Download 3.4 Justifying Constructions

3.4 Justifying Constructions
1) Copying An Angle (http://www.mathopenref.com/constcopyangle.html)
Proof
This construction works by creating two congruent triangles. The angle to be copied has the same measure in both
triangles
The image below is the final drawing above with the red items added.
Argument
Reason
1
Line segments AK, PM are congruent
Both drawn with the same compass width.
2
Line segments AJ, PL are congruent
Both drawn with the same compass width.
3
Line segments JK, LM are congruent
Both drawn with the same compass width.
4
Triangles ∆AJK and ∆PLM are congruent
Three sides congruent (SSS).
5
Angles BAC, RPQ are congruent.
CPCTC. Corresponding parts of congruent triangles are congruent
- Q.E.D
2) Bisecting An Angle (http://www.mathopenref.com/constbisectangle.html
Proof
This construction works by effectively building two congruent triangles. The image below is the final drawing above with
the red lines added and points A,B,C labelled.
Argument
Reason
1
QA is congruent to QB
They were both drawn with the same compass width
2
AC is congruent to BC
They were both drawn with the same compass width
3
∆QAC and ∆QBC are congruent
Three sides congruent (sss). QC is common to both.
4
Angles AQC, BQC are congruent
CPCTC. Corresponding parts of congruent triangles are congruent
5
The line QC bisects the angle PQR
Angles AQC, BQC are adjacent and congruent
- Q.E.D
3) Constructing a 45 degree Angle (http://www.mathopenref.com/constangle45.html)
Proof
This construction works by creating an isosceles right triangle, which is a 45-45-90 triangle. The image below is the final
drawing above with the red items added.
Argument
Reason
1
Line segment AB is
perpendicular to PQ.
Constructed that way. See Constructing the perpendicular bisector of a line.
2
Triangle APC is a right
triangle
Angle ACP is 90° (from step 1)
3
Line segments CP,CA
are congruent
Drawn with same compass width
4
Triangle ∆APC is
isosceles.
CP = AC
5
Angle APC has a
measure of 45°.
In isosceles triangle APC, base angles CPA and CAP are congruent. (See Isosceles
Triangles). The third angle ACP is 90° and the interior angles of a triangle always add to
180. So both base angles CPA and CAP are 45°.
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two 45° angle exercises. When you get to the page, use the browser print
command to print as many as you wish. The printed output is not copyright.
4) Constructing a 60 degree angle (http://www.mathopenref.com/constangle60.html)
Proof
This construction works by creating an equilateral triangle. Recall that an equilateral triangle has all three interior angles
60°. The image below is the final drawing above with the red items added.
Argument
Reason
1
Line segments AB, PB, PA are
congruent
All drawn with the same compass width.
2
Triangle APB is an equilateral
triangle
Equilateral triangles are those with all three sides the same length.
3
Angle APB has a measure of 60°
All three interior angles of an equilateral triangle have a measure of 60°. See
Equilateral triangle definition
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two 60° angle exercises. When you get to the page, use the browser print
command to print as many as you wish. The printed output is not copyright.
5) Constructing a 30 degree angle
Proof
The image below is the final drawing above with the red items added.
Argument
Reason
1
Line segments PT, TR, RS, PS, TS are congruent
(5 red lines)
All created with the same compass width.
2
∆PTR is an equilateral triangle, therefore angle
mTPS  60
All three interior angles of an equilateral triangle have a measure
of 60°. See Equilateral triangle definition
3
PR = PR
Reflexive Property
4
∆PTR  ∆PSR
SSS Property
5
TPR  SPR
CPCTC
6
mTPR  mSPR  mTPS
Angle Addition Postulate
7
2mSPR  60
Substitution
8
mSPR  30
Division Property of Equality
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two 30° angle exercises. When you get to the page, use the browser print
command to print as many as you wish. The printed output is not copyright.
Constructing a Hexagon
After doing
this
We start with
a line
segment AF.
This will
become one
side of the
hexagon.
Because we
are
constructing a
regular
hexagon, the
other five
sides will
have this
length also.
1. Set the
compass
point on A,
and set its
width to F.
The
compass
must remain
at this width
for the
remainder of
the
construction.
2. From
points A and
F, draw two
arcs so that
they intersect.
Mark this as
point O.
This is the
center of the
hexagon's
circumcircle.
Your work should look like this
Constructing a Hexagon
After doing
this
3. Move the
compass to O
and draw a
circle.
This is the
hexagon's
circumcircle the circle that
passes
through all six
vertices
4. Move the
compass on
to A and draw
an arc across
the circle.
This is the
next vertex of
the hexagon.
5. Move the
compass to
this arc and
draw an arc
across the
circle to
create the
next vertex.
Your work should look like this
Constructing a Hexagon
After doing
this
6. Continue
in this way
until you have
all six
vertices.
(Four new
ones plus the
points A and
F you started
with.)
7. Draw a
line between
each
successive
pairs of
vertices.
Your work should look like this
Constructing a Hexagon
After doing
this
Your work should look like this
8. Done.
These lines
form a regular
hexagon
where each
side is equal
in length to
AF.
Extra Credit Worksheet:
http://www.mathopenref.com/wshexagon.html
EXTRA CREDIT WORKSHEET
1) (a)
(b)
(c)
Construct a 45° angle on each end of the line below as in the example.
What is the precise name of the triangle that results?
What properties does this triangle have? (list 3).
NAME:____________________
Example
2) Perform the 60° angle construction twice to create a 120° angle from two 60° angles that are adjacent (share a
side) as in the example:
3)
Draw a triangle where two interior angles are both 30° as in the example below. What is the name of triangles
like this?
4)
(a)5)Construct two hexagons that share the side below. It should look like the figure on the right.
(b) Try to do it using only four circles and four arcs.
3.5 Inequalities in a Triangle
Shortest Distance to a Line
Corollary
The shortest distance from a point to a
line is measured
along the perpendicular segment from
the point to the line. (Lesson 3.3)
Shortest Distance to a Plane
Corollary
The shortest distance from a point to a
plane is the shortest line segment that
can be drawn from the point to the
plane.
Side-Angle Inequality Theorem
In a triangle, if one side is longer than
another side,
then the angle opposite the longer side
is larger than the angle opposite the
shorter side. Also the converse is true.
a
B
C
b
A
c
If c  b  a, then mC  mB  mA
If mC  mB  mA, then c  b  a,
http://www.mathopenref.com/trianglesideangle.html
IMPORTANT PROPERTY TO REMEMBER!
Addition Property of Inequality
If a > b and c > d, then a + c > b + d
Triangle Inequality Theorem
The Triangle Inequality Theorem states that the any side of a triangle must be less than the sum
of the other 2 sides, and greater than the absolute value of their difference.
|A - B| < C < A + B
|B – C| < A < B + C
|A - C| < B < A + C
http://www.mathsisfun.com/definitions/triangle-inequality-theorem.html
In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then
you know that the sides do not make up a triangle .
You only need to see if the two smaller sides are greater than the largest side! The interactive demonstration shows
that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also
illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight
line! You can't make a triangle! Also, the third side must be larger than the absolute value of the difference of the other
two sides.
Extra Credit Worksheets:
http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/5-The%20Triangle%20Inequality%20Theorem.pdf
http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/5-Inequalities%20in%20One%20Triangle.pdf