Ch 3: Parallel and Perpendicular Lines 3‐1 Properties of Parallel Lines 3‐2 Proving Lines Parallel 3‐3 Parallel and Perpendicular Lines 3‐4 Parallel Lines and the Triangle Angles‐Sum Theorem 3‐5 The Polygon Angle‐Sum Theorem 3‐6 Lines in the Coordinate Plane 3‐7 Slopes of Parallel and Perpendicular Lines 3.8 Constructing Parallel and Perpendicular Lines 3‐1 Properties of Parallel Lines: Focused Learning Target: I will be able to Standards: Geom 7.0 Identifying angles formed by two lines and a transversal Proving and using properties of parallel lines Vocabulary: Transversal Alternate Interior Angles Same‐Side Interior Angles Corresponding Angles Two‐Column Proof Alternate Exterior Angles Same‐side Exterior Angles 5 and 8 are alternate exterior angles 5 and 7 are same‐side exterior angels 5 6 56 78 7 8
Transcript
Ch 3: Parallel and Perpendicular Lines
3‐1 Properties of Parallel Lines
3‐2 Proving Lines Parallel
3‐3 Parallel and Perpendicular Lines
3‐4 Parallel Lines and the Triangle Angles‐Sum Theorem
3‐5 The Polygon Angle‐Sum Theorem
3‐6 Lines in the Coordinate Plane
3‐7 Slopes of Parallel and Perpendicular Lines
3.8 Constructing Parallel and Perpendicular Lines
3‐1 Properties of Parallel Lines: Focused Learning Target: I will be able to Standards: Geom 7.0
Identifying angles formed by two lines and a transversal
Proving and using properties of parallel lines
Vocabulary:
Transversal
Alternate Interior Angles
Same‐Side Interior Angles
Corresponding Angles
Two‐Column Proof
Alternate Exterior Angles
Same‐side Exterior Angles
5 and 8 are alternate exterior angles 5 and 7 are same‐side exterior angels
5 6 5 6 7 8 7 8
Identifying angles: I’ll do one: We’ll do one together: You try:
Using the diagram. Identify which angle forms a pair of alternate
interior angels with 1 . Identify which angle forms a pair of
alternate exterior with 4 .
Identify which angle forms a pair of
same side exterior angle with 5 . Identify which angel forms a pair of
corresponding angle with 2 .
Writing Two column proofs:
I’ll do one:
Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
We’ll do one together:
Given: ba ||
Prove: 1 and 3 are supplementary Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
You try one:
Given: ba ||
Prove: 1 and 4 are supplementary
Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
Using algebra to find angle measures
I’ll do one: We’ll do one: You try:
Find the value of x and y in the diagram. Justify each answer.
Find the value of a, b and c. Justify each answer. Given: ml ||
Find the values x and y. Justify each answer. Then find the measures of the angles.
3‐2 Proving Lines Parallel Focused Learning Target: I will be able to Standards: Geom 7.0
Using a transversal in proving lines parallel
I’ll do one: We’ll do one together: You try:
Which lines or segments are parallel? Justify your answer with a theorem or postulate.
Which lines or segments are parallel? Justify your answer with a theorem or postulate.
Which lines or segments are parallel? Justify your answer with a theorem or postulate.
Using Algebra I’ll do one: We’ll do one together: You try:
Find the value of x for which ml ||
Find the value of x for which ba ||
Find the value of x for which ml ||
3‐3 Parallel and Perpendicular Lines: Focused Learning Target: I will be able to
Relate parallel and perpendicular lines
Standards: Geom 7.0
I’ll do one:
Prove Theorem 3‐10
1. tr , ts 1. Given
2. 2.
3. 3.
4. 4.
5. 5.
6. 6. We’ll do one together:
Using Theorem 3‐11
1. ba , cb , and dc 1. Given
2. 2.
3. 3. We’ll do another one:
Prove Theorem 3‐11:
1. ba 1. Given
2. 2. Definition of right angles
3. cb || 3. Given
4. 4. Corresponding Angles Postulate
5. ca 5. Now that we have proven Theorems 3‐10 and 3‐11, we can use them in any proofs from now on! You Try:
1. rt , st 1.
2. sr || 2. You Try:
1. ba , cb 1.
2. ca || 2.
3. dc || 3.
4. da || 4.
Let’s try one more together:
1. pq || 1. Given
2. 21 2.
3. rp || 3.
4. 32 4.
5. 31 5.
6. rq || 6.
