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EQUALITY FROM PARALLEL LINES
The two basic methods are: *36. If two parallel lines are cut by a transversal, the corresponding angles are equal.
*37. If two parallel lines are cut by a transversal, the alternate-interior angles are equal.
*107. Angles having their sides parallel each to each are
The diagonals of a parallelogram bisect each other.
Two parallelograms are congruent if they have two adjacent sides and the included angle of the one equal respectively to two adjacent sides and the included angle of the other. *113. Parallel lines cutting off equal parts on one transversal cut off equal parts on any other transversal.
114. PROBLEM: Divide a straight line into any number of equal parts.
115. State two methods of proving one thing twice another.
116. The line joining the middle points of two sides of a triangle is parallel to the third side and equals one-half of it.
117. The line joining the middle points of the non-parallel sides of a trapezoid is parallel to the bases and equals one-half their sum.
EQUALITY FROM PARALLEL LINES
118. State five methods of proving angles equal. State four additional methods of proving lines equal.
EXERCISES 1. Parallel lines included between parallel lines are equal. 2. Two parallel lines are everywhere equally distant.
3. A line drawn through the vertex of an angle, perpendicular to its bisector, makes equal angles with the sides of the angle.
4. The bisector of an angle bisects also its vertical angle.
5. The median to the base of an isosceles triangle bisects the angle at the vertex.
6. If the non-parallel sides of a trapezoid are equal, the base angles are equal.
7. State and prove the converse.
8. If the diagonals of a parallelogram are perpendicular to each other, the figure is equilateral.
9. The diagonals of an equilateral parallelogram bisect the angles. 10. If from any point in the bisector of an angle a parallel to one of the sides be drawn, the bisector, the parallel, and the remaining side form an isosceles triangle.
11. If a line is drawn from the vertex of an isosceles A, parallel to the base, it bisects the exterior angle at the vertex.
12. If the bisector of an exterior angle of a triangle is parallel to the opposite side, the triangle is isosceles.
13. The diagonals of an isosceles trapezoid are equal.
14. If the non-parallel sides of an isosceles trapezoid are extended until they meet, an isosceles triangle is formed.
15. If a diagonal of a parallelogram bisects one angle, the figure is equilateral.
16. The line from the vertex of the right angle of a right triangle to the mid-point of the hypotenuse equals one-half the hypotenuse.
SPECIAL RIGHT TRIANGLES 120. The 60-30 right triangle and the 45° right triangle are useful in numerical exercises.
EXERCISES 1. A diagonal of a square makes two 45° right triangles.
2. An altitude of an equilateral triangle makes two 60-30 right triangles.
3. If one angle of a right triangle is 45°, the two arms are equal.
4. If the two arms of a right triangle are equal, each acute angle equals 45°
5. If angle A of right triangle ABC is 60° and angle C is a right angle, extend AC its own length to D; draw BD. Prove ZD= LA = 2 ABC.
6. If one angle of a right triangle is 60°, the hypotenuse is twice the shorter arm.
7. If the hypotenuse of a right triangle is twice the shorter arm, the included angle is 60°.
8. In right triangle ABC, the hypotenuse AB =4 and ZA = 60°; what is AC?
9. In triangle ABC, AB=6 and ZA =30°; what is the altitude from B to AC?
10. In triangle ABC, AB=6 and ZA 150°; what is the altitude from B to AC?
11. If each arm of an isosceles triangle is 10 and the altitude is 5, what is the vertex angle?
12. If the base of an isosceles triangle is 8 and the altitude is 4, what is the vertex angle?
13. In triangle ABC, AC=7 and ZA =135°; the altitude from B meets CA extended at D. If AD=5, what is BD?
14. In triangle ABC, AB=4 and ZA=120°; the altitude from B meets CA extended at D. What is AD?
EXERCISES ON PARALLELOGRAMS 1. The line joining the mid-points of two opposite sides of a parallelogram cuts off a parallelogram.
2. If E and F are the mid-points of sides AB and CD of parallelogram ABCD, then AECF is a parallelogram.
3. The lines joining the mid-points of the sides of a quadrilateral taken in order form a parallelogram.
4. In triangle ABC, D, E, and F are the mid-points respectively of AB, BC, and CA. Prove A DEF a parallelogram.
5. If E and F are the mid-points of AB and CD of quadrilateral ABCD;
and H and K are the mid-points of diagonals AC and BD, then EKFH is a parallelogram.
6. In triangle ABC, the medians AH and CK meet at O; E is the mid-point of A0, and F of Co. Prove EKHF a parallelogram.
The basic method is:
If one straight line meets another straight line, making two adjacent angles equal, the lines are perpendicular. (Art. 10.)
*122. Two points each equally distant from the ends of a line determine the perpendicular bisector of that line.
*123. PROBLEM: Construct the perpendicular bisector of a given straight line.
*124. PROBLEM: Construct a perpendicular to a given straight line from a point in the line.
*125. PROBLEM: Construct a perpendicular to a given straight line from a point outside the line.
*126. A line perpendicular to one of two parallel lines is perpendicular to the other.
127. The diagonals of a rhombus bisect each other at right angles.
128. State five methods of proving lines perpendicular.
1. The bisector of the vertical angle of an isosceles triangle is perpendicular to the base.
2. The median to the base of an isosceles triangle is perpendicular to the base.
3. The bisectors of two supplementary adjacent angles are perpendicular to each other.
4. If two isosceles triangles have the same base, the line joining their vertices is perpendicular to the base.
5. If a parallelogram has one angle a right angle, the figure is a rectangle.
6. A line parallel to the base of an isosceles triangle is perpendicular to the bisector of the vertex angle.
7. If an angle of a triangle equals the sum of the other two, this angle is a right angle.
8. If a base angle of an isosceles triangle is 45°, -the vertex angle is a right angle.
9. If an angle of a triangle is 60° and one including side is double the other, the triangle is a right triangle.
10. The line joining the middle points of the bases of an isosceles trapezoid is perpendicular to the bases.
11. The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base.
12. If the diagonals of a parallelogram are equal, the figure is a rectangle.
13. If the median to the base of a triangle is equal to one-half the base, the angle at the vertex is a right angle.
14. If the diagonals of a parallelogram are equal and perpendicular, the figure is a square.
15. The lines joining the middle points of the sides of a square, taken in order, form a square.
16. The bisectors of the angles of a parallelogram form a rectangle.