GUIDE

CARD

GUIDE CARD

A two-column geometric proof consists of a list of statements, and

the reasons that we know those statements are true. The

statements are listed in a column on the left, and the reasons for

which the statements can be made are listed in the right column.

Every step of the proof (that is, every conclusion that is made) is a

row in the two-column proof.

Writing a proof consists of a few different steps.

1. Draw the figure that illustrates what is to be proved. The figure

may already be drawn for you, or you may have to draw it

yourself.

2. List the given statements, and then list the conclusion to be

proved. Now you have a beginning and an end to the proof.

3. Mark the figure according to what you can deduce about it from the

information given. This is the step of the proof in which you actually

find out how the proof is to be made, and whether or not you are able

to prove what is asked. Congruent sides, angles, etc. should all be

marked so that you can see for yourself what must be written in the

proof to convince the reader that you are right in your conclusion.

4. Write the steps down carefully, without skipping even the simplest

one. Some of the first steps are often the given statements (but not

always), and the last step is the conclusion that you set out to prove.

A sample proof looks like this:

Given:

Segment AD bisects segment BC. Segment BC bisects segment AD.

Prove: Triangles ABM and DCM are congruent.

Notice that when the SAS postulate was used, the numbers in parentheses correspond to the numbers of the statements in which each side and angle was shown to be congruent. Anytime it is helpful to refer to certain parts of a proof, you can include the numbers of the appropriate statements in parentheses after the reason.

Here are some properties you can use as reasons in a proof: General:

Reflexive Property A quantity is congruent (equal) to itself. a = a

Symmetric Property If a = b, then b = a.

Transitive Property If a = b and b = c, then a = c.

Addition Postulate If equal quantities are added to equal quantities, the sums are equal.

Subtraction Postulate If equal quantities are subtracted from equal quantities, the differences are equal.

Multiplication Postulate

If equal quantities are multiplied by equal quantities, the products are equal. (also Doubles of equal quantities are equal.)

Division Postulate If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)

Substitution Postulate

A quantity may be substituted for its equal in any expression.

Partition Postulate

The whole is equal to the sum of its parts. Also: Betweeness of Points: AB + BC = AC Angle Addition Postulate: m<ABC + m<CBD = m<ABD

Construction Two points determine a straight line.

Construction From a given point on (or not on) a line, one and only one perpendicular can be drawn to

Angles:

Right Angles All right angles are congruent.

Straight Angles All straight angles are congruent.

Congruent Supplements Supplements of the same angle, or congruent angles, are

congruent.

Congruent Complements Complements of the same angle, or congruent angles, are

congruent.

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

Vertical Angles Vertical angles are congruent.

Triangle Sum The sum of the interior angles of a triangle is 180º.

Exterior Angle

The measure of an exterior angle of a triangle is equal to the

sum of the measures of the two non-adjacent interior angles.

The measure of an exterior angle of a triangle is greater than

either non-adjacent interior angle.

Base Angle Theorem

(Isosceles Triangle)

If two sides of a triangle are congruent, the angles opposite these

sides are congruent.

Base Angle Converse

(Isosceles Triangle)

If two angles of a triangle are congruent, the sides opposite these

angles are congruent.

Triangles:

Side-Side-Side (SSS) Congruence

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side (SAS) Congruence

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Angle-Side-Angle (ASA) Congruence

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Angle-Angle-Side (AAS) Congruence

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Hypotenuse-Leg (HL) Congruence (right triangle)

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.

CPCTC Corresponding parts of congruent triangles are congruent.

Angle-Angle (AA) Similarity

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

SSS for Similarity If the three sets of corresponding sides of two triangles are

in proportion, the triangles are similar.

SAS for Similarity

If an angle of one triangle is congruent to the

corresponding angle of another triangle and the lengths of

the sides including these angles are in proportion, the

triangles are similar.

Side Proportionality If two triangles are similar, the corresponding sides are in

proportion.

Mid-segment Theorem (also called mid-line)

The segment connecting the midpoints of two sides of a

triangle is parallel to the third side and is half as long.

Sum of Two Sides The sum of the lengths of any two sides of a triangle must

be greater than the third side

Longest Side

In a triangle, the longest side is across from the largest

angle.

In a triangle, the largest angle is across from the longest

side.