3‐4 Parallel Lines and the Triangle Sum Theorem: Focused Learning Target: I will be able to
Classify triangles based on their side and angle measures
Find the measures of angles in a triangle
Use exterior angles of triangles to find the measures of missing interior and exterior angles of a triangle
Standards: Geom 12.0 Geom 13.0
Vocabulary:
Equiangular Triangle
Acute Triangle
Right Triangle
Obtuse Triangle
Equilateral Triangle
Isosceles Triangle
Scalene Triangle
Exterior Angle of a Triangle
Remote Interior Angles of a Triangle
Triangles classified by their angles:
Triangles classified by their sides:
Each triangle can now be classified based on the characteristics of its angles and sides, like a first and last name. I’ll try one: We’ll try one: You Try one:
Some combinations of triangle classifications are not possible, can you think of any? No matter what type of triangle you have, they all have the same interior angle sum, which is described in the following theorem.
I’ll try one: We’ll try one together: You try:
Find the value of x. Then find the measures of the angles.
Find the values of x, y, and z.
Find the values of x, y, and z.
If every triangle has 3 interior angles, they must also have 3 exterior angles. An exterior angle of a triangle is an angle formed by extending a side. As a result, the two non‐adjacent interior angles of an exterior angle are called the remote interior angles.
An exterior angle of a triangle has an interesting relationship with its remote interior angles.
I’ll do one: We’ll try one: You Try one:
Find the missing angle measure.
Find the missing angle measure.
Find the missing angle measure.
Note: An equilateral polygon has all sides congruent. An equiangular polygon has all angles congruent. A regular polygon is both equilateral and equiangular.
3‐6 Lines in the Coordinate Plane: Focused Learning Target: I will be able to Standards: Geom 17.0
Graph lines given their equations
Write equations of lines
Vocabulary:
Slope‐intercept form
Standard form
X‐intercept
Point‐slope form
Graphing in slope‐intercept form:
I’ll do one:
Graph 3 4y x
1. Identify the slope (m) and the y‐intercept (b). m = b =
2. locate the y‐intercept and plot the point
3. Use the slope to find the next two points.
4. Draw the line through the points. -6 -4 -2 2 4 6 8
-4
-2
2
4
x
y
We’ll do one: You try:
Graph 2
43
y x
-6 -4 -2 2 4 6 8
-4
-2
2
4
x
y
m b
Graph: 2 3y x
m = b=
-4 -2 2 4 6
-4
-2
2
4
x
y
m b
You Try: You Try:
Graph: 2y x
-4 -2 2 4 6
-4
-2
2
4
x
y
Graph: 1
43
y x
-6 -4 -2 2 4 6 8
-6
-4
-2
2
4
x
y
Graphing Lines Using Intercepts: Standard form of a linear equation: a linear equation written in the form Ax By C , where A and B are not both
zero and A, B and C are all real numbers. Ex: 32 5
5x y is written in standard form.
x‐intercept: the x intercept is the point at which the line crosses the x‐axis.
I’ll do one: We’ll do one: You try:
-4 -2 2 4
-4
-2
2
4
x
y
-6 -4 -2 2 4
-4
-2
2
4
x
y
-6 -4 -2 2 4
-4
-2
2
4
x
y
x‐intercept = x‐intercept = x‐intercept =
To find the x & y‐intercepts algebraically:
1. Substitute 0 in place of x, then solve for y. The result is the y‐intercept.
2. Substitute 0 in place of y, then solve for x. The result is the x‐intercept.
I’ll do one: We’ll do one: You try:
Find the x & y‐intercepts:
2 5 10x y
Find the x & y‐intercepts:
3 5 30x y
Find the x & y‐intercepts:
7 4 21x y
Graph an equation by finding the x & y‐intercepts:
I’ll do one: We’ll do one together:
Graph: 3 5 15x y
-8 -6 -4 -2 2 4 6 8
-8
-6-4-2
246
8
x
y
Graph: 4 3 24x y
-8 -6 -4 -2 2 4 6 8
-8
-6-4-2
246
8
x
y
You Try:
2 3 12x y
-8 -6 -4 -2 2 4 6 8
-8
-6-4-2
246
8
x
y
To transform an equation of a line into slope‐intercept form, solve for y:
I’ll do one: We’ll do one together: You try:
Write the equation in slope‐
intercept form: 3 9 36x y
Write the equation in slope‐
intercept form: 5 7 42x y
Write the equation in slope‐
intercept form: 2 5 20x y
Writing the equation of a line given two points on the line: find the slope of the line, then use the slope and one of the points and plug them into point‐slope form of the line. This will give you the y‐intercept.
I’ll do one: We’ll do one together: You try:
Write the equation of the line that passes through (2, 3) and (5, 5)
Write the equation of the line that passes through (3, ‐2) and (4, 5)
Write the equation of the line that passes through (5, 7) and (18, 13)
Equations of horizontal and vertical lines: horizontal and vertical lines are special cases. Draw the line. When drawn, one of the coordinates will repeat. The repeating coordinate is the equation.
