Download GEOMETRY R Unit 2: Angles and Parallel Lines

GEOMETRY R
Unit 2: Angles and Parallel Lines
Day
Friday
9/16
Classwork
Unit 1 Test
Homework
Monday
9/19
Angle Relationships
HW 2.1
Tuesday
9/20
Angle Relationships with Transversals
HW 2.2
Wednesday
9/21
Proving Lines are Parallel and
Constructing Parallel Lines
HW 2.3
Thursday
9/22
Interior and Exterior Angle Theorems
HW 2.4
Friday
9/23
More Unknown Angle Problems with
Justifications
Unit 2 Quiz 1
HW 2.5
Monday
9/26
Unknown Angle Proofs
HW 2.6
Tuesday
9/27
Proofs with Auxiliary Lines
HW 2.7
Wednesday
9/28
Review
Unit 2 Quiz 2
Review Packet
Thursday
10/29
Review
Study!
Friday
9/30
Unit 2 Test
Angles [1]
Vocabulary Term
Example
Adjacent Angles - Two angles in the same plane that
share a common vertex and a common ray, but share
no common interior points.
Linear Pair - Two adjacent angles with non-common
sides that are opposite rays.
Complementary Angles – Two angles with measures
that have a sum of 90 degrees.
Supplementary Angles – Two angles with measures
that have a sum of 180 degrees.
Angle Addition Postulate: If two adjacent angles
combine to form a whole angle, the measure of the
whole angle equals the sum of the measures of the two
adjacent angles.
Supplement Theorem – Two angles that form a linear
pair are supplementary angles.
Examples:
1. Find the measure of ∠𝑎.
2. Find the measure of ∠𝑓.
𝐴
𝐵
3. Find the measure of ∠𝑐.
4. Find the value of x.
5. Two angles are complementary. One angle is six less than twice the other angle. Find both angles.
Vertical Angles - Two non-adjacent angles formed by
two intersecting lines.
Vertical Angles Theorem - Vertical angles are
congruent.
Examples:
1. If m1  x2  4 x and m2  x  36 , find m3 .
3
1
2
2. Find the value of x and y.
Find the values of the missing variables.
3.
4.
5.
2x + 1
3x + 10
8
x + 10
9
y—3
y
6.
3x + 6
x - 10
7.
8.
x - 11
y
ANGLES AND PARALLEL LINES [2]
Vocabulary
Examples
Transversal - A line that intersects two or
more coplanar lines.
Corresponding Angles - Non-adjacent angles
that lie on the same side of the transversal,
one exterior and one interior.
Alternate Interior Angles - Non-adjacent
interior angles that lie on opposite sides of a
transversal.
Alternate Exterior Angles - Non-adjacent
exterior angles that lie on opposite sides
of a transversal.
Theorems
Words
Corresponding Angles
Postulate
If 2 parallel lines are cut by a
transversal, then corresponding
angles are congruent.
Alternate Interior
Angles Theorem
If 2 parallel lines are cut by a
transversal, then alternate
interior angles are congruent.
Consecutive Interior
Angles Theorem
If 2 parallel lines are cut by a
transversal, then consecutive
interior angles are supplementary
Alternate Exterior
Angles Theorem
If 2 parallel lines are cut by a
transversal, then alternate
exterior angles are congruent.
Diagram
Finding Values of Variables
The special relationships between the angles formed by two parallel lines and a transversal can be used to
find unknown values.
1. If m∠4 = 2x – 17 and m∠1 = 85, find x.
2. Find y if m∠3 = 4y + 30 and m∠7 = 7y + 6.
3. If m∠1 = 4x + 7 and m∠5 = 5x – 13, find x.
4. Find y if m∠7 = 68 and m∠6 = 3y – 2.
An _________________________________is sometimes useful when solving for
unknown angles. In this figure, we can use the auxiliary line to find the
measure of ∠𝑊. What is the measure of ∠𝑊?
Exercises
Find the unknown angles that are labeled. Justify each step.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Find the unknown (labeled) angles. Give reasons for your solutions.
1.
2.
3.
4.
PROVING LINES ARE PARALLEL [3]
Words
Diagram
Converse of the
Corresponding Angles
Postulate
If 2 lines are cut by a transversal
such that corresponding angles
are congruent, then the lines are
parallel.
Converse of the
Alternate Interior
Angles Theorem
If 2 lines are cut by a transversal
such that alternate interior angles
are congruent, then the lines are
parallel.
Converse of the
Consecutive Interior
Angles Theorem
If 2 lines are cut by a transversal
such that consecutive interior
angles are supplementary, then
the lines are parallel.
Converse of the
Alternate Exterior
Angles Theorem
If 2 lines are cut by a transversal
such that alternate exterior angles
are congruent, then the lines are
parallel.
Converse of the
Perpendicular
Transversal Theorem
If 2 lines are perpendicular to the
same line, then they are parallel to
each other.
Examples
1. Given the following information, is it possible to prove that any of the lines shown are parallel?
Justify your reasoning.
a. ∠2 ≅ ∠8
c. ∠1 ≅ ∠15
b. ∠3 ≅ ∠11
d. ∠8 ≅ ∠6
e. ∠12 ≅ ∠14
2. Find m∠MRQ so that a||b. Show your work.
3. Use the converse of the corresponding angles postulate to construct a line parallel to ⃡𝐴𝐵 , passing
through point C.
C
A
B
⃡ ,
4. Use the converse of the alternate interior angles theorem to construct a line parallel to 𝐴𝐵
passing through point C.
