1. Focus Graph and label the following points on a coordinate grid.
P(-1, -1), Q(0, 4), R(-3, 5), S(2, 5), and T(3, -4)
Name three noncollinear points.
Name three collinear points.
Name two intersecting lines
Lesson 4: Measuring Segments and Angles

2. Measuring Segments and Angles
Lesson 1-4
Measuring Segments
and Angles
Lesson 4: Measuring Segments and Angles

3. The Ruler Postulate (1-5)
The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that:
Any two chosen points can be paired with 0 and 1.
The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points.
Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │
Lesson 4: Measuring Segments and Angles

4. Ruler Postulate : Example
Find the distance between P and K.
Note: The coordinates are the numbers on the ruler or number line!
The capital letters are the names of the points.
Therefore, the coordinates of points P and K are 3 and -2 respectively.
Substituting the coordinates in the formula │a – b │
PK =
| | = 5
Remember : Distance is always positive
Lesson 4: Measuring Segments and Angles

5. Lesson 4: Measuring Segments and Angles
Between
Definition: X is between A and B if AX + XB = AB.
AX + XB = AB
AX + XB > AB
Lesson 4: Measuring Segments and Angles

6. The Segment Addition Postulate
If C is between A and B, then AC + CB = AB.
Example:
If AC = x , CB = 2x and AB = 12, then, find x, AC and CB. 2x x 12
Step 1: Draw a figure
Step 2: Label fig. with given info.
AC + CB = AB
x x = 12
3x = 12
x = 4
Step 3: Write an equation
x = 4
AC = 4
CB = 8
Step 4: Solve and find all the answers
Lesson 4: Measuring Segments and Angles

7. Lesson 4: Measuring Segments and Angles
Congruent Segments
Definition:
Segments with equal lengths. (congruent symbol: )
Congruent segments can be marked with dashes.
If numbers are equal the objects are congruent.
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Correct notation:
Incorrect notation:
Lesson 4: Measuring Segments and Angles

8. Lesson 4: Measuring Segments and Angles
Midpoint
Definition:
A point that divides a segment into
two congruent segments
Formulas:
On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is
In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is
Lesson 4: Measuring Segments and Angles

9. Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
Now find the midpoint on the number line.
Lesson 4: Measuring Segments and Angles

10. Lesson 4: Measuring Segments and Angles
Segment Bisector
Definition:
Any segment, line or plane that divides a segment into two congruent parts is called segment bisector.
Lesson 4: Measuring Segments and Angles

11. Lesson 4: Measuring Segments and Angles
Angle and Points
ray
vertex
ray
Angles can have points in the interior, in the exterior or on the angle. A E D B C
Points A, B and C are on the angle. D is in the interior and E is in the exterior.
B is the vertex.
Lesson 4: Measuring Segments and Angles

12. Lesson 4: Measuring Segments and Angles
Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide
Using 3 points:
vertex must be the middle letter
This angle can be named as
Using 1 point:
using only vertex letter
* Use this method is permitted when the vertex point is the vertex of one and only one angle.
Since B is the vertex of only this angle, this can also be called A C B
Lesson 4: Measuring Segments and Angles

13. Lesson 4: Measuring Segments and Angles
Naming an Angle - continued
Using a number:
A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as A B 2 C
* The “1 letter” name is unacceptable when …
more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.
Lesson 4: Measuring Segments and Angles

14. Lesson 4: Measuring Segments and Angles
Example
K is the vertex of more than one angle.
Therefore, there is NO in this diagram.
There is
Lesson 4: Measuring Segments and Angles

15. Lesson 4: Measuring Segments and Angles
4 Types of Angles
Acute Angle:
an angle whose measure is less than 90.
Right Angle:
an angle whose measure is exactly 90 .
Obtuse Angle:
an angle whose measure is between
90 and 180.
Straight Angle:
an angle that is exactly 180 .
Lesson 4: Measuring Segments and Angles

16. Lesson 4: Measuring Segments and Angles
Measuring Angles
Just as we can measure segments, we can also measure angles.
We use units called degrees to measure angles.
A circle measures _____
A (semi) half-circle measures _____
A quarter-circle measures _____
One degree is the angle measure of 1/360th of a circle.
360º ?
180º ? ?
90º
Lesson 4: Measuring Segments and Angles

17. Lesson 4: Measuring Segments and Angles
Adding Angles
m1 + m2 = mADC also.
Therefore, mADC = 58.
Lesson 4: Measuring Segments and Angles

18. Lesson 4: Measuring Segments and Angles
Angle Addition Postulate
Postulate:
The sum of the two smaller angles will always equal the measure of the larger angle.
MRK
KRW
MRW
Lesson 4: Measuring Segments and Angles

19. Lesson 4: Measuring Segments and Angles
Example: Angle Addition
K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.
First, draw it!
3x + x + 6 = 90
4x + 6 = 90
– 6 = –6
4x = 84
x = 21 3x
x+6
Are we done?
mMRK = 3x = 3•21 = 63º
Lesson 4: Measuring Segments and Angles

20. Lesson 4: Measuring Segments and Angles
Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.
Example:
Since 4   6, is an angle bisector. 5 3
Lesson 4: Measuring Segments and Angles

21. Lesson 4: Measuring Segments and Angles
Congruent Angles
Definition:
If two angles have the same measure, then they are congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is  3 5
Example:
3   5.
Lesson 4: Measuring Segments and Angles

22. Lesson 4: Measuring Segments and Angles
Example
Draw your own diagram and answer this question:
If is the angle bisector of PMY and mPML = 87, then find:
mPMY = _______
mLMY = _______
Lesson 4: Measuring Segments and Angles