Download 12.1 Congruence Through Construction

12.1 Congruence Through Construction
Definitions
Definition
AB ∼
= CD if and only if AB = CD
Remember, mAB = AB.
Definitions
Definition
AB ∼
= CD if and only if AB = CD
Remember, mAB = AB.
Definition
∼ ∠DEF if and only if m(∠ABC) = m(∠DEF)
∠ABC =
Congruence Postulates
Postulate
Side-Side-Side
If three sides of a triangle are congruent to the corresponding sides in
another triangle, then the triangles are congruent.
Congruence Postulates
Postulate
Side-Angle-Side
If two sides an the included angle of a triangle are congruent to the
corresponding sides and angle in another triangle, then the triangles
are congruent.
Congruence Postulates
Postulate
Hypotenuse-Leg
If the hypotenuse and leg of one right triangle are congruent to the
hypotenuse and corresponding leg of another triangle, then the two
triangles are congruent.
Euclidean Tools
A compass looks like our compass, but it has no markings on it.
So we can’t set it to draw circles with predetermined radii. Also,
when we pick up the compass, it collapses, so we cannot copy a
circle by picking up the compass and drawing another. We can
only draw circles when given a center and a point on the
circumference.
Euclidean Tools
A compass looks like our compass, but it has no markings on it.
So we can’t set it to draw circles with predetermined radii. Also,
when we pick up the compass, it collapses, so we cannot copy a
circle by picking up the compass and drawing another. We can
only draw circles when given a center and a point on the
circumference.
The straightedge is like a ruler with no markings. We can make
straight lines as long as we choose using any two points, or we
can extend an existing line segment as long as we want.
What We Can Construct
Create a line through two points
What We Can Construct
Create a line through two points
Create a point at the intersection of lines
What We Can Construct
Create a line through two points
Create a point at the intersection of lines
Question
What is the difference between intersecting and concurrent lines?
What We Can Construct
Create a line through two points
Create a point at the intersection of lines
Question
What is the difference between intersecting and concurrent lines?
Create a circle with two points where one is the center and the
other is any point on the circumference
What We Can Construct
Create a line through two points
Create a point at the intersection of lines
Question
What is the difference between intersecting and concurrent lines?
Create a circle with two points where one is the center and the
other is any point on the circumference
Create point(s) of intersection of lines and circles
Create point(s) of intersection of two circles
What We Can Construct
Create a line through two points
Create a point at the intersection of lines
Question
What is the difference between intersecting and concurrent lines?
Create a circle with two points where one is the center and the
other is any point on the circumference
Create point(s) of intersection of lines and circles
Create point(s) of intersection of two circles For these two, how
many points could there be?
Question
For the intersection of lines, circles or one of each, how many points
of intersection could there be?
What We Can Construct
Bisect an angle
What We Can Construct
Bisect an angle
Find the perpendicular bisector of a segment
What We Can Construct
Bisect an angle
Find the perpendicular bisector of a segment
Construct regular polygons
What We Can Construct
Bisect an angle
Find the perpendicular bisector of a segment
Construct regular polygons
Circumscribe regular polygons
What We Can Construct
Bisect an angle
Find the perpendicular bisector of a segment
Construct regular polygons
Circumscribe regular polygons
Circumscribe some other polygons
Terms We’ll Need
Definition
A perpendicular bisector is a line that passes through the midpoint of
another line segment and the intersection forms four right angles.
Terms We’ll Need
Definition
A perpendicular bisector is a line that passes through the midpoint of
another line segment and the intersection forms four right angles.
Definition
The altitude of a triangle is a line segment that begins at a vertex of a
triangle and is perpendicular to the opposite side.
Terms We’ll Need
Definition
A perpendicular bisector is a line that passes through the midpoint of
another line segment and the intersection forms four right angles.
Definition
The altitude of a triangle is a line segment that begins at a vertex of a
triangle and is perpendicular to the opposite side.
Definition
An angle bisector is a a line that passes through the vertex of an angle
and divides the angle into two equal angles.
Circumscribing Polygons
Definition
Circumscribing a polygon means we draw a circle that passes through
all of the vertices of the polygon.
Circumscribing Polygons
Definition
Circumscribing a polygon means we draw a circle that passes through
all of the vertices of the polygon.
Definition
The point at which the perpendicular bisectors of the sides of a
triangle meet is the circumcenter. The circle we draw that passes
through each vertex is called the circumcircle.
When Can We Circumscribe a Quadrilateral?
Theorem
a. If a circle can be circumscribed about a convex quadrilateral, then
the opposite angles are supplementary.
b. If the opposite angles of a quadrilateral are supplementary, then a
circle can be circumscribed about the quadrilateral.