lesson 8: proving theorems about lines and angles 6/*5 t...

SIMILARITY, CONGRUENCE, AND PROOFSLesson 8: Proving Theorems About Lines and Angles

NAME:

CCGPS Analytic Geometry Teacher Resource U1-468

© Walch Education

Metalbro is a construction company involved with building a new skyscraper in Dubai. The diagram below is a rough sketch of a crane that Metalbro workers are using to build the skyscraper. The vertical line represents the support tower and the other line represents the boom. For safety reasons, the boom cannot be more than 15º beyond the horizontal in either direction. A horizontal line forms a 90º angle with the support tower. A straight line forms a 180º angle.

4

3 2

1

1. What are the safety requirements for m 1∠ ?




5. Use your findings to fill in the table.

Based on lower boundary of 1∠

Based on upper boundary of 1∠

m 1∠m 2∠m 3∠m 4∠

6. What do you notice about these angles?

Lesson 1.8.1: Proving the Vertical Angles TheoremWarm-Up 1.8.1


Instruction

CCGPS Analytic Geometry Teacher Resource © Walch EducationU1-473

• Straight angles are angles with rays in opposite directions—in other words, straight angles are straight lines.

Straight angle Not a straight angle

BC

D

BCD∠ is a straight angle. Points B, C, and D lie on the same line.

PQ

R

PQR∠ is not a straight angle. Points P, Q, and R do not lie on the same line.

• Adjacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points.

• Nonadjacent angles have no common vertex or common side, or have shared interior points.

Adjacent angles Nonadjacent angles

B

A

C

D

A

B

D

C

E

F

P

Q R

S

ABC∠ is adjacent to CBD∠ . They share vertex B and

! "!BC .

ABC∠ and CBD∠ have no common interior points.

ABE∠ is not adjacent to FCD∠ . They do not have a common vertex.

PQS∠ is not adjacent to PQR∠ . They share common interior points within PQS∠ .


Instruction


© Walch Education

• Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angle.

Linear pair Not a linear pair

A

B

C

D

ABC∠ and CBD∠ are a linear pair. They are adjacent angles with non-shared sides, creating a straight angle.

A

B

D

C

E

F

ABE∠ and FCD∠ are not a linear pair. They are not adjacent angles.

• Vertical angles are nonadjacent angles formed by two pairs of opposite rays.

TheoremVertical Angles Theorem

Vertical angles are congruent.

Vertical angles Not vertical angles

A

B

C

DE

A

B

C D

E

ABC∠ and EBD∠ are vertical angles. ABC EBD∠ ≅∠ABE∠ and CBD∠ are vertical angles. ABE CBD∠ ≅∠

ABC∠ and EBD∠ are not vertical angles. ! "!BC and

! "!BD are not opposite rays.

They do not form one straight line.


Instruction


PostulateAngle Addition Postulate

If D is in the interior of ABC∠ , then m ABD m DBC m ABC∠ + ∠ = ∠ .

If m ABD m DBC m ABC∠ + ∠ = ∠ , then D is in the interior of ABC∠ .

B

C

D

A

• Informally, the Angle Addition Postulate means that the measure of the larger angle is made up of the sum of the two smaller angles inside it.

• Supplementary angles are two angles whose sum is 180º.

• Supplementary angles can form a linear pair or be nonadjacent.

• In the following diagram, the angles form a linear pair.

m ABD m DBC 180∠ + ∠ =

B

C

D

A

P

QR

25º

T

U

V

155º

• The next diagram shows a pair of supplementary angles that are nonadjacent.

m PQR m TUV 180∠ + ∠ =

B

C

D

A

P

QR

25º

T

U

V

155º

TheoremSupplement Theorem

If two angles form a linear pair, then they are supplementary.


Instruction


© Walch Education

• Angles have the same congruence properties that segments do.

TheoremCongruence of angles is reflexive, symmetric, and transitive.

• Reflexive Property: 1 1∠ ≅∠

• Symmetric Property: If 1 2∠ ≅∠ , then 2 1∠ ≅∠ .

• Transitive Property: If 1 2∠ ≅∠ and 2 3∠ ≅∠ , then 1 3∠ ≅∠ .

TheoremAngles supplementary to the same angle or to congruent angles are congruent.

If m m1 2 180∠ + ∠ = and m m2 3 180∠ + ∠ = , then 1 3∠ ≅∠ .

• Perpendicular lines form four adjacent and congruent right angles.

TheoremIf two congruent angles form a linear pair, then they are right angles.

If two angles are congruent and supplementary, then each angle is a right angle.

• The symbol for indicating perpendicular lines in a diagram is a box at one of the right angles, as shown below.

Q

S

RP

• The symbol for writing perpendicular lines is ⊥ , and is read as “is perpendicular to.”


Instruction


• In the diagram, ! "# ! "#SQ PR⊥ .

• Rays and segments can also be perpendicular.

• In a pair of perpendicular lines, rays, or segments, only one right angle box is needed to indicate perpendicular lines.

• Remember that perpendicular bisectors are lines that intersect a segment at its midpoint at a right angle; they are perpendicular to the segment.

• Any point along the perpendicular bisector is equidistant from the endpoints of the segment that it bisects.

TheoremPerpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment.

