-- --- ---
E
c
DI II II I II II II
aL F
Theorem 5.20.
REASONS
1. Point plotting postulate.
2. Postulate 2.
3. Given.
4. Definition of perpendicular.
:'~.!.L.:D..rn_njrion.llf:..LLI:n'Trsil 6. Given.
7. Right angles are congruent.
8. Given.
9. S.A.S.
10. Corresponding sides of are ~.
11. Given.
12. Theorem 3.5.
13. Definition of straight angle.
14. Base angles of an isosceles b- are ~.
15. Theorem 5.19.
16. Reason 10.
17. Theorem 3.5 (from 1 and 16).
18. S.A.S. or S.S.S.
---
Theorem 5.21
5.34. If the measure of one acute angle of a right triangle equals 30, the length of the side opposite this angle is one- half the length of the hypotenuse. The proof of this theorem is left as an exercise for the student. (Hint: Ex- tend AB to D, making mBD = mAB.
Draw CD. Prove CB bisects AD of equilateraI6ADG.)
Exercises
1. Given: CD 1- AB;
BC 1- AC;
Prove: LA
~LBCD.
c
A D
Theorem 5.21.
c
A
B
Ex. 1.
2. G;ven:
-
__n 00 -
- ---
ivJ----..-.----.
Prove:
TM 1- LM;
TK 1- LK;
TM ~
TK.
TL bisects LKLM.
3. Given: AD 1- DC;
BC 1- DC;
M is the midpoint of DC;
AM ~ BM.
Prove: AD ~ Be.
T L
K Ex. 2.
:V~:
Ex.3.
176
FUNDAMENTALS OF COLLEGE GEOMETRY4. Given: SQ bisects RL at T;
RS 1- SQ; LQ 1- SQ.
Prove: RL bisects SQ at T.
5. Given: AE
1- BC;CD 1- AB;
AE
==CD.
BA
==BC.
Prove:
6. Given: L, M, R, T are collinear;
RS
1- LS; LM ==TR;
NM
1-TN; LL
==LT.
Prove: RS
==MN.
7. Given: RT
==ST;
RS
1-TQ.
Prove: RQ == SQ.
R~
Qs ~
LEx.4.
C E
A!;;>B
Ex. 5.
s
L
N Ex. 6.
T
R
Q Ex. 7.
T
s
8. Given:
Prove:
9. Prove:
10. Prove:
II. Prove:
PARALLEL AND PERPENDICULAR LINES
177
CD IIBE;
..DAC
1-CD;
mLA = 60.
m(AB) = tm(AE) .
E
If the bisector of an exterior angle of a triangle is parallel to the opposite side, the triangle is isosceles.
The bisectors of the base angles of an isosceles triangle intersect at a point equidistant from the ends of the base.
Any point on the bisector of an angle is equidistant from the sides of the angle.
A B C
Ex. 8.
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Summary Tests
Test 1
COMPLETION STATEMENTS
1. The sum of the measures of the angles of any triangle is
2. Angles in the same half-plane of the transversal and between lines are
3. Two lines parallel to the same line are to each other.
4. The measure of an exterior angle of a triangle is than the measure of either nonadjacent interior angle.
5. A proof in which all other possibilities are proved wrong is called 6. A line cutting two or more lines is called a(n)
7. If two isosceles triangles have a common base, the line joining their vertices is to the base.
8. A line parallel to the base of an isosceles triangle cutting the other sides cuts off an triangle.
9. A triangle is if two of its altitudes are congruent.
10. The statement that through a point not on a given line there is one andj only one line perpendicular to that given line asserts the and \
properties of that line.
11. The acute angles of a right triangle are
12. If two parallel lines are cut by a transversal, the interior angles on the same.
side of the transversal are .
13. If the sum of the measures of any two angles of a triangle equals the:
measure of the third angle, the triangle is a(n) triangle.:
14. If from any point of the bisector of an angle a line is drawn parallel to:
one side of the angle, the triangle formed is a(n) triangle.
