Download Block_12 - Math GR. 9-12

Geometry: Points, Lines, Planes,
and Angles
MA.912.G.1.2 Construct congruent segments
and angles, angle bisectors, and parallel and
perpendicular lines using a straight edge and
compass or a drawing program, explaining
and justifying the process used.
Block 12
Congruent figures
• In geometry, two figures are congruent if
they have the same shape and size.
Congruent figures
• More formally, two sets of points are called
congruent if, and only if, one can be
transformed into the other by an isometry,
i.e., a combination of translations, rotations
and reflections.
Examples of congruent figures
• What kind of transformations you need to
confirm that the figures are congruent?
Congruent figures
• Rosette window in
the cathedral of
Notre Dame in
Paris, northern
part.
• It contains a lot of
congruent figures:
circles, segments
etc.
Rose windows
• Rose windows were
created by Gothic
architects using only a
compass and
straightedge as the
tools in their design.
Rose windows
• The precision of the
carving of the stone,
and the artistry of the
colors and design of the
glass, make rose
windows among the
most magnificent of
architectural
achievement.
Example of figure with a lot of
congruent figures
• Modular origami
Congruent segments
• Congruent segments are segments of the
same length
Congruent segments
• We can construct a segment congruent to a
given one by using a straight edge and
compass
Congruent Angles
• Definition: Angles are congruent if they have
the same angle measure.
Congruent Angles
• They can be at any orientation on the plane. In
the figure above, there are three congruent
angles.
• Note they are pointing in different directions.
If you drag an endpoint, the other angles will
change to remain congruent with the one you
are changing.
Congruent Angles
• For angles, 'congruent' is similar to saying
'equals'. You could say "the measure of angle
A is equal to the measure of angle B". But in
geometry, the correct way to say it is "angles A
and B are congruent".
• To be congruent the only requirement is that
the angle measure be the same, the length of
the two arms making up the angle is
irrelevant.
Constructing congruent angles
• It is possible to construct an angle that is
congruent to a given angle with a compass
and straightedge alone
• Follow the instructions in Handout
• Justify the process
Congruent triangles
• Definition: Polygons are congruent when they
have the same number of sides, and all
corresponding sides and interior angles are
congruent.
• The polygons will have the same shape and
size, but one may be a rotated, or be the
mirror image of the other.
• This is also true for triangles
Congruent triangles
• Congruent polygons have the same area
• Congruent triangles has the same area
• There is a rule named: CPCTC
"Corresponding Parts of Congruent Triangles are
Congruent"
Congruent triangles
• There are several ways to justify if the
triangles are congruent without checking all
the sides and all the angles:
Congruent triangles
1. SSS (side side side)
All three corresponding sides are equal in
length.
2. SAS (side angle side)
A pair of corresponding sides and the included
angle are equal.
3. ASA (angle side angle)
A pair of corresponding angles and the
included side are equal.
Congruent triangles
4. AAS (angle angle side)
A pair of corresponding angles and a nonincluded side are equal.
5. HL (hypotenuse leg of a right triangle)
Two right triangles are congruent if the
hypotenuse and one leg are equal.
Congruent triangles
Congruent triangles
We can not use the rules:
AAA does not work (this rule tells only that the
triangles are similar)
Congruent triangles
SSA does not always work (you can have two
possibilities for triangles)
Tiling of the plane
• There are well known
tiling of the plane with
congruent figures
• For many years there
was a question: can we
tile a plane with convex
pentagons?
• Until now we have 14
such types of tiling (see
next slide)
Review and questions
Tiling a plane with congruent regular polygons
• When we can create tessellation of the plane
with congruent, regular polygons?