Measure and Classify Angles Objectives: To define, classify, draw, name, and measure various angles To use the Protractor and Angle Addition Postulates To construct congruent angles and angle bisectors with compass and straightedge To convert angle measurement between degrees and radians

Vocabulary Angle Obtuse  Vertex Right  Sides Straight  Acute  As a group, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions. Angle Obtuse  Vertex Right  Sides Straight  Acute  Congruent ’s

A “Rabbit Ear” antenna is a physical model of an angle An angle consists of two different rays (sides) that share a common endpoint (vertex). Angle ABC, ABC, or B Vertex Sides A “Rabbit Ear” antenna is a physical model of an angle

Angle An angle consists of two different rays (sides) that share a common endpoint (vertex). Angle ABC, ABC, or B

Example 1 How many angles can be seen in the diagram? Name all the angles.

Click me to learn more about measuring angles How Big is an Angle? Is the angle between the two hands of the wristwatch smaller than the angle between the hands of the large clock? Both clocks read 9:36 Click me to learn more about measuring angles

Measure of an Angle The measure of an angle is the smallest amount of rotation about the vertex from one side to the other, measured in degrees. Can be any value between 0 and 180 Measured with a protractor

Classifying Angles Surely you are familiar with all of my angular friends by now.

C Comes Before S…

Example 1a Given that <1 is a complement of <2 and m<1 = 68°, find m<2. Given that <3 is a supplement of <4 and m<3 = 56°, find m<4.

Example 2 Let <A and <B be complementary angles and let m<A = (2x2 + 35)° and m<B = (x + 10)°. What is (are) the value(s) of x? What are the measures of the angles?

Linear Pairs of Angles

Linear Pairs of Angles Two adjacent angles form a linear pair if their noncommon sides are opposite rays. The angles in a linear pair are supplementary.

Vertical Angles

Vertical Angles Two nonadjacent angles are vertical angles if their sides form two pairs of opposite rays. Vertical angles are formed by two intersecting lines. GSP

Example 3 Identify all of the linear pairs of angles and all of the vertical angles in the figure.

Example 4: SAT In the figure and , what is the value of x?

How To Use a Protractor The measure of this angle is written:

Example 3 What is the measure of DOZ?

Example 3 You basically used the Angle Addition Postulate to get the measure of the angle, where mDOG + mGOZ = mDOZ.

Angle Addition Postulate If P is in the interior of RST, then mRST = mRSP + mPST.

Example 4 Given that mLKN = 145°, find mLKM and mMKN.

markings to your picture. Congruent Angles Two angles are congruent angles if they have the same measure. Add the appropriate markings to your picture.

Angle Bisector An angle bisector is a ray that divides an angle into two congruent angles.

Example 5 In the diagram, YW bisects XYZ, and mXYW = 18°. Find mXYZ.

Example 6 In the diagram, OE bisects angle LON. Find the value of x and the measure of each angle.

Use Midpoint and Distance Formulas 1.3 Use Midpoint and Distance Formulas Objectives: To define midpoint and segment bisector To use the Midpoint and Distance Formulas To construct a segment bisector with a compass and straightedge

Midpoint The midpoint of a segment is the point on the segment that divides, or bisects, it into two congruent segments.

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

Example 1 Find DM if M is the midpoint of segment DA, DM = 4x – 1, and MA = 3x + 3.

Example 3 Segment OP lies on a real number line with point O at –9 and point P at 3. Where is the midpoint of the segment? What if the endpoints of segment OP were at x1 and x2?

In the Coordinate Plane We could extend the previous exercise by putting the segment in the coordinate plane. Now we have two dimensions and two sets of coordinates. Each of these would have to be averaged to find the coordinates of the midpoint.

1.3 The Midpoint Formula If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the midpoint M of AB has coordinates

The Midpoint Formula The coordinates of the midpoint of a segment are basically the averages of the x- and y-coordinates of the endpoints

Example 4 Find the midpoint of the segment with endpoints at (-1, 5) and (3, 8).

Example 5 The midpoint C of IN has coordinates (4, -3). Find the coordinates of point I if point N is at (10, 2).

Example 6 Use the Midpoint Formula multiple times to find the coordinates of the points that divide AB into four congruent segments.

Parts of a Right Triangle Which segment is the longest in any right triangle?

The Pythagorean Theorem In a right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then c2 = a2 + b2.

Example 7 How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall?

The Distance Formula Sometimes instead of finding a segment’s midpoint, you want to find it’s length. Notice how every non-vertical or non-horizontal segment in the coordinate plane can be turned into the hypotenuse of a right triangle.

Example 8 Graph AB with A(2, 1) and B(7, 8). Add segments to your drawing to create right triangle ABC. Now use the Pythagorean Theorem to find AB.

Distance Formula In the previous problem, you found the length of a segment by connecting it to a right triangle on graph paper and then applying the Pythagorean Theorem. But what if the points are too far apart to be conveniently graphed on a piece of ordinary graph paper? For example, what is the distance between the points (15, 37) and (42, 73)? What we need is a formula!

The Distance Formula To find the distance between points A and B shown at the right, you can simply count the squares on the side AC and the squares on side BC, then use the Pythagorean Theorem to find AB. But if the distances are too great to count conveniently, there is a simple way to find the lengths. Just use the Ruler Postulate.

The Distance Formula You can find the horizontal distance subtracting the x-coordinates of points A and B: AC = |7 – 2| = 5. Similarly, to find the vertical distance BC, subtract the y-coordinates of points A and B: BC = |8 – 1| = 7. Now you can use the Pythagorean Theorem to find AB.

Example 9 Generalize this result and come up with a formula for the distance between any two points (x1, y1) and (x2, y2).

The Distance Formula If the coordinates of points A and B are (x1, y1) and (x2, y2), then

Example 10 To the nearest tenth of a unit, what is the approximate length of RS, with endpoints R(3, 1) and S(-1, -5)?

Example 11 A coordinate grid is placed over a map. City A is located at (-3, 2) and City B is located at (4, 8). If City C is at the midpoint between City A and City B, what is the approximate distance in coordinate units from City A to City C?

Example 12 Points on a 3-Dimensional coordinate grid can be located with coordinates of the form (x, y, z). Finding the midpoint of a segment or the length of a segment in 3-D is analogous to finding them in 2-D, you just have 3 coordinates with which to work.

Example 12 Find the midpoint and the length of the segment with endpoints (2, 5, 8) and (-3, 1, 2).