Geometry Chapter 1 Note-Taking Guide Name ________________________ Foundations of Geometry Per ____ Date _________________

1.1 A line segment (or segment or seg) -

1.1 A ray-

1.1 Opposite rays-

1.1 Postulate (also known as an axiom)- 1.1 Name 2 planes in the diagram:

Where do the two planes intersect? Name a line in plane T: Name a line in plane N:

Name the intersection of those two lines: Name three collinear points: Name three non-collinear points: Name four coplanar points: Name four non-coplanar points: Name a pair of opposite rays:

1.1 Ruler postulate: points on a line can be put into a one-to-one correspondence with the real numbers (which are called coordinates).

1.1 The distance between two points is the absolute value of the difference of the coordinates.

AB = |𝑎𝑎 − 𝑏𝑏| = |𝑏𝑏 − 𝑎𝑎|

The measure (m) of a segment is equal to the distance between its two endpoints.

𝑚𝑚𝐴𝐴𝐴𝐴����= AB =

1.1 Segment Addition Postulate –

1.1

1.1

1.1 Given that M is between N and O. If NM = 17, MO = 3x – 5, and NO = 5x + 2, then find x and the measure of each segment.

1.1 Angle Addition Postulate – Group Breakdown: Solve for x and y.

1.1 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 115∘,𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷 = 48∘, 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑚𝑚∠𝐷𝐷𝐷𝐷𝐷𝐷.

1.1 Reflection:

1.2 Constructing congruent segments: 1. Start by drawing a line to create the congruent segment on. 2. Create your starting point on the line.

3. Use your compass to measure the segment you want to “copy” by putting the sharp end on one

endpoint and the pencil on the other endpoint.

4. Place the sharp point of the compass on your starting point, then lightly draw a curve on the

line to create the endpoint for the congruent segment.

1.2 SAMPLE: Construct a segment congruent to 𝑀𝑀𝑀𝑀�����. Construct a segment that measures MN + PQ

1.2 Constructing congruent angles: 1. Draw a starting line. Create a point on the line that will be the vertex of your new angle.

2. Go to the original angle you are trying to “Copy”. Draw an arc so it crosses both sides of the angle. DO

NOT MOVE THE SETTING ON THE COMPASS!

2. 3. 4. 3. Go to the starting line and vertex point, use the setting to draw an arc. 4. Go back to the original angle. Remember where the arc crossed the two sides? Put the sharp point on

the crossing on one side and place the pencil on the crossing for the other side (I draw a little arc). DO NOT MOVE THE SETTING ON THE COMPASS!

5. Go to the starting line, put the sharp point where the arc crosses the starting line. Draw an arc that crosses the first arc drawn.

5. 6. 6. Connect the vertex point to the point where the two arcs cross. This arc should be a “copy” or

congruent to the original angle.

1.2 Constructing a Perpendicular Bisector: 1. Place the sharp point of the compass on one endpoint. 2. Make sure the pencil is farther than half but less the whole length of the segment. 3. Draw a semi-circle that starts above the endpoint, crosses the segment and ends below the

endpoint. 4. DO NOT CHANGE THE COMPASS SETTING! 5. Place the sharp point of the compass on the other endpoint. 6. Draw a semi-circle that starts above the endpoint, crosses the segment and ends below the

endpoint. 7. The semi-circles will intersect at two places. Use a ruler to connect those two points and draw

a line. 8. The line will cross the original segment at the midpoint.

1.2 Construct the Perpendicular Bisector of 𝑃𝑃𝑃𝑃����

1.2 Constructing an angle bisector:

1. Draw an arc of any size so it crosses both sides of the angle. 2. From one of those intersections draw another arc inside of the angle. KEEP THE COMPASS

SETTING! 3. From the other intersection draw the arc inside the angle so it crosses the last arc.

Connect the intersection of those arcs to the vertex of the angle. 1.2

Bisect both angles above.

1.2 Congruent angles - are two angles that are equal in measure. Like congruent segments, angles have special markings to show which angles are congruent. These marks are called arc marks.

1.2

1.2 Types of angles based on measure:

1. Acute Angle: 2. Right Angle: 3. Obtuse Angle:

1.2 A ray that divides an angle into two congruent angles is called an angle bisector.

Given: bisects KM JKL

( )4 6m JKM x= + °

( )7 12m LKM x= − ° Find m JKL

1.2 Adjacent angles -

1.2 Linear Pair -

1.2 Complementary angles -

1.2 Supplementary Angles -

1.2 Vertical Angles -

1.2 Name the relationship between the angles: 1. ∠5 𝑎𝑎𝑓𝑓𝑓𝑓 ∠7 2. ∠5 𝑎𝑎𝑓𝑓𝑓𝑓 ∠8 3. ∠6 𝑎𝑎𝑓𝑓𝑓𝑓 ∠5 4. ∠6 𝑎𝑎𝑓𝑓𝑓𝑓 ∠8

1.2 An angle is 10° more than 3 times the measure of

its complement. Find the measure of the complement.

An angle is 3 degrees less than twice the measure of its complement. Find the measure of its complement.

1.3 The coordinate plane is a plane that is divided into four regions by a horizontal axis (x-axis) and a vertical axis (y-axis). The location, or coordinate, of a point is given by an ordered pair (x,y).

1.3 The midpoint of any segment (or two points) in the coordinate plane can be calculated by the

midpoint formula:

1.3 Find the midpoint between (-2, -1) and (4, 2) The midpoint of a segment is (-1, 1) and one endpoint is (-6, -1). Find the coordinate of the other endpoint. The midpoint of a segment is (4, -3) and one endpoint is (2, 2). Find the coordinate of the other endpoint.

1.3 In a coordinate plane, the distance (or length of a segment) d between two points (x1, y1) and (x2, y2) can be calculated by distance formula:

1.3 Find mME if M = (3, 2) and E = (-3, -1).

1.3 Find AB and CD. Determine if AB CD≅ . Find FE and GH. Determine if FE GH≅ .

1.3

1.4

1.4

B.

1.4

1.4 Reflection:

1.4

1.5

1.5

1.5

1.5

1.5

1.5

1.6 Deductive Reasoning:

1.6 Law of Detachment

1.6

1.6

1.6