I’ll do one We’ll do one together: You try:
Write the equation of the horizontal line that passes through (5, 7).
Write the equation of the vertical line that passes through (3, ‐5).
Write the equation of the horizontal line that passes through (0, 7).
3‐7 Slopes of Parallel and Perpendicular Lines: Focused Learning Target: I will be able to
Relate slope and parallel lines
Relate slope and perpendicular lines
Write converses of conditional statements
Slopes of Parallel Lines:
If two nonvertical lines are parallel, their slopes are equal.
If the slopes of two distinct nonvertical lines are equal, then the lines are parallel
Any two vertical lines are parallel
You can test whether nonvertical lines are parallel by comparing slopes.
Example 1: Checking for Parallel Lines I’ll do one: We’ll do one together: You try:
Are lines l1 and l2 parallel? Explain.
Are lines l1 and l2 parallel? Explain.
Are lines l1 and l2 parallel? Explain.
Slope‐intercept form allows you to compare slopes easily in order to decide whether lines are parallel. Slope intercept form is y=mx+b where m is the slope.
Example 2: Writing equations of parallel lines: I’ll do one: We’ll do one together: you try:
Write an equation for the line
parallel to y =1
2x + 1 that contains
(6, ‐1).
Write an equation for the line parallel to ‐4y = 8x + 3 that contains (‐3, 5).
Write an equation for the line parallel to y = ‐4x+3 that contains (1,‐2).
Slopes of perpendicular lines:
If two nonvertical lines are perpendicular, the product of their slopes is ‐1. If the slopes of two lines have a product of ‐1, the lines are perpendicular. Any horizontal line and vertical line are perpendicular.
To find the slope of a line containing the points (x1, y1) and (x2,y2), use the formula: m = y2‐y1 x2‐x1
Example 3: Checking for Perpendicular Lines: I’ll do one: We’ll do one together: You try:
Lines l1 and l2 are neither vertical nor horizontal. Are they perpendicular? Explain.
Lines l1 and l2 are neither vertical nor horizontal. Are they perpendicular? Explain.
Lines l1 and l2 are neither vertical nor horizontal. Are they perpendicular? Explain.
You can write an equation for a line perpendicular t a given line. Example 4: Writing Equations for Perpendicular Lines I’ll do one: We’ll do one together: You try:
Write an equation for the line
perpendicular to XY
that contains
point Z.
XY
: 3x + 2y = ‐6, Z(3, 2)
Write an equation for the line
perpendicular to XY
that contains
point Z.
XY
: y =3
4x + 22, Z(12, 8)
Write an equation for the line
perpendicular to XY
that contains
point Z.
XY
: ‐x + y = 0, Z(‐2, ‐1)
3‐8 Constructing Parallel and Perpendicular Lines: Focused Learning Target: I will be able to Standards: Geom 16.0
Construct parallel lines
Construct perpendicular lines
You can use what you know about parallel lines, transversals, and corresponding angles to construct parallel lines.
To construct the perpendicular to a given line through a given point not on the line:
1. Open your compass to a size greater than the distance from Q to l. With the compass point on point Q, draw an arc that intersects l at two points. Label the points E and F
2. Place the compass on point E and make an arc
3. Keep the same compass setting. With the compass tip on F, draw an arc that intersects the arc from Step 2. Label the point of intersection G.
4. Draw line QG
I’ll do one: We’ll do one together: You try:
Construct a line perpendicular to
line l through point Q.
Construct a line perpendicular to
line l through point Q.
Construct a line perpendicular to
line l through point Q.
To construct the perpendicular to a given line at a given point on the line: 1. Put the compass point on point T. Draw arcs intersecting l in two points. Label the points A and B. 2. Open the compass wider. With the compass tip on A, draw an arc above point T. 3. Without changing the compass sitting, place the compass point on point B. Draw an arc that intersects the
arc from Step 2. Label the point of intersection C. 4. Draw line CT
I’ll do one: We’ll do one together: You Try:
Construct a line perpendicular to
line l at point T.
Construct a line perpendicular to
line l at point T.
Construct a line perpendicular to
line l at point T.
To construct a line parallel to a given line and through a given point not on the line:
1. Label two points H and J on line l. Draw line HK 2. Construct 1 with vertex at K so that 1m = KHJm and the two angles are corresponding angles. Label
the line you just constructed m
I’ll do one: We’ll do one together: You try:
Construct a line parallel to line l and through point K.
Construct a line parallel to line l and
through point K.
Construct a line parallel to line l and
through point K.
Now you can put your constructions together to construct polygons I’ll do one:
Construct a quadrilateral with one pair of parallel sides of lengths a and the other pair length b.
We’ll do one together:
Construct a right triangle with leg lengths of b and c.
