Download Triangle Congruence Unit

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TEKS Focus:
(5)(A) Investigate patterns to make conjectures about
geometric relationships, including angles formed by
parallel lines cut by a transversal, criteria required
for triangle congruence, special segments of
triangles, diagonals of quadrilaterals, interior and
exterior angles of polygons, and special segments
and angles of circles choosing from a variety of
tools.
(1)(C) Select tools, including real objects,
manipulatives paper and pencil, and technology as
appropriate, and techniques, including mental math,
estimations, and number sense as appropriate, to
solve problems.
(1)(E) Create and use representations to organize,
record, and communicate mathematical ideas.
(1)(F) Analyze mathematical relationships to connect
and communicate mathematical ideas.
In the glossary, a polygon is defined as a
closed plane figure formed by three or
more line segments that intersect only at
their endpoints.
Each segment that forms a polygon is a
side of the polygon. The common
endpoint of two sides is a vertex of the
polygon. A segment that connects any two
nonconsecutive vertices is a diagonal.
This polygon is ABCDE or AEDCB or many other options.
You may start at any letter and go in a circular motion
either clockwise or counter-clockwise.
You can name a
polygon by the
number of its sides.
The table shows the
names of some
common polygons.
Example: 1
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of sides.
polygon, hexagon
not a polygon
polygon, heptagon
polygon, nonagon
not a polygon
not a polygon
All the sides are congruent in an
equilateral polygon. All the angles are
congruent in an equiangular polygon.
A regular polygon is one that is both
equilateral and equiangular. If a
polygon is not regular, it is called
irregular.
A polygon is concave if any part of a
diagonal contains points in the exterior of
the polygon. If no diagonal contains points
in the exterior, then the polygon is convex.
A regular polygon is always convex.
Example: 2
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
irregular, convex
regular, convex
irregular, concave
irregular, concave
regular, convex
To find the sum of the interior angle measures of a
convex polygon, draw all possible diagonals from one
vertex of the polygon. This creates a set of triangles. The
sum of the angle measures of all the triangles equals the
sum of the angle measures of the polygon.
Remember!
By the Triangle Sum Theorem, the sum of the
interior angle measures of a triangle is 180°.
(1) 180°= 180°
(2) 180°=360°
(3) 180°=540°
(4) 180°=720°
(n-2)
(n-2) 180°
In each convex polygon, the number of triangles
formed is two less than the number of sides n.
So the sum of the angle measures of all these
triangles is (n — 2)180°.
Example: 3
Find the sum of the interior angle measures of a
convex heptagon.
(n – 2)180°
Polygon  Sum Thm.
(7 – 2)180°
A heptagon has 7 sides, so
substitute 7 for n.
900°
Simplify.
Example: 4
Find the measure of each interior angle of a
regular 16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(16 – 2)180° = 2520°
Substitute 16 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.
Example: 5
Find the sum of the interior angle measures of
a convex 15-gon.
(n – 2)180°
Polygon  Sum Thm.
(15 – 2)180°
A 15-gon has 15 sides, so
substitute 15 for n.
2340°
Simplify.
Example: 6
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540°
Polygon  Sum Thm.
mA + mB + mC + mD + mE = 540°
Polygon  Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
Substitute.
135c = 540
Combine like terms.
c=4
Divide both sides by 135.
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
Example: 7
Find the measure of each interior angle of a
regular decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
(10 – 2)180° = 1440°
Polygon  Sum Thm.
Substitute 10 for n and
simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 10.
In the polygons below, an exterior angle has
been measured at each vertex. Notice that in
each case, the sum of the exterior angle
measures is 360°.
Remember!
An exterior angle is formed by one side of a
polygon and the extension of a consecutive side.
Example: 8
Find the measure of each exterior angle of a
regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°.
measure of one ext.  =
Polygon  Sum Thm.
A regular 20-gon has 20
 ext. s, so divide the
sum by 20.
The measure of each exterior angle of a regular 20-gon is 18°.
Example: 9
Find the measure of each exterior angle of a
regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
Polygon  Sum Thm.
measure of one ext.
A regular dodecagon has
12  ext. s, so divide the
sum by 12.
The measure of each exterior angle of a regular dodecagon is 30°.
Example: 10
Find the value of b in polygon
FGHJKL.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
Polygon Ext.  Sum Thm.
120b = 360
b=3
Combine like terms.
Divide both sides by 120.
Example: 11
Find the value of r in
polygon JKLM.
4r° + 7r° + 5r° + 8r° = 360°
24r = 360
r = 15
Polygon Ext.  Sum Thm.
Combine like terms.
Divide both sides by 24.