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

A C

D

E

B

If ! "#DE is the perpendicular bisector of AC , then DA = DC.

If DA = DC, then ! "#DE is the perpendicular bisector of AC .

• Complementary angles are two angles whose sum is 90º.

• Complementary angles can form a right angle or be nonadjacent.

• The following diagram shows a pair of nonadjacent complementary angles.


Instruction


© Walch Education

m B m E 90∠ + ∠ =

RQ

P

35º

A

55ºE

12

S

BC

F

D

RQ

P

35º

A

55ºE

12

S

BC

F

D

• The next diagram shows a pair of adjacent complementary angles labeled with numbers.

m m1 2 90∠ + ∠ =

RQ

P

35º

A

55ºE

12

S

BC

F

D

TheoremComplement Theorem

If the non-shared sides of two adjacent angles form a right angle, then the angles are complementary.

Angles complementary to the same angle or to congruent angles are congruent.

Common Errors/Misconceptions

• not recognizing the theorem that is being used or that needs to be used

• setting expressions equal to each other rather than using the Complement or Supplement Theorems

• mislabeling angles with a single letter when that letter is the vertex for adjacent angles

• not recognizing adjacent and nonadjacent angles


Instruction


Example 1

Look at the following diagram. List pairs of supplementary angles, pairs of vertical angles, and a pair of opposite rays.

12 3

45 67 8A

BC

D

E

F

1. List pairs of supplementary angles.

Supplementary angles have a sum of 180º.

5∠ and 6∠ are adjacent supplementary angles. They form a linear pair.






Guided Practice 1.8.1


Instruction


© Walch Education

Example 3

In the diagram below, ! "#AC and

! "#BD are intersecting lines. If m x1 3 14∠ = + and m x2 9 22∠ = + , find

m 3∠ and m 4∠ .

12

34

A B

CD

1. Use the Supplement Theorem.

Since ! "#BD is a straight line, m m1 2 180∠ + ∠ = .

2. Use substitution to find the value of x.

Substitute the measures of 1∠ and 2∠ into the equation m m1 2 180∠ + ∠ = .

m x1 3 14∠ = +

m x2 9 22∠ = +

m m1 2 180∠ + ∠ = Supplement Theorem

(3x + 14) + (9x + 22) = 180 Substitute 3x + 14 and 9x + 22 for m 1∠ and m 2∠ .

12x + 36 = 180 Combine like terms.12x = 144 Subtract 36 from both sides.x = 12 Divide both sides by 12.


Instruction


5. Use the Subtraction Property.

Since m m2 2∠ = ∠ , these measures can be subtracted out of the equation m m m m1 2 2 3∠ + ∠ = ∠ + ∠ .

This leaves m m1 3∠ = ∠ .

6. Use the definition of congruence.

Since m m1 3∠ = ∠ , by the definition of congruence, 1 3∠ ≅∠ .

1∠ and 3∠ are vertical angles and they are congruent. This proof also shows that angles supplementary to the same angle are congruent.

Example 5

In the diagram below, ! "#DB is the perpendicular bisector of AC . If AD = 4x – 1 and DC = x + 11, what

are the values of AD and DC ?

A C

D

B

1. Use the Perpendicular Bisector Theorem to determine the values of AD and DC.

If a point is on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment being bisected. That means AD = DC.


NAME:


Maresol is retiling the backsplash over her kitchen stove, and has to cut square ceramic tiles into 4 congruent triangles to create the pattern she wants. If she doesn’t cut each square into perfectly equal triangles, the tiles won’t fit together properly. Before she cuts the first tile, she uses a pencil to draw two segments on the tile. The segments form perpendicular bisectors. Use what you know about triangle congruency and perpendicular bisectors to prove that the 4 triangles are congruent.

A B

CD

E

Problem-Based Task 1.8.1: Cutting Kitchen Tiles


NAME:


© Walch Education

continued

Use the following diagram to solve problems 1–4.

1 2345

1. List two pairs of adjacent angles and two pairs of nonadjacent angles.

2. List two pairs of supplementary angles. Write a statement about those angles using the Supplement Theorem.

3. List a pair of vertical angles. Write a statement about those angles using the Vertical Angles Theorem.

4. List a pair of complementary angles. Write a statement about those angles using the Complement Theorem.

Practice 1.8.1: Proving the Vertical Angles Theorem


NAME:


In the diagram that follows, ! "#AC and

! "#BD intersect. Use this information to solve for the measures of

the unknown angles in problems 5 and 6. Show and justify your work.

1 234

A

B

C

D

5. Find m 4∠ if m x1 3 4∠ = + and m x2 2 4∠ = − .

6. Find m 1∠ if m x1 13 7∠ = + and m x3 7 49∠ = + .

continued


NAME:


© Walch Education

Use the diagram that follows to solve problems 7 and 8.

1

432

7. Find m 1∠ given the following: 3∠ and 4∠ are complementary, 2 3∠ ≅∠ , m x2 6 24∠ = + , and m x4 5∠ = .

8. Find m 1∠ if 2∠ and 3∠ are complementary, m x1 4 23∠ = − , and m x4 38∠ = + .

continued

lesson 8: proving theorems about lines and angles 6/*5 t...

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