Altitude Rule

The altitude to the hypotenuse of a right triangle is the

mean proportional between the segments into which it

divides the hypotenuse.

Leg Rule

Each leg of a right triangle is the mean proportional

between the hypotenuse and the projection of the leg on

the hypotenuse.

The Same Angle Supplements Theorem states that if two angles are supplementary to the same angle then the two angles are congruent. Prove this theorem.

Given : and are supplementary angles. and are supplementary angles.

Prove :

Statement Reason

1. and are supplementary

and are supplementary 1. Given

2.

2. Definition of supplementary angles

3. 3. Substitution

4. 4. Subtraction

5. 5. angles have = measures

The Vertical Angles Theorem states that vertical angles are congruent. Prove this theorem.

Given : Lines and intersect.

Prove:

Statement Reason

1. Lines and intersect 1. Given

2. and are a linear pair

and are a linear pair 2. Definition of a Linear Pair

3. and are supplementary

and are supplementary 3. Linear Pair Postulate

4.

4. Definition of Supplementary Angles

5. 5. Substitution

6. 6. Subtraction

7. 7. angles have = measures

ACTIVITY

CARD

If you’re ready, let’s start the

adventure!

GENERAL INSTRUCTIONS: Since our entire topic is all about writing statement and its reason so, first, in this following activity let us connect it in real life wherein we are going to distinguish its cause and effect. ACTIVITY 1 Instructions: Complete the table below.

Cause Effect

1.The girl played in the mud

2.I never brush my teeth

3. The doctor put a cast on his arm.

4.I flipped the light switch on

5. We were late for work

6.Julia’s cat died

7. Laura eat too much candies

8.she’s healthy

9.Mila is sad

10. we had to cancel the match

Are you ready for the next activity?

Okay, if you are, let’s now proceed to the

next.

ACTIVITY 2

Instructions: Match the following phrases to complete a sentence.

CAUSE EFFECT

People killed too many of them so now he have lung cancer

We ran out of milk we went to the grocery store

Powerful winds blew the roof she is well loved by her pupils

off the house

You should brush your teeth a star explodes

James is smoking cigarettes he was hungry

When nuclear fusion stop or starts and a thief broke into the room

Mrs. Ramos is a kind teacher you won’t get cavities

The room was left unlocked alligator almost became extinct

Keegan skipped lunch the whole family had nowhere to

live

Heavy rain traffic

ACTIVITY 3 Instructions: Fill in the blanks

1. There was flooding because of ____________________________________________

2. The air get cold so ______________________________________________________

3. Jim is scared because ____________________________________________________

4. The baby was crying, so __________________________________________________

5. I learned to play the drums quickly because __________________________________

6. Mara didn’t follow the recipe correctly, so ___________________________________

7. Some believe dinosaurs died out because ____________________________________

8. I had to mop since ______________________________________________________

9. John made a rude comment, so ____________________________________________

10. We went to the mall since, _______________________________________________

Assessme

nt

CARD

Make a two-column proof for the following

problems. Choose your answers from the box below each problem.

The Right Angle Theorem states that if two angles are right angles, then the angles are congruent. Prove this theorem.

To prove this theorem, set up your own drawing and name some angles so that you have specific angles to talk about.

Given : and are right angles

Prove :

Statement Reason

1. and are right angles 1. Given

2. and 2.

3. 3. Transitive

4. 4. angles have = measures

Definition of right angles

angles have = measures

Write a two-column proof for the following:

Given: and

Prove:

Statement Reason

1. 1. Given

2. and are right angles 2.

3. 3. Right Angle Theorem

4. 4.

5. 5. Subtraction

Definition of Perpendicular Lines Given

Write a two-column proof for the following:

Given: is supplementary to , is supplementary to ,

Prove:

Statement Reason

1. is supplementary to , is supplementary to 1. Given

2. , 2.

3. 3. Substitution

4. 4.

5. 5. Substitution

6. 6.

Definition of Supplementary Angles Given

Subtraction

Enrichmen

t

CARD

Identify which pair of triangles below does NOT illustrate an angle angle side (AAS) relationship.

4.

Proof

5.

Proof

6.

Proof

7.

