Basic Geometry
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Definitions Point
A location in space Line
A geometric figure made up of infinitely many points Line segment
A line segment is a part of a line that has two end points. Ray
A ray is part of a line. It has one end point and extends to infinity in one direction. Plane
A flat surface that is infinitely large and with zero thickness extends endlessly in all Directions Congruent line segments
Line segments that have the same length Midpoint
The point that divides a line segment into two congruent
Definitions Angle
A geometric figure formed by two rays with a common ndpoint A
B C
The Relation between the Degree and Radian
If an angle measured in degree is D and in radian is !" then the relation between D and ! is given by D = (180°/π ) * R R = (π/180°)*D
#$ample In the radian system of angular measurement, the measure of one revolution is !"
#$ample% &hat is the radian measure of '()* '( + !-./(0 1 !-2 Convert ."3 radians into degrees* ."3 ./()-40 1 /(" )
Types of Angles A5ute angle
An angle whose measure is less than #$% and greater than $% Right angle
An angle whose measure is #$% 6btuse angle
An angle whose measure is greater than #$% but less than &'$ % 7traight angle
An angle whose measure is &'$% Refle$ angle
An angle whose measure is more than &'$% but less than ()
%$ Types of Lines Interse5ting Lines
Two or more lines that meet at a point are called intersecting lines. Parallel Lines
Type of lines that never meet. Perpendi5ular Lines
Lines intersect to form right angles. 78ew Lines
Lines that do not lie on the same plane. Co planer Lines
Lines that lie on the same plane. Transversal Line
A line that intersect two other coplanar Lines.
Transversal
Angle Relationships Ad9a5ent Angles A
Any two angles that share a *ommon vertex *ommon side
D 1 B
2 C
:erti5al Angles
+ertical angles are the angles that are opposite each other when two lines intersect. +ertical angles are not ad,acent. They are always e-ual in measure. air of Angles /A0" D* 1 and /0* " AD1 are vertical angles
A
D
E
B
C
Angle Relationships Complementary Angles
Two Angles are *omplementary if they add up to #$ degrees 2ne of the complementary angles is said to be the complement of the other.
7upplementary Angles
Two Angles are 3upplementary if they add up to &'$ degrees 2ne of the supplementary angles is said to be the supplement of the other.
Angle Relationships Congruent angles
Two angles that have the same measure are called congruent angles.
45
45
Corresponding Angles
airs of angles that are in similar positions. *orresponding angles have the same degree measurement. air of angle /A "*1 and /0"D1 are corresponding angles.
C A
B
D
Angle Relationships Alternate Interior Angles
airs of interior angles on opposite sides of the transversal are called alternate interior angles. Alternate interior angles have the same degree measurement.
C A
D
B
air of Angles/A" D1 and /0"*1 are alternate interior angles. Alternate #$terior Angles
airs of exterior angles on opposite sides of the transversal are called alternate exterior angles. Alternate exterior angles have the same degree measurement. air of Angles/A" D1 and /0"*1 are alternate exterior angles.
C
A
B
D
Angle Relationships Angle bise5tor 24˚
A ray that divides an angle into two congruent angles
24˚
Linear Pair
It is a pair of ad,acent angles that form a straight line.
56˚
124˚
Problem
Line $ and y are parallel, then find%
/i1 air of vertically opposite angles. /ii1 air of linear pairs. /iii1 air of alternate interior angles. /iv1 air of alternate exterior angles. /v1 air of corresponding angles.
Identifi5ation of Angles Line x and y are parallel" then /a"c1 /b"d1 /g"e1 /f"h1 are air of vertically opposite angles. /a"b1 /b"c1 /d"c1 /a"d1 /g"f1 /g"h1 /h"e1 /f"e1 are linear pairs. /b"f1 / c"g1 are alternate interior angles. /a"e1 /d"h1 are alternate exterior angles /a"g1 /b"h1 /d"f1 /c"e1 are corresponding angles.
Problem In the figure below, name all pairs of ad9a5ent angles"
7olution Angles AD*" *D. Angles AD*" *D0. Angles AD" D0. Angles *D" D0. These are the pairs of ad9a5ent angles"
Problem ;rom the following figure"
Determine whether each set of points are collinear /a1 A" 0" * /b1 A" 0" 4 /c1 0" D" 4 /d1 A" * Determine whether each set of points are coplanar /a1 A" 0" /b1 D" 4" *" /c1 0" *" " 4 /d1A" 0" *"
7olution /a1 A" 0" * /b1 A" 0" 4 /c1 0" D" 4 /d1 A" * In this A"0"* and 0"D"4 are *ollinear /a1 A" 0" /b1 D" 0" *" /c1 0" *" " 4 /d1 A" 0" " * In this A"0" and A"0""* are coplanar
Problem In the figure below, L. is parallel to L" Angle L.A< 1 w 1 .=) and Angle LC< 1 > 1.3()" Then find Angle A
a1') b1#5 c1'6 d175
7olution Let us draw a parallel line D. Angle A0* 8 angle A0D 9 angle *0D. angle A0D 8 angle w: /Alternate interior angles1 angle *0D 8 angle z: /Alternate interior angles1 ; 9 w: 8 z 9 z: 8 &'$ w 8 &'$ > &65 8 55% Also" z: 8 &'$ > z 8 &'$ > &?$ 8 ?$% @ence" Angle A? 1 == @ 3( 1 =)
Problem ;ind the value for y and > in the figure below
a0 (,. b0 .(, = 50 .2, d0 B, .(
7olution 5z 9 &( and #( (z are Alternate interior angles 5z 9 &( 8 #( (z 5z 9 (z 8 '$ 'z 8 '$ z 8 &$ 3o" 5z 9&( 8 5B&$ 9 &( 8 )( Also y and 5z 9 &( are complementary angles y 9 5z 9&( 8 #$ y 9 )( 8 #$ y 8 67 There fore, y 1 B and > 1 .("
Problem
;ind the value for R, 7 and T in the figure below
a1 77.5" &5" &?$ 01 )5.6" &$" &6) *1 )6.&" &5" &5$ D17$.6" &$" &(#
7olution Alternate xterior AnglesC /&?$ and T1 T 8 &?$% Linear air and supplementaryC &?$% 9 /?3 6$ 1% 8 &'$% &6$ 9 ?3 8 &'$ ?3 8 &'$ > &6$ ?3 8 )$ 3 8 &5 *orresponding anglesC 6! &5 8 &?$% 6! 8 &?$ 9&5 6! 8 &55 ! 8 77.5 Therefore, R 1 BB"=, 7 1 .=, T 1 .3(
Problem 6bserve the given figure and find whi5h of the statements are true or false*
/a1 3egments A* and D are parallel. /b1 3egment D intersects segment 0. /c1 All perpendicular lines intersect. /d1 Line D is perpendicular to @. /e1 Line 4@ intersects line 0 and D.