J
15. Two planes are if their intersection is a null set.!
178
j
angles.
17. Two planes perpendicular to the same line are
18. Through a point outside a plane (how many?) lines can be drawn parallel to the plane.
19. 1\'0 right triangle can have a(n) angle.
20. Two lines perpendicular to the same plane are 21. The two exterior angles at a vertex of a triangle are
therefore angles.
22. The geometry which does not assume Playfair's postulate is sometimes
called geometry.
to each other.
angles and are
Test 2
TRUE-FALSE STATEMENTS
1. An isosceles triangle has three acute angles.
2. A line which bisects the exterior angle at the vertex of an isosceles triangle is parallel to the base.
3. The median of a triangle is perpendicular to the base.
4. If two lines are cut by a transversal, the alternate exterior angles are supplementary.
5. The perpendicular bisectors of two sides of a triangle are parallel to each other.
6. In an acute triangle the sum of the measures of any two angles must be
0'-
7. If any two angles of a triangleb'~' are congruent, the third angle is congruent.
8. I f two parallel planes are cut by a third plane, the lines of intersection are skew lines.
9. To prove the existence of some thing, it is necessary only to prove that there is at least one of the things.
10. The acute angles of a right triangle are supplementary.
11. The expressions "exactly one" and "at most one" mean the same thing.
12. Two planes perpendicular to the same plane are parallel.
]3. A plane which cuts one of two parallel planes cuts the other also.
]4. Two lines perpendicular to the same line are parallel to each other.
]5. Two lines parallel to the same line are parallel to each other.
]6. Two lines parallel to the same plane are parallel to each other.
]7. Two lines skew to the same line are skew to each other.
18. An exterior angle of a triangle has a measure greater than that of any interior angle of the triangle.
]9. If two lines are cut by a transversal, there are exactly four pairs of alter- nate interior angles formed.
179
180
FUNDAMENTALS OF COLLEGE GEOMETRY20. If [, m, and n are three lines such that [
.1m and m
.1n, then [
.1n.
21. An exterior angle of a triangle is the supplement of at least one interior angle of the triangle.
22. In a right triangle with an acute angle whose measure is 30, the measure, of the hypotenuse is one-half the measure of the side opposite the 30
angle. .
23. When two parallel lines are cut by a transversal the two interior angles o~
the same side of the transversal are complementary.
24. If [, m, and n are lines, [11m, [
.1n, then n
.1m.
25. If [, m, n, and p are lines, [II m, n
.1 [,P
.1m, and n #- p, then n
IIp. . 26. Line [ passes through P and is parallel to line m if and only if PEL an<
[n m=.0.
27. If transversal t intersects line [ at A and line m at B, then t n
{A, B}, where A #- B.
Test 3
PROBLEMS 125°
1-8. Solve for mLo::
Prob.I. 111m;rlls.
~
l
40°
m 58°
0' Prob.3. 111m.
Lh
Prob.5.
---
Prob.4.
Prob. 6.
- ---
Test 4
EXERCISES 50°
SUMMARY TESTS
181
Prob. 7.
~
Prob.8.
1. Supply the reasons for the statements in the following proof:
D
Given: AC ==
BC; CD
==CEo
Prove: DF .1 AB.ST ATEMENTS
REASONS
A F B
Ex. I.
1. AC == BC; CD == CEo 2. mLA
=
rnLB: mLCDE=
mLCED.
3. mLAFD
= mLFEB + mLB.
4. mLFEB = mLCED.
5. mLFEB = mLCDE.
6. :. mLAFD
= mLCDE+mLA.
7. mLAFD+mLCDE+mLA = 180.
8. mLAFD+ rnLAFD = 180.
9. mLAFD = 90.
10. :.l5F -L AB
--- .
2. 1.
3.
4.
5.
6.
7.
8.
9.
10.
----
----
R
Ex.2.
2 Given: RS == LS;
SP
bisects L TSL.Prove:
SP
IIRL.
182
T B
Ex. 3.