prove that, if in a triangle two angles are equal, the opposite sides are also equal and the triangle is isosceles.

3. What is the measure of an inscribed angle? State and prove.

4. Upon a given straight line to construct a segment such that any angle inscribed in it shall have a given magnitude.

5. To find a fourth proportional to three given lines.

6. Define Similar Polygons. Draw two polygons mutually equiangular, but not similar; also two polygons having proportional sides, but not similar. In what cases are triangles similar.

7. Prove that any two parallelograms of the same base and altitude are equivalent.

8. Prove: (a.) That similar triangles are to each other as the squares of their homologous sides. (b.) Prove that of similar polygons.

ས.

1. Prove that the perpendicular from the centre of a circle upon a chord bisects the chord and the arc subtended by the chord.

2. To circumscribe a circle about a given triangle.

3. Prove that two angles are to each other in the ratio of two arcs described from their vertices as centres with equal radii.

4. Prove that a line drawn through two sides of a triangle parallel to the third side divides those two sides into proportional parts.

5. State and prove the proportion which exists between the parts of two chords which cut each other in a circle. State what proportion exists when two secants are drawn from a point without the circle.

6. Prove that two regular polygons of the same number of sides are similar.

7. Prove that similar triangles are to each other as the squares of their homologous sides.

8. Show how the area of a polygon circumscribed about a circle may be found; then how the area of a circle may be found; then prove that circles are to each other as the squares of their radii.

VI.

1. Prove that if two opposite sides of a quadrilateral are equal and parallel, the other two sides are also equal and parallel.

2. To describe a circle of which the circumference shall pass through three given points not in a straight line.

3. To find a fourth proportional to three given lines by a geometrical construction.

4. Prove that a perpendicular dropped in a right triangle from the vertex of the right angle to the hypothenuse divides the triangle into two triangles which are similar to each other and to the whole triangle.

5. To find a mean proportional between two given lines.

6. To circumscribe about a circle a regular polygon similar to a given inscribed regular polygon.

7. Similar polygons are to each other as the squares of their homologous sides. What is the ratio between the areas of two circles?

8. Prove that the area of a circle of which r is the radius is equal to π r2.

VII.

1. Prove that if two triangles have two sides of the one respectively equal to two sides of the other, while the included angles are unequal, the third sides will be unequal, and the greater third side will belong to that triangle which has the greater included angle.

2. Prove that the greater of two chords in a circle is subtended by the greater arc; and the converse.

3. Find the common measure of these two lines, and express their ratio in numbers:

4. To divide one side of a triangle into two parts proportional to the other two sides. (Solve and prove.)

5. The perimeters of similar polygons are to each other in what ratio? (State and prove.)

6. To circumscribe a circle about a given regular polygon. (Solve and prove.)

7. Prove that the line which joins the middle points of the two sides of a trapezoid which are not parallel is parallel to the two parallel sides and equal to half their sum. What is the area of a trapezoid?