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 non- included 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

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?