C
A
B
ANGLES OF TRIANGLES [4 & 5]
The sum of the interior angles of a triangle is 180 degrees.
The acute angles of a right triangle are complementary.
Each angle of an equilateral triangle measures 60 degrees.
The sum of the interior angles of a quadrilateral is 360 degrees.
Examples
1.
The vertex angle of an isosceles triangle exceeds the measure of a base angle by
30. Find the measure of the vertex angle.
2.
Classify triangle ABC based on its sides and angles if 𝑚∠A = (9x)°, 𝑚∠B=(3x-6)°,
and 𝑚∠C = (11x + 2)°.
Theorem
Words
The Exterior Angle
Theorem
An exterior angle of a
triangle equals the sum of
its two remote interior
angles
Proof of Exterior Angle Theorem:
Example
Examples
1.
Find x:
x2
x  12
5x
Find ma and mb in the figure to the right. Justify your
results.
2.
In each figure, determine the measures of the unknown (labeled) angles. Give reasons for each step of
your calculations.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
UNKNOWN ANGLE PROOFS [6]
Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction is
called inductive reasoning. These conclusions lack logical certainty. The system of reasoning that uses
facts, rules, definitions, or properties to reach logical conclusions is called deductive reasoning.
1. Angles that form a linear pair are supplementary. Use this fact to prove that vertical angles are
equal in measure.
2. Given the diagram at the right, prove that 𝑚∠𝑤 + 𝑚∠𝑥 + 𝑚∠𝑧 = 180°.
3. Given the diagram at the right, prove that 𝑚∠𝑤 = 𝑚∠𝑦 + 𝑚∠𝑧.
4. In the diagram at the right, prove that 𝑚∠𝑦 + 𝑚∠𝑧 = 𝑚∠𝑤 + 𝑚∠𝑥.
̅̅̅̅ ∥ 𝐶𝐷
̅̅̅̅ and 𝐵𝐶
̅̅̅̅ ∥ ̅̅̅̅
5. In the figure at the right, 𝐴𝐵
𝐷𝐸 .
Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐶𝐷𝐸.
6. In the figure at the right, prove that the sum of the angles
marked by arrows is 900°.
̅̅̅̅ ⊥ 𝐸𝐹
̅̅̅̅ .
7. In the figure at the right, prove that 𝐷𝐶
Z
8. In the labeled figure on the right, prove that 𝑚 || 𝑛.
9. In the diagram on the right, prove that the sum of the
angles marked by arrows is 360°.
10. In the diagram at the right, prove that 𝑚∠𝑎 + 𝑚∠𝑑 − 𝑚∠𝑏 = 180.
a
d
b
c
MORE UNKNOWN ANGLE PROBLEMS [7]
A proof is a logical argument in which each statement you make is supported by a statement that is
accepted as true. A theorem is a statement or conjecture that can be proven true by undefined terms,
definitions, and postulates.
̅̅̅̅ ∥ ̅̅̅̅
1. In the figure on the right, ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐷𝐸 and 𝐵𝐶
𝐸𝐹 . Prove that B  E .
̅̅̅̅ and ̅̅̅̅
(Hint: Extend 𝐵𝐶
𝐸𝐷 )
̅̅̅̅ ∥ 𝐶𝐷
̅̅̅̅
2. Given: 𝐴𝐵
Prove: 𝑚∠𝑧 = 𝑚∠𝑥 + 𝑚∠𝑦
3. In the figure, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸.
Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐶𝐷𝐸.
4. In the figure, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸.
Prove that 𝑚∠𝐵 + 𝑚∠𝐷 = 180.
5. In the figure, prove that 𝑚∠𝐴𝐷𝐶 = 𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶.
6. Use the theorems involving parallel lines discussed to prove that the three angles of a triangle
sum to 180°. For this proof, you will need to draw an auxiliary line, parallel to one of the
triangle’s sides and passing through the vertex opposite that side. Add any necessary labels and
write out your proof.
7. A theorem states that in a plane, if a line is perpendicular to one of two parallel lines, then it is
perpendicular to the other of the two parallel lines.
Prove this theorem. (a) Construct and label an appropriate figure, (b) state the given information
and the theorem to be proved, then (c) list the necessary steps to demonstrate the proof.
8. In the figure, 𝐴𝐵 || 𝐷𝐸 and 𝐵𝐶 || 𝐸𝐹.
Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹.
9. In the figure, 𝐴𝐵 || 𝐶𝐷.
Prove that 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐴 + 𝑚∠𝐶.
MORE EUCLIDEAN PROOFS
1. Given: ∠1 ≅ ∠2
Prove: 𝑙𝑖𝑛𝑒 𝑙 ∥ 𝑙𝑖𝑛𝑒 𝑚
̅̅̅̅̅ ∥ ̅̅̅̅
2. Given:𝑊𝑋
𝑌𝑍 , ∠2 ≅ ∠3
̅̅̅̅̅ ∥ 𝑋𝑍
̅̅̅̅
Prove: 𝑊𝑌
3. Given: ∠1 ≅ ∠6, ∠3 ≅ ∠6
Prove: ̅̅̅̅
𝐷𝐶 ∥ ̅̅̅̅
𝐴𝐵
D
4. Given: AE bisects DAC
2  B
Prove: AE || BC
A
E
2
B
5. Given: CB bisects
1
ACD ;
3
C
1 3
A
3
Prove: AB CD
1
2
C
6. Given:
Prove:
1 2
3 4
4
D
3
k
2
1
B
4
m