Envision Topic 1 Geometry

Bell Ringer

Bell Ringer

1.1

Points, Lines, & Planes

Objective: Students will be able to name geometric
objects and sketch intersections of lines & planes

Definitions & Diagrams

Point

Diagram

Naming

- has no dimension
- represented by a dot

Line

Diagram

Naming

- has one dimension
- represented by a line
with two arrowheads
- extends WITHOUT ending
- Through any 2 points,
there is exactly one line.
- You can use any 2 points
on a line to name it.

Definitions & Diagrams

Plane

Diagram

Naming

- has two dimensions
- represented by a shape
that looks like a floor
- extends WITHOUT ending
- Through any 3 points not
on the same line there
is exactly one plane.
- You can use any 3 points
that are not on the same
line to to name it.

Definitions & Diagrams

Collinear
- describes points that
lie on the same line

Diagram

Coplanar
- describes points that
lie in the same plane

Example 1 - Naming Points, Lines, & planes

a. Give two other names for

and plane .

b. Name three points that are collinear.

c. Name four points that are coplanar.

Definitions & Diagrams

In geometry, terms that can be described using words such as
point or line are called defined terms.

Line Segment

Diagram

Naming

- AKA “segment”
- consists of two endpoints
and all points on the line
between the endpoints

Ray

Diagram

Naming

- consists of one endpoint
and extends indefinitely
in ONE direction
- Note: and are DIFFERENT!

Definitions & Diagrams

Opposite Rays

Diagram

- two rays with a common endpoint
that form a straight line
- if point C lies on between
A and B, then and are
Opposite rays

Segments and rays are collinear when they lie on the same line.
So, opposite rays are collinear. Lines, segments, and rays are
coplanar when they lie in the same plane.

Example 2 - Naming Segments, Rays, & Opposite Rays

a. Give another name for .

b. Name all rays with endpoint .

c. Which rays are opposite rays?

Example 3 - Naming Segments, Rays, & Opposite Rays

a. Give another name for .

b. Are and the same ray? Explain.

c. Are and the same ray? Explain.

Sketching Intersections

Two or more geometric figures intersect when they have one or
more points in common. The intersection of the figures is the

set of points the figures have in common.

The intersection of two

different lines is a point.

The intersection of two

different planes is a line.

Example 4 - Sketching Intersections of Lines & Planes

a. Sketch a plane
and a line that
is in the plane.

b. Sketch a plane
and a line that
does not intersect
the plane.

c. Sketch a plane
and a line that
intersects the
plane at a point.

Show me what you learned today!

a. Name the intersection of and line k.

b. Name the intersection of plane A and plane B.

c. Name the intersection of line k and plane A.

Drag and Drop

a. ​ ​

Placeholder

b. ​

Placeholder

c. ​

Placeholder

Drag these tiles and drop them in the correct blank above

______

M

k

Q

line k

line PQ

Line k

P

Plane B

PlaneA

Example 1 - Measuring Segments

Use a ruler to determine the length of the segment.

a.

b.

c.
d.

Constructing Congruent Segments

A construction is a geometric drawing

that uses a limited set of tools,

usually a compass and a straightedge.

Today we will construct a line
segment that is congruent to a

given line segment.

Constructing Congruent Segments

Step 3 - Copy length.

Place the compass at point C. Mark point D on the new segment
using the other end of the compass.

NOTE: You can erase the excess part of the segment to the
right of point D, but it’s not necessary.

Definitions/Postulates

Midpoints & Segment Bisectors

The midpoint of a segment is the
point that divides the segment
into two congruent segments.

A segment bisector is a point, ray,
line, line segment, or plane that
intersects the segment at its midpoint.
A midpoint or a segment bisector
bisects a segment.

M is the midpoint of .
So,

and

.

is a segment bisector of

So,

and

Example 2 - Using Algebra with Segment Lengths

a. Point M is the midpoint of . Find the length of .

Example 3 - Using the Midpoint Formula

a. The endpoints of are
and . Find the coordinates
of the midpoint M.

b. The midpoint of is .
One endpoint is . Find the
coordinates of endpoint K.

Example 3 - Using the Midpoint Formula

e. The midpoint of is .
One endpoint is . Find the
coordinates of endpoint U.

f. The midpoint of is
One endpoint is . Find the
coordinates of endpoint V.

Formula

The Distance Formula

The Distance Formula is related to the Pythagorean Theorem,
which you will see again when you work with right angles.
(You may have already seen it in a previous math class!)

Distance Formula

Pythagorean Theorem

Note: The red box shown in the triangle indicates
that the marked angle is a right angle.

Bisecting a Segment - Midpoint

Step 2 - Fold the paper.

Fold the paper so that point B is on top of point A.

Make a crease.

Bell Ringer:

Example 4 - Modeling with Mathematics

b. You are building patio in your school’s courtyard. In the
diagram, the coordinates represent the four vertices of the patio.
Each unit in the coordinate plane represents one foot. Find the
area of the patio.

Definitions

Naming Angles

Angle
-A set of points
consisting of two
different rays
that have the
same endpoint

Vertex
-The common endpoint of
the rays of an angle
(the “corner” point)

Sides
-The rays of an angle

Ways to Name an Angle
1. Use its vertex (if
there is only ONE angle)

2. Use a point on each
ray and the vertex
(vertex MUST be in the
middle)

3. Use a number

Example 1 - Naming Angles

a.

A lighthouse keeper measures the angles formed by the
lighthouse at point M and three boats. Name three different
angles shown in the diagram.

Postulate/Definition

Measuring Angles

Protractor Postulate
Consider and a point A on one
side of . The rays of the form
can be matched one to one with
the real numbers from 0 to 180.

The measure of , which can be
written as , is equal to the
absolute value of the difference between
the real numbers matched with
and on a protractor.

Definition

Congruent Angles
-Angles that have the same measure

Note: In diagrams, matching arcs indicate congruent angles.
When there is more than one pair of congruent angles, use
multiple arcs.

Symbols

Words

The measure of angle G is equal

to the measure of angle S

Angle G is congruent to angle S

Postulate

Angle Addition Postulate

Words

If P is in the interior of

,

then the measure of

is equal

to the sum of the measures of

and

Symbols

If P is in the interior of

,

Example 4 - Finding Angle Measures

c. Given that

is a right angle,

find

and

Copying an Angle

Step 3 - Draw an arc

Label B, C, and E. Draw an arc with radius BC and center E.

Label the intersection F.

Bell RInger

1.

2.

Complementary:

Supplementary:

Adjacent:

Example 1 - Identifying Pairs of Angles

b. In the figure, name a pair of complementary angles, a
pair of supplementary angles, and a pair of adjacent angles.

Example 3 - Using Algebra to find Angles

a.

Example 4 - Identifying Angle Pairs

c. Do any of the numbered angles in the
figure form a linear pair? Explain.

d. Which angles are vertical angles? Explain.

Example 2 - Using Algebra with Segment Lengths

b. Identify the segment
bisector of . Then find MQ.

c. Identify the segment
bisector of . Then find RS.

1.6

Describing Pairs of Angles

Objective: Students will be able to identify
relationships among angles.

Complementary

Supplementary

d

Definitions

Complementary and Supplementary Angles

-Two positive angles whose
measures have a sum of .
-Each angle is a complement
of the other.

-Two positive angles whose
measures have a sum of .
-Each angle is a supplement
of the other.