Review - Midpoint and Distance formula (6, 7) M A B B M • (4.5, 4) (-9, 4) (-5, 4) (-1, 4) C (3, 1) What is the distance from A to B? What is the midpoint? The measure of the lengths of segment AM and MB are equal. So the two segments are congruent because they are the same size. What would be the endpoint of the segment of AB if the midpoint was at (-1,4)? What about if we look at a segment that is slanted what tools do we have to find the length of the segment? How about the midpoint? How could we use the distance formula to prove that segment CM is congruent to MD? C • (0,-5)
Questions on Homework 1.3??
Warm up 1. The sum of two numbers is 90 and one number is 4 times the other. Write an equation and solve to find the numbers. ANSWER x + 4x = 90; 18, 72 2. Find m ABD. What kind of angle is it? If you finish your daily quiz early start the warm up. I should not hear any talking or I will take your quiz everyone has a task and should be working. At this point pt your homework in the right hand corner of your test. Remember you will get a 5 out of ten if I just see answers. You need to be showing your work. ANSWER 180° , straight
Questions on 1.4?
Describe Angle Pair Relationships 1.5 Lesson Describe Angle Pair Relationships
Complementary Angles Examples: 2 angles that add to equal 90. Direct students to the table in their notes Ask students to come up with their own definitions given these examples Way to remember: “It is right to give complements”
Supplementary Angles 2 angles are supplementary if the sum of their angle measure is 180. Example: These two angles are supplementary. Note that these two angles can be "pasted" together to form a straight line! Direct students to the table in their notes Ask students to come up with their own definitions given these examples One of the supplementary angles is said to be the supplement of the other. Way to remember: “S stand for straight”
Adjacent Angles 2 angles next to each other that share a common vertex and side, but have no common interior points. Ask students to come up with their own definitions given these examples A &B are adjacent angles
share a common vertex and side, they are adjacent. GUIDED PRACTICE for Example 1 In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1. Because FGK and HGK share a common vertex and side, they are adjacent. Because 41° + 49° = 90°, FGK and GKL are complementary angles. Direct student to the guided practice Because 49° + 131° = 180°, HGK and GKL are supplementary angles.
GUIDED PRACTICE for Example 1 Are KGH and LKG adjacent angles? Explain. 2. Nope! While they do share a common. Adjacent angles do not have common interior points.
EXAMPLE 2 Find measures of a complement and a supplement a. Given that 1 is a complement of 2 and m 1 = 68°, find m 2. SOLUTION a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 2 = 90° – m 1 = 90° – 68° = 22
EXAMPLE 2 Find measures of a complement and a supplement b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3. SOLUTION b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°
EXAMPLE 3 Find angle measures Sports When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD. Remember in geometry we take are knowledge of the diagram and properties of the angle pairs to set up an equation.
EXAMPLE 3 Find angle measures SOLUTION STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°. m BCE + m ∠ ECD = 180° Write equation. (4x+ 8)° + (x + 2)° = 180° Substitute. 5x + 10 = 180 Combine like terms. 5x = 170 Subtract 10 from each side. x = 34 Divide each side by 5.
EXAMPLE 3 Find angle measures STEP 2 Evaluate: the original expressions when x = 34. m BCE = (4x + 8)° = (4 34 + 8)° = 144° m ECD = (x + 2)° = ( 34 + 2)° = 36° The angle measures are 144° and 36°. ANSWER
Angles Formed by the Intersection of 2 Lines Click on picture!! What angles always are equal, which ones add to equal 180 degree? What are properties of linear pairs (adjacent angles) and vertical angles? How might this be useful? Time saving? When would the vertical and adjacent angles all be equal? Click Me!
Linear Pair A linear pair is formed by two angles that are adjacent (share a leg) and supplementary (add up to 180°) “forms a line”
“When you draw over the 2 angles it forms an X” Vertical Angles A pair of non-adjacent angles formed by the intersection of two straight lines “When you draw over the 2 angles it forms an X”
EXAMPLE 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. SOLUTION To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER To find vertical angles, look or angles formed by intersecting lines. 1 and 5 are vertical angles. ANSWER
Example 5 Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.
Example 6 Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles.
Wrap Up sketches the angle pairs I describe. be sure to make up angles measures that would fit the description. There are millions of answers that could be correct. If it is not possible, write not possible.