Proof

Reference

CARD

examples.yourdictionary.com/cause-and-effect-

examples.html#comment-1741397603

www.grammar-quizzes.com/19-2.html

edhelper.com/cause_and_effect.html

www.grammar-quizzes.com/8-7.html

edhelper.com/language/reading_comprehension_skills202.html

http://www.ck12.org/geometry/Two-Column-Proofs/lesson/Two-Column-Proofs-Intermediate/

http://m.sparknotes.com/math/geometry3/geometricproofs/problems_1.html http://www.regentsprep.org/regents/math/geometry/gp4/PracCongTri.html

Answer Card

Activity 1

1. She got dirty

2. I got cavities

3. He broke his arm

4. The light turned on

5. The alarm didn’t work

6. Julia forgot to feed her cat

7. She got tooth cavity

8. She eats fruits and vegetables

9. She got a bad day

10. It rained so hard

Activity 2

1.Alligator almost become extinct

2.We went to grocery

3.The family gad nowhere to live

4.You wont get cavities

5.So now he have lung cancer

6.A star explodes

7.She is well loved by her pupils

8.And a thief broke into the room

9.He was hungry

10. Traffic

Activity 3

1. Heavy rains

2. She put on her jacket

3. Its dark

4. The father gave him milk

5. I took advance lesson

6. The food didn’t taste good

7. A meteor hit the earth

8. My juice spilled on the floor

9. Elise got mad

10. We don’t have classes

Statement Reason

1. and are right angles

1. Given

2. and 2. Definition of right angles

3. 3. Transitive

4. 4. angles have = measures

ASSESSMENT 1

Statement Reason

1. 1. Given

2. and are right angles

2. Definition of a Perpendicular Lines

3. 3. Right Angle Theorem

4. 4. Given

5. 5. Subtraction

ASSESSMENT 2

Statement Reason

1. is supplementary to , is supplementary to

1. Given

2. , 2. Definition of Supplementary Angles

3. 3. Substitution

4. 4. Given

5. 5. Substitution

6. 6. Subtraction

ASSESSMENT 3

ENRICHMENT 1

Enrichment 2

Statements Reasons

1.

1. Given

2. 2. Perpendicular lines meet to form right angles.

3. 3. Right angles are congruent.

4.

4. (ASA) If two angles and the included side of one triangle are

congruent to the corresponding parts of a second triangle, the

triangles are congruent.

5. 5. (CPCTC) Corresponding parts of congruent triangles are

congruent.

Statements Reasons

1.

1. Given

2.

2. Two angles that form a linear pair are

supplementary.

3.

3. Supplements of the same angle, or

congruent angles, are congruent.

4.

4. Reflexive property: A quantity

is congruent to itself.

5. 5. (SAS) If two sides and the

included angle of one triangle are

congruent to the corresponding

parts of a second triangle, the

triangles are congruent.

6.

6. (CPCTC) Corresponding parts

of congruent triangles are

congruent.

7.

7. An angle bisector is a ray whose

endpoint is the vertex of the angle

and which divides the angle into

two congruent angles.

Enrichment 3

Statements Reasons

1.

;

1. Given

2.

2. Reflexive property - a quantity is

congruent to itself.

3.

3. (ASA) If two angles and the included

side of one triangle are congruent to the

corresponding parts of a second triangle,

the triangles are congruent.

4. 4. (CPCTC) Corresponding

parts of congruent triangles

are congruent.

5. AB = BE ; DB = BC 5. Congruent segments are

segments of equal length.

6. AB = AD + DB

EB = EC + CB

6. Betweeness of Points: The

whole is equal to the sum of

its parts.

7. AD + DB = EC + CB 7. Substitution.

8. AD = EC 8. Subtraction

9. 9. Congruent segments are

segments of equal length.

Statements Reasons

1.

;

1. Given

2.

2. Congruent angles are angles of

equal measure.

3. 3. Reflexive property - a quantity

is equal to itself.

4.

4. Angle Addition Postulate: The

whole is equal to the sum of its

parts.

5. 5. Substitution.

6.

6. Subtraction

7. 7. Congruent angles are angles of

equal measure.

8. 8. Midpoint of a line segment is the

point on that line segment that

divides the segment into two

congruent segments.

9. 9. (SAS) If two sides and the

included angle of one triangle are

congruent to the corresponding parts

of a second triangle, the triangles are

congruent.

All in all, if you got…

Perfect Score

7-9

5-6

4 and below