7olution 3egments A* and D are parallel.
> True
3egment D intersects segment 0. > 4alse All perpendicular lines intersect.
> True
Line D is parallel to @.
> 4alse
Line 4@ intersects line 0 and D.
> 4alse
Problem
The supplement of an angle x is )$% less than twice the supplement of the complement of the angle x. 4ind the measure of Angle x. /a1 ($ /b1 #$ /c1 6$ /d1 7$ /e1 ?$
7olution #$>x 8 complement of angle &'$>x 8 supplement of angle &'$> /#$>x1 8 #$9x 8 supplement of complement of the angle. iven that" The supplement of an angle is )$% less than twice the supplement of complement of the angle. Then" &'$>x 8 6/#$9x1 > )$ &'$>x 8 &'$96x >)$ >x 8 6x >)$ >x9)$ 8 6x )$ 8 (x $ 1 ()
Problem Twice the complement of angles x is (5% less than the supplement of angle x. 4ind angle x. /a1 ($ /b1 ?$ /c1 (5 /d1 ?5 /e1 5$
7olution ivenC Twice the complement of angles x is (5% less than the supplement of angle x Let the angle be x. *omplement of angle 8 #$> x 3upplement of angle 8 &'$ > x 6/#$ > x1 9 (5 8 &'$ > x 6&5 > 6x 8 &'$ >x $ 1 2=)
Problem @alf the supplement of an angle is ($% less than twice the complement of the angle. 4ind the measure of the angle. /a1 ($ /b1 5$ /c1 (5 /d1 ?5 /e1 ?$
7olution E@alf the supplement of an angle is ($% less than twice the complement of the angle.E Let the angle be x. /&F61B/&'$>x1 8 6/#$>x1 > ($ /G1 B /&'$ x1 8 &'$ 6x ($ /G1 B /&'$1 > /G1 B x 8 &5$ 6x 6x /G1 B x 8 &5$ > #$ (xF6 8 )$ (x 8 &6$ x 8 &6$F( $ 1 3() is the angle
Problem ;hat are the measures of a linear pair of angles if the measure of one angle is five times that of the otherH
a1 )$" &5$ b1 &5$"($ c1 &6$"($ d1 &6$" ?5
7olution Let the measure of the angles be x and y. Then" -uationsC x 9 y 8 &'$ /Linear air1 x 8 5y 3ubstitute for ExE and solve for EyEC 5y 9 y 8 &'$ )y 8 &'$ y 1 2()
3olve for ExEC x 8 5y x 8 5B($ $ 1 .=()
ui> uestions &. @ow many degrees in a -uarter turnH 6. ame all pairs of supplementary angles in this diagram. (. oint 0 lies on segment A*. A0 8 &$ and 0* 8 '. 4ind A*. ?. oint 0 lies in the interior of angle A2*. The measure of angle A20 8 5$ degrees and the measure of angle A2* 8 7$ degrees. 4ind the measure of angle 02*. 5. /i1 *onvert each of the degree into radians. /a1 &6$% /b1 76% /ii1 *onvert each of the radian measure into degree /a17JF? /b1#JF6
7olutions &. @ow many degrees in a -uarter turnH #$K 6. ame all pairs of supplementary angles in this diagram. /0"*1 /*"D1 /D"A1 /A"01 are supplementary angles. (. oint 0 lies on segment A*. A0 8 &$ and 0* 8 '. 4ind A*. A* 8 &' ?. oint 0 lies in the interior of angle A2*. The measure of angle A20 8 5$ degrees and the measure of angle A2* 8 7$ degrees. 4ind the measure of angle 02*. Angle 02* 8 6$K 5. /i1 *onvert each of the degree into radians. /a1 &6$ 8 6JF( /b1 76 8 /JF&'$1 B 76 8 6JF5 /ii1 *onvert each of the radian measure into degree /a17JF? 8 /&'$FJ1 B /7JF?1 8 (&5% /b1#JF6 8 '&
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